Kevin Knudson: Welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, a professor of mathematics at the University of Florida. And I'm joined by my other host.

Evelyn Lamb: Hi, I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, where we just got a lovely few inches of snow last night. So I've developed a theory that podcasts cause snow here. Although it could be the other way. Maybe snow causes podcasts.

KK: Maybe.

EL: It’s hard to tell.

KK: I don't know. It's 85 degrees for today. Sorry.

EL: Yeah. I meant to say don't tell me what the weather is in Florida.

KK: It’s very nice.

EL: It’s too painful.

KK: It’s very nice. Yes, speaking of painful, we were having our pre-banter about I had a little hand surgery yesterday, and I have this ridiculous wrap on my right hand, and it's making me kind of useless today. So I have to do anything left handed, and to control everything on the computer left handed. But you know, it'll resolve my issue on my finger. That'll be good. So anyway, today, we are pleased to welcome Dave Kung. Dave, why don't you introduce yourself?

Dave Kung: Hi, there. I'm Dave Kung. I'm a mathematician by training. I spent 21 years at St. Mary's College of Maryland, and I've recently moved on from there, and I work at the Dana Center, the Charles A. Dana Center, at the University of Texas at Austin. I work with Uri Treisman down there on math ed policy.

KK: It’s very cool. So you got a really serious taco upgrade.

DK: Definitely. I'm still living in Maryland, but I get to visit to Texas every few months.

KK: Are you gonna relocate there? Or are you just gonna stay in Maryland?

DK: I’ll be here for now. Yeah.

KK: I mean, it seems like the sort of work that you could do remotely, it’s policy work mostly, right?

DK: A lot of policy work. There's going to be a fair amount of travel once that's more of a thing, but not all of it will be to Texas. Some of it will actually be in DC, in which case, I'm pretty close.

EL: Yeah, you’re right there.

KK: Yeah. Yeah. Well, they do really great work at the Dana center. I've been involved a little bit with the math pathways business, and it is really vital stuff. And of course, Uri is, like, well, he's the pied piper or something else. I don't know. But when you hear him talk about it, he's an evangelist, you really you can't help but like him.

DK: Yeah. And making sure that we know that students have the right math at the right time with the right supports, we're far from that goal right now, but we can get closer.

EL: Yeah, very important work for all mathematicians to care about.

KK: Yeah, yeah. Okay, but I think we're going to talk a little bit higher-level than math pathways today. So we asked you on to have a favorite theorem. What is it?

DK: My favorite theorem is the Banach-Tarski theorem, which is usually labeled the Banach-Tarski paradox.

KK: Yes. Yeah. So what is let's hear it. Well, let's let our listeners know.

DK: So the Banach-Tarski paradox says the following thing: that you can take a ball, think of a sort of a solid ball, and you can split it up into a finite number of pieces — we’ll come back to that word pieces in a bit — but you can split it up into a finite number of pieces, and then just move those pieces and end up with two balls the same size and the same shape as the original. It is incredibly paradoxical. And I remember hearing this theorem a long time ago, and it just sort of blew my mind.

EL: Yeah, it was one — I think I was an undergraduate, I don't think I'd even taken, like, a real analysis class. But I heard about this and read this book, there's this book about it that I think it's called The Pea and the Sun. Because another statement of — I mean, once you can make two of the same size things out of one, you can make kind of anything out of anything.

DK: Just repeat that process. That's the other statement, you take a pea and you do it enough times. And if you do it, well, you can reassemble them to form a sun. Absurd.

EL: Yeah, I just, I mean, reading it, there was a lot that went over my head at the time, because of what my mathematical background was, but at first, I was like, Okay, this means that math is irrevocably broken. And then after actually reading it, it’s like, okay, it doesn't mean math is broken. So maybe, maybe you should talk a little more about why it doesn't mean math is broken. If you have that perspective. Maybe you do think math is broken.

DK: It certainly feels like. I mean, I think my first reaction was: Cool. Let's do that with gold, right? We'll just take a small piece of gold and split it up and keep doing that. I think there's a lot in this theorem, right? And so I mean, it's one thing to understand it, sort of at a deep sense why it's not absurd. And I think it helps me to think about just sort of, you know, once you know a little bit about infinity and the fact that there are different sizes of infinity. And once you know that somehow the even numbers, the even integers, have the same size as all the integers, which is already sort of weird, this feels a little bit like that. I mean, you could sort of somehow take the odd integers and the even integers, and each one of those is the same size as the full integers. But some of the integers themselves, it's weird to be able to split the integers up into two things, which are the same size as itself.

EL: Right.

DK: And so fundamentally, this is about infinity. And the reason this is a little bit more than that, well, first of all, obviously, the ball here is not countable, right? We're not dealing with a countable number of points. This is uncountable. So now we're talking about the continuum in terms of the cardinality of the points. But I think then the surprising thing is that it works out geometrically. So it's not just about cardinality, but you can do this geometrically. And so you can actually define these sets. And, you know, the word pieces, when we say you can split it up into a number of finite number of pieces. I think the record is somewhere under 10 pieces.

EL: It might be just like five or something. It’s been a while.

DK: But the word pieces is doing a whole lot of work in that statement.

EL: Yes.

DK: And these pieces are not something you could ever do with like a knife and fork or, you know, even define easily. It requires the axiom of choice to define these pieces.

KK: Uh-oh!

DK: So it's really high-level mathematics, to understand how to do these pieces, but the fact that you could do these pieces, and then geometrically it works to just reassemble them, to just rotate and translate these pieces, and get back two balls the size of the origina, that’s just astonishing.

KK: Yeah, that's why I've always had a problem with this. I mean, I, I can read it and understand it and go, yes, you can follow every logical step. But you're right, it doesn't work visually, if you think about it. So can you describe these pieces at all?

DK: They are screwed up. So the analogy I like to make is, you know, if you're working on the interval from zero to one, you know, so first of all, the Banach Tarski paradox does not work in in one dimension. But in terms of these pieces, if you're thinking about the interval from zero to one, you could think of the rational numbers as a piece of that, right? And so it's a piece in the sense that it's part of the whole, it's not a piece in the sense that you could cut it out with a butter knife, or you could model this with a stick of butter or something like that, right? You have to hit 1/2, 1/3, 2/3, you have to hit, you know, 97/101 in there, right? All of those are rational numbers, but you can certainly think of all of the rational points in the interval from zero to one as a single piece. You can talk about them, you can define them, and then you can talk about moving them. And so you can think about it that way, right? So you have all those rational numbers, and you can think of that as one piece. And it certainly is a lot more complicated than that. You know, when you think about Banach, Tarski, one of the things I love to do is with students is to go back and think about what it would mean for a set to be non-measurable. So we can measure things like intervals, but it doesn't take too much mathematics to sort of dive into the fact that there have to be sets — if you have a sense of measure, which works on intervals and things like that, and you want it to have other properties, like when you take two disjoint sets, the measure of the two disjoint sets together should be the sum of the two measures, like really basic properties, it doesn't take too much to be able to prove that there are sets that are non measurable. And once you can prove that, like Oh, then then the world gets really screwed up. Because in the world, in our everyday living, everything seems measurable, in some sense. Like even if you have some screwed up sculpture, you could measure its volume. You dunk it in water and see how the water level rises, right? I mean, it is, it has a measure. And the idea that there is no way to measure something is just incredibly counterintuitive. But once you get that, then it's it's it's a little bit more of a leap, but to understand at a fundamental level that you can define pieces that are so screwed up that you could just rearrange them and get two copies of the original, it’s fantastic.

EL: It’s interesting that you, when you first introduced this, you said it's something about infinity and I remember what — so now I'm thinking I might have taken a real analysis class before I had read this book, because I remember when I read the book, thinking, Oh, this is telling me something about non-measurability and, like, really giving me a concept of what non-measurability does to things. And that's that's how I viewed it. It's a statement about how important measurability is.

DK: And just to be really clear, if these pieces were measurable, right, then we would just have some volume or something like that. And there's no way to double the volume. Right? So you can't you just can't do that. So clearly these pieces, at least some of them have to be non measurable.

KK: Right? So much for your alchemy idea, right?

DK: No more doing this with gold. Okay.

EL: It won’t work on atoms.

DK: There’s this fundamental idea that's permeates mathematics that we can continue to divide things, right? You can get things as small as you want. You see this — this is basically what calculus is all based on, infinitesimally small things. And it's just a reminder that the real world does not work like that. You take a gold atoms, you could keep splitting something up. But eventually, you’ve got one gold atom in each piece and you can't go any smaller than that.

KK: Well, you could, but then

DK: It wouldn't be gold anymore.

KK: Right. All right. So you sort of hinted at this, but why do you love this theorem so much?

DK: I love that it makes you question so many things. I mean, I love paradoxes in general, right? paradoxes are these moments when there's, there's so much cognitive conflict going on and cognitive dissonance, that it forces you, in order to resolve that cognitive dissonance, you have really have to question some other fundamental aspect of the world. So it's something that you were thinking before, is not true, or this paradox is like totally crap, right? So something like that. But in this case, the theorem is true, right? The Banach-Tarski paradox is true. And so it forces you to just go back and question some fundamental ideas about the world. And I love that there are statements like that, that can force you to go back and question so many things. I think we as humans need to do better at this. There are so many things that we just accept, as, as we take for granted, right, and we take them for granted as if they are true with a capital T. And in this case, it's about, like, you can measure all things. But of course, in our in our everyday lives, it's things about the world, it's things about people, it's things about politics, it's things about, you know, topical issues. And we grow up, and it's so normal that we think these things are just part of the world with a capital T on truth. And I love those moments that force us to go back and question those those fundamental “truths,” which all of a sudden turn out to be assumptions, some of which may not be right, or some of which we might want to reassess.

KK: So maybe you're arguing that we should study more mathematics to make the world better.

DK: We certainly should do that, Kevin.

KK: To build a better citizenry, right?

DK: Certainly if we all understood mathematics better, we would have been better off during this pandemic. That's certainly true.

EL: I have a question. So as you mentioned, this is called a paradox often. Do you think it is a paradox?

DK: Yeah, that's a great question. You know, paradox has a couple of different meanings. One is sort of this deep philosophical meaning, like, is it really something which is somehow both true and false simultaneously? And, and this is not in that sense, a paradox. It is a paradox in a sort of weaker sort of more everyday sense of that word where it really throws us into into some cognitive dissonance that forces us to question other things. We can't hold both true that like everything can be measurable and things have volume and you can take a ball, split it into six pieces, rearrange them and get two balls. Those are fundamentally in conflict, and one of them has to go.

KK: It does rely on choice, though, right? So there's something to argue about there. You know, there are those people who deny the axiom of choice.

DK: They're few and far between in the math community, but they're out there. Yeah, they're not measure zero, so to speak. So yeah, so it does, it does use the axiom of choice. So this idea that you can, you know, you can make infinitely many choices. And you can see in the proof where you have to do that. So you end up with uncountably many sets and you choose one point from each one of those uncountably many sets, and that's part of the way you get one of the sets that creates the the Banach-Tarski paradox.

KK: Yeah.

EL: I guess I'm a little surprised that the Banach-Tarski paradox hasn't made more people reject the axiom of choice, to say like, Okay, well, clearly, you can't do this. So, therefore, what is this relying on? Well, it's relying on the axiom of choice.

KK: Well, there's a lot of things about like that, like the Tychonoff product theorem. So that's equivalent to the axiom of choice. So do you want your product of compact spaces to be compact or not, you know, I mean, I don't know.

DK: Yes, I think Evelyn, you're hitting upon this fundamental thing: at a very deep level, math gets weird. Yeah. Right. And you can have things. I mean, you see that in, you know, in Gödel’s work, you see like, well, is that true? Well, you know, it can either be true or false. What's your pleasure today? Right? To the axioms or we can reject it. And, and fundamentally, it kind of doesn't matter. But you know, go ahead like, is there an infinity between the integers and the real line? Like, oh, you know, take your Take your pick?

EL: Yeah, what seems more more useful to you right now? Or what sounds like more fun to play with?

KK: Sure.

DK: And in some sense, that that is kind of the beauty of math, because so much of what mathematicians do is based on wherever you want to start. It’s theoretical, like, oh, well, if we start here, this is where we get. If we start at a different place, we get this other thing. And so you can hold those both in your head, despite the fact that maybe you can't have the hold them simultaneously. Like, either the axiom of choice, you take it to be true or you take it to be false, but you can't take it to really be both. Then things break down. But you can you can do a lot of mathematics either way.

KK: Yeah. So another thing we like to do on this podcast is invite our guests to pair their theorem with something. So I'm curious, what have you chosen to pair with this paradox?

DK: So there's this piece that I played, I think I first played this when I was in high school. I'm a violinist and Evelyn and I share this as string players, but it's called the Enigma Variations. It's by Edward Elgar. And it's a really interesting piece with sort of a fascinating story behind it. And the story tells, or explains the name, enigma. So the idea is that Edward Elgar was sitting there, and he played this little melody, and he was sitting at the keyboard — this is in, like, 1898, late 19th century. And he started riffing on this on this melody, right? And he's like, oh, okay, like, we could play it this way, or this way. And then he started putting names on it, like, Oh, my friend would play it this way. Or, you know, my wife, she would play it in this way. And then he sort of just kept taking that further and further, and he ended up writing a bunch of different movements based on how different people in his life, he imagined would play this particular melody. And one of the it's sort of ingenious if you want something to go viral, I guess. But he never, he never lays out what the what the melody was. And so that's the enigma part of it. Somehow out there somewhere is this melody, and all you're hearing is different people's take on it. He does tell you who the people are, and so some of these movements have initials telling you who some of them are. Some are explicit with names. But all you have is that, right? And it's beautiful music and when you listen to it, some of the some of the movements are really fast loud and really exciting. And others are just incredibly slow and languishing. And it's hard to imagine that all of that could sort of in some sense be based on a single melody, and somehow it is. That's what I'd like to pair it with, the Enigma Variations.

KK: Do you have a favorite so that we can insert a clip?

DK: Um, let's see. I do. So probably the most famous of these is called Nimrod. It's an incredibly slow movement. I've played it a bunch of times. It's the sort of thing that, you know, if you want to get people to cry in a movie you could play a little bit of Nimrod. I don't think I've ever successfully been on stage playing this without crying. It is that emotional come and it's also, for me it's more emotional if you play it slower, so it's incredibly evocative. [Clip of the Nimrod variation]

KK: Do you still play?

DK: I do. I do. Well, in theory. We’re in this pandemic where I haven’t been able to play much. But yeah, I play in a local community orchestra and play with my daughter and yeah, I've had some had some fun over during the pandemic, by myself playing, picking up some old pieces and playing them.

KK: Very cool.

EL: Yeah, I was thinking probably everyone has heard Nimrod without realizing it, because it's in the background. If it's not Adagio, for strings, it's Nimrod in the background of that, you know, swelling, emotional scene, your farewell or someone’s dying or whatever. You know, it’s there.

DK: And when you hear it, you can hear little hints of Pomp and Circumstance, which we hear all the time during graduations. And so you can see like, yeah, those are, those are similar sort of in structure, in the melodies and the harmonies and how they fit together,

KK: Right, plus it’s public domain by this point, so you can just throw it into movies pretty easily. Right? That's true, too.

EL: Yeah, I guess you're the practical one here.

DK: Yeah, that's important when you're doing things like podcasts. Yeah, yeah.

KK: Yeah. All right. So um, that's a really good pairing. I like that a lot. So we also like to give our guests a chance to promote anything they want to promote. So So we've talked about the Dana Center a little bit. Anything you want to pitch? Where can we find you online?

DK: You can find me at the at the date of the Charles Dana center, it's UT Dana Center. dot.i.com. I think maybe .edu I'm not sure actually sure. [Editor’s note: it’s neither! It’s actually https://www.utdanacenter.org/], but it’s a pretty easy web search to find us. You know, we're trying to make sure that mathematicians, that everybody has an opportunity to see themselves as a mathematician, and that everybody has access to the right math for them at the right time with the right supports. And so much of the math community, so much of our curriculum is grounded in things that are really old. T algebra, geometry, algebra II sequence was decided in 1892 by a group of 10 white men in the northeast who decided that that would be the right thing. And this focus on calculus comes out of the Cold War, it comes out of the need to produce a small number of engineers who are going to work pencil and paper, and then with massive computers that they could program, to win the Cold War, to beat the Soviets into space and to do all of that. That's no longer the world we live in. And we need to expand what we think of as mathematics. And it's not that calculus isn't important, it still is, but maybe not sending everybody in that direction would be better. And so some students would be much better off if they took statistics. Other students, Quantitative Literacy would be great. We're swimming in so much data, we don't know what to do with and the careers out there that deal with data are taking off like you wouldn't believe. What are the tools we need to give students so that they can deal with that?

KK: Yeah.

DK: All right. So all of these things are just updates that we need to do for their math curriculum. And and updating a system that's so complicated, with so many moving parts is difficult. And so that's part of what we work on at the Dana Center. And if anybody's out there who wants to help work on those things and at the same time, make sure all of these systems are equitable, because we know they haven't been in the past, and make sure all students have access to high-quality math instruction independent of your zip code, or who your parents are your economic circumstances. Those are the sorts of things that we work on at the Dana Center.

KK: Yeah, that's really important work. I mean, it's funny to hear you talk about this, because I've been saying the same thing. You know, the math degree that we still give out more or less is what I got 30 years ago, which isn't different than what they were handing out 30 years before that. I mean, things have changed. And we're still sort of stuck that way. It's unfortunate. And they're still doing it on the high schools. You know, my son when he was going through high school, they just marched him in lockstep through algebra, geometry. And he's a musician. I mean, it's great, you know, it's good for him, but I don’t know.

DK: And even some of these topics that are still important, like, I think algebraic thinking is still important. I think a lot of people do algebraic thinking out there all the time. It's just, it's not pencil and paper. It's not symbol manipulation. The most popular place that people use it, their algebraic thinking is when they're working with spreadsheets, right? Writing formulas in spreadsheets is incredibly algebraic. You have these placeholders that are really just variables, things change, some things stay the same. It's highly algebraic. And we could be teaching algebra using that so that everybody, every student coming out of algebra would have a basis for understanding how algebra is going to be used. You know, it's something like 98% of employers want their employees to be able to use spreadsheets. Well, there's a perfect example. We could we could tweak the curriculum to take something that right now is seen as kind of useless and make it very useful and in fact vital. And it's still teaching essentially the same core ideas, but it's really approaching them from a very different perspective.

KK: Yeah, yeah. All right.

DK: And as long as you give me the space Kevin, you know talking about mathematics and music, so I I have to take a little space. I did this set of 12 lectures for the Great Courses on math and music. And it's a sort of a tour through the listening of music. So like when when we record something, and that music eventually gets into your ear, like every step on that process involves mathematics. And over the course of 12 lectures, we talk about rhythm, we talk about harmonics, we talk about tuning, we talk about lots of, we even talk about the digital side of it. So there's a ton of mathematics that goes into CDs, there's a lot of mathematics that goes into compression that we're using now, so that we can, you know, actually hear each other from this far away. And all of that, there's just lots of mathematics embedded in that. Yeah, so it's really fun, it was just a real honor to be able to do that series. So you can find that out there in the Great Courses.

KK: Yeah. Do you talk about why you can't have perfect tuning on a piano or anything like that?

DK: Yeah, in fact, I got to demonstrate tuning. So they brought in a baby grand for me to do this on. And so we tuned an octave and pulled it out of tune a little bit and you could hear the beats. Again, like these things that I saw in high school and in a trig class, but I had no idea they were applicable, right? So I mean, you can actually hear — there are trig identities that tell you, if you play two notes that have really similar frequencies, that's kind of equivalent to something that goes a wah-wah-wah, it has beats in it. And you can actually do that. And so I played that and tuned it, so that was perfect. And then I took a fifth and we did the mathematics to figure out if you want everything to be sort of in tune, how out of tune do your fifths need to be right, which is, you know, fascinating thing. And I know, Evelyn, you've written about this, what was the column called, like, the saddest thing you know about the natural numbers?

EL: Yeah. Something like that.

DK: Like three is not a power of two or something like that.

EL: It is so sad.

DK: It is sad. And, you know, there are mathematical facts at the heart of that, and so much that we hear in music. And I get notes every week from somebody who has had these nagging questions in the back of their head about how music and math are related. And they watch these 12 lectures, and they're just so thrilled to sort of unpack some of that.

EL: Yeah, it's neat. And it's also it's not just, like, the sound waves and the math, which is definitely part of it. But there's also this extra perception in our ears and our brains that is involved and has some math, but that has some, basically, I don't know, wizardry that our brains do to be like, Oh, when I hear this kind of thing, I often hear this kind of thing with it, so it probably came together, I just missed something there.

DK: There’s a there's a whole section on auditory illusions, which are really fascinating, ways in which our brain can trick itself or, or like, because it's been useful in the past, like, our brain does certain things. And so you can intentionally use that to have these auditory illusions, which are really just fascinating. And my favorite fact about this is that is that essentially, your ear is doing a Fourier transform. When you are listening, your ear is doing this Fourier transform. And you know, just sort of physically doing it and breaking down the sound into its constituent frequencies. And that is just a phenomenally cool idea.

KK: That our brains are that sophisticated.

DK: That our brains somehow involved evolved to do something that we didn't describe mathematically until the 19th century, right? Like, oh, now we know what our brains did just sort of by, you know, random chance and little tweaks to random changes in a DNA sequence. And somehow we got to like, oh, oh, yeah. And that's now called Fourier transform.

EL: Yeah. All right. It's a lot of fun. I feel like we could almost do a whole nother podcast episode about one of these facts.

KK: Probably.

DK: That would be so much fun.

KK: Yeah. Okay. Yeah. Well, maybe we’ll have you back for part two sometime.

DK: That'd be great. Yeah.

KK: Well, this has been great fun, Dave. I'm surprised Banach-Tarski made it this long without being somebody's favorite. This is pretty good.

EL: Yeah. Yeah.

KK: Because we've had we've had repeats before, but but this one, I'm surprised. So thanks for joining us. And yeah.

DK: Thank you guys so much. I love the work that you do, and I really appreciate it.

On this episode of My Favorite Theorem, we welcomed Dave Kung from the Dana Center at the University of Texas at Austin to talk about the Banach-Tarski paradox/theorem. Here are some links you might enjoy:
Kung's website and Twitter account
The Dana Center website
Leonard Wapner's book The Pea and the Sun about the Banach-Tarski paradox
A shorter article by Max Levy explaining the theorem
A primer on the axiom of choice from the Stanford Encyclopedia of Philosophy

The Tychonoff product theorem
Kung's course How Music and Mathematics Relate from the Great Courses

Evelyn's article The Saddest Thing I Know about the Integers, mourning the fact that no power of 3 is also a power of 2

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm your host Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. I almost forgot my name there for a second.

EL: It happens.

KK: I realized, like, I was hesitating, and I was like, “Who am I again?” Yeah, you know — so our listeners don't know, but it's 5:30 where I am, which, you know, doesn't sound late. But I've been at work all day, and now I'm tired.

EL: Yeah. Well, you should have made up something. You know, just tried on a different name for fun just to see.

KK: Well, yeah, so even my parents had the deal that if I was a boy, my dad got to name me. So he went with Kevin Patrick. And if I was a girl, my mother was going to get to name me. And should I tell you what I would have been?

EL: Yeah.

KK: Kandi. Kay Knudson.

EL: Yikes!

KK: Now, I'll let you work out why that would have been terrible for lots of reasons. Already, there are multiple axes along which that is terrible.

EL: Great. Yeah, well, my name if I had been a boy ended up with my younger brother. So it was kind of not that interesting. I mean, if you knew my family, you would be like, Okay, well, that's boring. Anyway, yeah. We are very happy today to have Priyam Patel on the show. So yeah, Priyam, could you introduce yourself a little bit?

Priyam Patel: Sure. So my name is Priyam. I am an assistant professor at the University of Utah, and I have been here for three years. Before that I was around everywhere, it feels like, for my postdoc. I was at UCSB for a few years, before that at Purdue for a few years, And I did my PhD at Rutgers, which now feels like ages ago.

EL: Yeah, you’ve been in, like, every region of the country, though, I guess not central timezone, because Indiana is right on the west edge of Eastern.

KK: That’s right.

PP: Yeah. So I was never in the Central time zone. And that's why — in the summer in Indiana, the sun sets at, like, 10:30pm. It's really bizarre.

KK: You could call that Central Daylight if you wanted to, right?

PP: Yeah. Something like that.

EL: Yeah. And as you mentioned, you've been at Utah for about three years. And you you first got here in fall 2019, and I was gone for most of the fall 2019. And then of course, we all know what happened in 2020. So part of the reason I wanted to invite you is because I feel like I should know you better because you've lived here for three years. But, like, with the weirdness of the past three years, I feel like I haven't gotten to talk with you that much. And so of course, obviously the best way to do this is, like, on a podcast that we want to just broadcast to the entire world.

PP: Yeah, perfect. So no private conversation over drinks. Just put me on the podcast.

EL: Yeah. Excellent. So So yes, I'm excited to get to chat with you. And yeah, hopefully we can do this over drinks in a real venue at some point.

KK: Wait a minute, what happened in 2020?

EL: I tried to block it out.

PP: Nothing at all.

EL: For some parts of it, really nothing.

PP: It feels like a whole blur since then. So

KK: I’m not convinced it isn’t still 2020 somehow.

PP: Yeah, yeah.

KK: Alright. Anyway, I'm being weird today, and I apologize. So let’s get to math. So Priyam, you have a favorite theorem. Which is it?

PP: Yeah. So I chose the Brouwer fixed point theorem, which I learned has been done twice already on this podcast.

EL: Yes, I'm very excited to hear more about it because in our emails, you mentioned some aspects of that I wasn't aware of. And so this is very exciting. And this is when people, when we email with people, they’re always like, “well has this been used?” And we're like, “It doesn't matter if it has, you can use it anyway.” We like to talk about theorems because it is interesting, just the different relationships people have with the same math. So for anyone who hasn't been you know, avidly listening and taking notes on every single episode we've done since 2017, can you tell us what the Brouwer fixed point theorem is?

PP: Yeah, so I'm just going to state it for the closed disk because that's the only context that I'm going to talk about it in. But basically, if you take in the plane in our two if you take the closed unit disk, then the theorem says that every continuous map from the disk to itself necessarily has a fixed point. So should I go into detail about what a continuous map? Would that help?

EL: Yeah. Or at least intuitively.

KK: Sure.

PP: So I actually did listen to a lot of the previous podcast episodes while I was preparing. And I like this idea of if you take the unit desk, and you, like, kind of shake it around a little bit, and everything kind of moves in a nice smooth fashion where things don't get sent, like, really far away — so if in a little neighborhood, you’re wiggling, one point is not just going to pop out and end up somewhere else, right? I like that idea of continuity. So if you're wiggling around the disk, the unit disk, and you use any continuous map, somehow one of the points has to stay fixed, so it gets sent to itself. And that's kind of surprising. It feels like if you just move things around enough, something, everything, should get moved off of itself. But in fact, that can't happen. So that's kind of my interpretation of Brouwer’s fixed point theorem.

EL: Yeah. And it's like I guess I always imagine it made of rubber or something. Because you are allowed to, like, stretch and smush a little bit. It doesn’t — because otherwise, you might think, Oh, the only thing you can do is rotate it. So of course, that central point will be fixed. But you could do a lot of other things.

PP: Yeah, absolutely.

EL: And fix some different point.

PP: Yeah, so I think Evelyn has a great point, like, you can spread things out, like you're making it out of like stretchy fabric or material, you can spread things out in one part of the circle, in the unit disk, and then, you know, string things together in another part and that's okay. It's like, you know, just kind of smoothly moving around is the way I think about it.

KK: Yeah, yeah. But something stays put.

PP: Something stays put, which is kind of strange sometimes, actually. And there's like, so many proofs of this theorem, I feel like, and so many different perspectives for proving it. But I do have a favorite proof of that, actually.

KK: Okay, good. Let’s hear it.

PP: So it's unfair, because it uses some algebraic topology. So o be able to get to this point in this in your math life, where you're like, Yeah, this is the proof I like the best, you have to learn some algebraic topology. But essentially, the idea is that when you're in topology, in the field of topology, you're trying to understand when two objects that are made out of bendy, squishable material that you can stretch and shrink, when two of those are really the same. So if you have, let's say, a circle, or a really oblong wiggly circle, those two are the same. It doesn't really matter if one is really beautiful and perfectly symmetric. It's really the same space in topology. So two things that are not the same topologically are the closed unit disk, and just the outer boundary, which is just a circle. Okay, so there's an a thing called an algebraic invariant that you can compute called the fundamental group, that tells you that topologically, formally, these two spaces really aren't the same. And essentially, there's a proof that says, If there wasn't a fixed point, then you could basically take the entire closed unit disc, and shrink every point in the desk to the boundary. This is called a retract. You’re basically saying like, I'm going to retract the entire closed unit disc to just the circle. And retracts are supposed to give you the same fundamental group. And you already know that those two things aren't the same. And so that's my favorite version of this group. And I can slow down on any part of that if you'd like more details.

EL: Yeah, that's really nice. Well, I think maybe a good way to see this is like, you know, that example of turning the circle around, you know, like a record spinning on a record player or something. Like if you took away that central point, everything else can move. And you can also imagine pulling that rubber all the way to the edge, making it into a bike tire or something else like that. (Which is actually topologically different.)

PP: Right, but as soon as you puncture it. So Evelyn's basically saying, let's just take out the center point. But what corresponds to the origin in R2? Well actually, once you do that, there's no contradiction that you derive, right? You can have every point moving. And in fact, that punctured disk and the circle are the same topologically. That retract that you can use to just pull everything to the boundary shows you, actually, that they're the same topologically. So it's just that one— it’s amazing how much like one point can make such a huge difference, right?

EL: Yeah.

PP: Adding in that one point. But yeah, so that's my favorite proof. It's fancy in some ways, but once you know the basic material that leads up to it, it's like a three-line proof, right? Which is kind of incredible.

EL: Yeah, but it's maybe a little bit like, what is the phrase, like using a sledgehammer to kill a mosquito.

PP: Oh yeah.

EL: Once you’ve built all of this fundamental group, then sure, you could just whack that.

PP: Yeah. And it's so funny because in math, typically I am the opposite of a hammer-striker, right? I never use the hammer. I want to understand the nitty-gritty of why you can just explain this using elementary math or something like that, right? But for some reason, when I saw this proof in, like, Hatcher’s algebraic topology book, I was like, Oh my gosh, that just like makes perfect sense to me, like I totally get why now. So it definitely is one of those use a hammer use a sledgehammer to kill a mosquito type of approaches.

KK: I’m actually teaching algebraic topology this year, and so I that is the proof I use for the Bouwer fixed-point theorem. But I use the sledgehammer to prove that every polynomial of odd degree with real coefficients has a root. And I use to use the Lefschetz fixed point theorem to do it. Let’s use the biggest sledgehammer we can find!

PP: It’s awesome. I mean, honestly, if you want for this podcast, if you want to talk about a fun theorem that's called theorem at the end, right? Not just some result, but that's actually named, a lot of the ones you come up with in topology are the fixed-point theorems, right? And I was like, Oh, this one's actually my favorite. And there's a reason for it. So yeah, that makes sense.

EL: So what else do you love about this theorem?

PP: Yeah. So this actually was inspired from talking to a few grad students the other day, but I realized that, you know, I gave them this task, which says, Can you classify all of the isometries of hyperbolic 2-space. Now, that's already a fancy sentence to say. So I can break down all of what that means. But in fact, one of the key ingredients for the approach that is my favorite to solving that problem is to use the Brouwer fixed-point theorem. So I can start off by talking about what hyperbolic space is, and like what metric spaces are. And from there, I can explain what an isometry. It’s kind of similar to a continuous map, but it has a lot more structure and preserves a lot more structure.

So let's start off with just hyperbolic space, shall we? Okay, so the way I think about anything that is a non Euclidean geometry, which hyperbolic geometry is one of those, I have to start thinking about, well, Euclidean geometry first, right? And Euclidean geometry, when I think of that, I think axioms, right? There's Euclid axioms and they're written down. You don't need to know what they are. But the last one is the one that people started saying, let's try to break it and see what happens to these models of geometry that we're sort of studying, right? Like, could we come up with a different interpretation than just Euclidean space. And so if you break the parallel postulate, there's a few different types of geometries you can get that satisfy all the other ones, but they don't satisfy the parallel postulate. And hyperbolic geometry is one of them. So what is a geometry, right? It's a space, like we just talked about, let's say the closed unit disk. And to me geometry, you're studying rigid things like distances, angles. And so you want to have a notion of measuring distance on whatever space you choose. So since we're going to talk about the power fixed point theorem, of course, my space is going to be the closed unit disk. In fact, I'll just start off with the open unit disk for now. So let's just get rid of the boundary. So if I start off with the open unit disk, that is my space. And there is a way of measuring distance on that space. So you can say, oh, put in these two points, I want to know the distance between them. There's a way of measuring distance on it where if you want to go from the center point at the origin, out to one of the boundary points, let's say just (0,1), or (1,0) in the plane, it actually takes you an infinite amount of distance to get there. Okay, so in hyperbolic space, this model of hyperbolic space, which is called the Poincaré disk model, the boundary of the disk is sort of off at infinity. And as you get close to that sort of boundary at infinity, points are getting really, really, really far away. That's what it means to get, you know, closer and closer to infinity, is distances go really big. And so that's the idea of what the Poincaré disk model of hyperbolic to space is. And, of course, if it weren't a podcast, I'd be showing like tons of pictures right now.

EL: Yeah, it is quite attractive. It's just a lovely, appealing model.

PP: Yeah. And like, you can look up all these amazing pictures by Escher. There are these famous paintings where Escher uses the upper half plane model or the disc model, and shows how, like, a bat, or whatever the figure is that he uses to tessellate, a bat of the same area drawn in different parts of the hyperbolic plane can look to our eyes, very different, right? And that's again, coming back to this notion of as you move out towards the boundary of the unit disk, distances are getting really big so the bat would have to look really small to your eye to to have the same area as a bat in the center of the disk. So I highly encourage listeners to Google just hyperbolic space or Escher's hyperbolic paintings, right. And you'll come up with so many things.

EL: Yeah, well, fun fact is that my Twitter profile picture is a tiling of the hyperbolic plane with the Poincaré. with like, a picture of me in it.

PP: Yeah. So I love it. I know Evelyn really loves all of these, like, hyperbolic geometry, topology type thing. So yeah, that's also partly why I chose this topic.

EL: Yeah, you’re definitely speaking my language.

PP: Okay, so that's the idea of what hyperbolic space is. There's so many more things you need to do to sort of gain the intuition of what it feels like to live in hyperbolic space, right? And those are the kinds of things that you build over years of studying it in your life. But the real thing I want to talk about is isometries of the hyperbolic disk model.

So what is an isometry? It's a map of the space to itself. So kind of like that jiggling that we were talking about. But where all of the distances between any two pairs of points remain the same. So a great example that Evelyn already talked about was this rotation around the origin, right? If you rotate around the center point, all of the points, pick any two of them, they actually see the same distance apart. And that's not an easy thing to see, partly because I never told you how to measure distances, right? That completely relies on how we decided to define that, which is the metric on the space, that notion of measuring distances. But if you knew it, and you wrote it down explicitly, you could actually calculate that that rotation is distance-preserving in the Poincaré disk model of hyperbolic space. So that's the idea of what an isometry is.

Okay, so now we want to try to get fancy. And usually, what you do when you have a space is you say, I'm going to try to understand all the isometries of this space. So where does this notion come from? In topology, if we're not talking about geometric structure, and we just kind of care about a space and it's all blobby, and can be stretched and shrunken, we think about all of the symmetries, right? All the topological symmetries of the space. When we're talking about isometries, what we're actually talking about is geometric symmetries of the space, all different ways of moving around this space, where distance hasn't really changed, so you're preserving the geometric structure. It turns out that if you take all of the isometries, you end up with a group. It has a really nice structure, you can compose two of them. But that's not really even important for today, what you really care about always is, can I classify all things of this type? And there are infinitely many of them. It's really hard to classify things when there's a whole infinite set.

EL: Right.

PP: I’m not just putting things in bins, like these are red marbles, and these are blue ones and these are green ones, right? So amazingly, it actually turns out that isometries of H2, the hyperbolic plane, or disk model, only fall into three flavors. They're either elliptic, which are very similar to the rotations that we talked about, or they are the rotations basically. There are parabolics, and there are loxadromics, or sometimes called hyperbolics, which doesn't make sense, because it's confusing. When you're talking about hyperbolic space, calling something a hyperbolic isometry, when you mean a certain type is confusing. So I'll just call it loxadromic, right? And so there's a few things you need to know, like what could the isometry even look like? How could I possibly get equations of these isometries? So you have to work a little bit. But it turns out that you can write down a general formula for a generic isometry of the Poincaré disk model, or the upper half plane model of hyperbolic space. So just as an example, I'm going to switch models. And in fact, when I say I'm going to switch a model of hyperbolic space, what I mean is, I'm just going to go to a different space with another metric on it, but it ends up being the same geometrically. There is a nice map between the two of them where all the geometry is preserved. So I like sometimes the upper half-plane model, because it's really easy to like write down what the isometries are.

EL: Yeah.

PP: So what I'll say is, and I, again, I'd write this down on the board, if I could. But imagine it, close your eyes and imagine it. I'll take four real numbers a, b, c, and d. And all, really orientation-preserving, but let's sweep that under the rug. Well, orientation-preserving isometries of the upper half plane model look like (az+b)/(cz+d), where z is a complex number. Okay, the criteria you need to make sure you satisfy is that ad−bc is 1. Okay, there's another interpretation actually in terms of matrices. Put those four things into a matrix [Editor’s note it’s hard to write a matrix in Word. The top row is the numbers a and b. The bottom row is the numbers c and d.] Well, you're seeing I want ad−bc to be equal to 1, that's SL(2,R). If you multiply the top and the bottom by negative one in the top of the bottom, you're not changing the transformation. So you have to mod out by plus or minus the identity. So we're really looking at PSL(2,R) protect devised special linear space.

EL: Yeah. I actually I do love that because the first time you see these hyperbolic isometries, the az plus b, over cz+d,, and then they're like, oh yeah, ad−bc, you just have this almost spidey sense tingling of like, Okay, that's like a determinant. Why are we doing this? There must be some relationship here with linear algebra.

PP: Yeah. And your spidey sense is totally on point. So I love I love that connection. And I sort of, you know, I always wonder how much detail to go into with these things, since I'm not writing at the board. But I love that, right? Because anybody listening to the podcast should be like, Okay, wait, that's the determinant, just like you did. So, yeah, so there's a little bit of complex analysis that goes into deriving the fact that these are actually isometries, that they map the hyperbolic plane to itself and so on. But once you have a nice general formula, you start to use Brouwer’s fixed-point theorem. So I, this is part of my life, I go back and forth between the models all the time. So I'm sorry if this is getting annoying, but I'm going to go back to the disk model for a second.

Okay. And well, let's think for a second. Right now, what I have is the open disk. And I have a boundary at infinity. And often, when you're working with spaces, you can sort of complete up the space by adding in the boundary. Okay, this is the sort of completion of the Poincaré disk model of hyperbolic space. So now, if I take the hyperbolic disk model with its boundary, I have the closed unit disk, right? And an isometry is way better than a continuous map, but in particular, it is continuous. And so Brouwer’s fixed-point theorem says no matter what map I'm talking about from the disk to itself, the closed disk to itself, I have a fixed point. So if you have a formula, and you know there has to be a fixed point, you should try to solve for those fixed points, right?

KK: Right.

PP: And so I'm going to pop back to the hyperbolic plane model, because that's where we have our nice formula. So I'm just going to try to solve az+b/cz+d=z, right? This is a function, your input is z. Even though it seems weird, because you're like, wait, I'm multiplying by a, then adding b, dividing by… this is really complicated, right? But there are certain points for which you put it in, and depending on a, b, c, and d, you spit out the same number, right? The same complex number. So this is what I tell people to do. This is what my students did the other day. They said, How am I supposed to approach this? And that's the classic proof, you use the Brouwer fixed-point theorem, and you start to solve for az+b/cz+d=z. The key point that you use after you're doing all the algebraic manipulations is that ad−bc is always 1. So a ton of stuff cancels out as you're solving, right? It becomes a very nice equation. But what am I doing, actually? If I cross-multiply, right, multiply by cz+d on both sides, and then move everything over to one side, I'm getting a degree two polynomial in the z coordinate. How do you solve any degree two polynomial? You use the quadratic equation! So the bane of some people's existence when they’re going through high school, they’re like, I'll never use this? Well, first, if you become a mathematician, you're definitely going to use it.

EL: Yes.

PP: But I love when things that you learn when you're so mathematically young still come into play when you're doing really sophisticated math. I think that's really cool. So okay, we write down the formula. And basically, what it comes down to is what's underneath the square root, right? What is the discriminant? Because when you take the square root of a negative number, you get imaginary things, right? Imaginary numbers, complex numbers with non-trivial imaginary part. If the thing under the under the square root is zero, well, you're just getting one root, right? And then if the thing under the square root is not zero, but it's positive, you're just getting two real roots. Okay, so let's think about that. We have three categories: one real root, two complex roots, or two real roots, right?

The thing is that in the upper half plane model, the thing I never talked about, was that you need the imaginary part to be positive. Okay, so the actual categories are one real fixed point, one complex fixed points in the hyperbolic plane, or two real fixed points. Really, when you're going between the models, what ends up happening is in the upper half plane model, the boundary at infinity is the real number line, it's the x-axis in the complex plane. So that ends up being the boundary of the circle, the unit disk. So we're talking about three cases: one fixed point on the boundary, two fixed points on the boundary, or one fixed point on the interior, which is the complex one. And that's it. That's literally the classification. Because if you have anything more than three fixed points, you can show that your transformation was the Do Nothing transformation. It's the identity.

KK: Right.

PP: So it's not just that the Brouwer fixed-point theorem tells you that you can find fixed points, that there is one, you can actually classify all of the isometries based on these three categories, which I think is like, just incredible. And if you want, I can give you a little bit of a geometric interpretation of what the three isometry is the classes of isometry.

KK: Sure!

EL: Yeah, but I would like to pause and say one of the last classes I taught when I was at the University of Utah was an undergraduate, like introduction to topology class. So we touched on some of this a little bit, and it's like, I kind of want to go back and teach it now and use this for that part of it.

PP: Yeah, it's kind of amazing. Even though it has some sophisticated things going on, you can tell some advanced undergrad students about this stuff, and really show them a lot of beautiful pictures. So when I was in Santa Barbara, I did teach a non-Euclidean geometry class. And, you know, of course, I have to do all the other geometries justice as well.

EL: Eh, do you really?

PP: I know, I know. But I mean, when you get to hyperbolic geometry, though, it's kind of like it's limitless, right? The amount of stuff you can kind of tell and teach students. So I do love that aspect of it. But it's not such an advanced sort of theorem that you're just like, What is this using? Where does it come from? It's like, you need to know the Brouwer fixed-point theorem, you need to know the basics of the model of hyperbolic geometry you're thinking about, and that's basically it, right? Okay, so let's talk about the three sort of classic, I guess I would say, the canonical examples that people give for each of the three categories. So for anybody who's listening that's really into math, and that knows a little bit about algebra, every isometry is actually conjugate to one of these. But they're like the model. They're sort of like, the best-behaved one in each category. What do they look like?

Okay, so for the elliptic ones, we're going to start there, because Evelyn's already told us what they look like, right? In the disk model, they’re all just rotations about the origin. Technically, the fixed point could be anywhere, and it's still kind of a rotation around that fixed point. But again, up to this conjugation, you can move that fixed point to the origin. And then so now you're really just asking if I want to just understand the canonical form of this, I just am going to try to understand isometries of the disk that fix the origin. And I'm going to get that these are rotations, right? You can actually go through and derive the formulas, you know, you say I solved for the root, it's complex. Here it is, let me write down what this might mean. And you can really see which Möbius transformations you're talking about. But that's the canonical way that we think about elliptic ones. And I think the elliptic word has to do with that like sort of rotation, right? But don't quote me on that, because I'm not good at words. I'm good at the math and pictures, but not great at words.

Okay, so let's go to something more interesting. So what are the parabolic ones? These are the ones where you have one fixed point on the boundary. And I'm going to go ahead and use the upper half-plane model again, because they think it's a little bit prettier to see the parabolic one there. So I didn't really say what the upper half-plane model was. So let me go ahead and do that. So the upper half-plane model of hyperbolic space, is you take the entire plane, but then you only think about the upper half part, right? So where the y-coordinate is strictly greater than zero. In the disk model, when we approached the boundary, distances got really big, right? What is the boundary for the upper half plane model? It's the real axis in the complex plane, so the x-axis, and the point all the way out at infinity in the plane itself. So in R2, there's actually an infinity, the one point at infinity, that has to get thrown in there. Okay, so what we can do is say, let's have fun and say that the fixed point in the hyperbolic plane model is infinity.

KK: Sure.

PP: So I'm going to take a straight line, it goes from zero, the origin, (0,0) in the complex plane, and it's just going to go straight up to infinity. Okay, if infinity is fixed, and you have to map these sort of straight lines to straight lines, what you can come up with I mean, I'm waving my hands here, you have to actually do some like algebra and manipulate everything and sort of make sure you're reducing this the right way. But what it turns out to say is that these are all translations. So the maps az+b/cz+d, well, c is 0, d is 1. And it's really just z+b, or something like that. Okay, it's like a translation by some by some number. That's approximately what a parabolic transformation looks like. So translation is something we understand from Euclidean geometry, right? It's just that the way that it affects points in — so what do I want to say? Translation in the Euclidean plane, we understand with that metric. We have the upper half-plane with a different metric. It turns out in this case, that translation is still an isometry. But you have to remember, distances look different. So when you're going to see parabolics in the disk model, things get a little bit more complicated. You have to talk about things called horocycles. And that, I would say, it's better to just look up.

EL: Yeah.

PP: This is where a picture would be very, very useful, right? Okay, and now, the queen of them all is the loxodromic isometries. So this is where we start to see a lot of the connections between hyperbolic geometry and dynamics. So when you have a loxodromic isometry, there are two fixed points on the boundary. With a little bit of work, again, what you can see is that if you take those two fixed points on the boundary, there's actually a, sort of like a shortest line segment going from one to the other. Lines look different in hyperbolic space because the metric is different. But essentially, the way that this isometry acts on the disk model is that one of those fixed points acts as a source. The other one acts as a sink, and everything in the disk model is getting taken away from the source and being pulled towards the sink, actually along the axis — that geodesic axis that you have between the two fixed points, it’s acting as translation along that. So this is a phenomenon in dynamics, more generally called north-south dynamics. You have this source and a sink, and things are moving from the source to the sink in a north-south sort of way. And, yeah, that's my favorite one. Of course, it's the most complicated one. It's the one that comes up the most when you're studying surfaces. So yeah, that that sort of is my — I know, that's just my favorite. That's my favorite type of isometry. I think it makes sense when you're working in hyperbolic geometry, because it comes up all the time.

EL: Yeah, well, that's so fun. So another thing we like to do on the podcast is force our guests to pair their theorem with something in the real world.

KK: “Force.”

EL: So what have you chosen as your pairing today? “Invite” our guests to do that.

PP: Yeah, I actually didn't think about this one so hard, because I have a natural pairing in my life, which is climbing. So I love to climb. So I got into rock climbing when I was a postdoc in Santa Barbara.

KK: Good place.

PP: In fact, one of my most favorite, yeah, it is a good place for climbing. And it just so happened that my mathematical grandfather, Mike Freedman actually, is in Santa Barbara as well. He is a very good rock climber. And he took me on my first like, sort of ropes climbing outside adventure. It was a lot more intense than I thought it was going to be. But he's an intense guy. So I kind of knew what I was getting myself into. But it was such a moment of growth for me. And there are a lot of mathematicians that are attracted to climbing. And there's a reason, right? It's problem-solving. But like, with the physical component put in there, right? So when I'm not problem-solving in my office, or at my home office, I'm usually in the gym, problem-solving climbing problems with my friends.

EL: Yeah, well, and that's, I have, I've like, done one little rock climbing thing. I've never done it, but they actually call them problems, right? Like, figuring out a route is called a problem.

PP: Yeah, absolutely. Yeah, it's very heady you know. Of course, if you ask a mathematician to choose a physical sport to get into, they're like, oh climbing, then I can still use my brain all the time.

KK: When I was a postdoc, I did some. There was a climbing gym in Evanston. And yeah, and you're right. It's very good for your brain. And I always thought that too, but man, my fingers just, it hurt so bad. And I was mostly a cyclist at the time, which was also good because it doesn't require you to really use your brain, like riding a bike is so automatic that I could go out for a ride and think about math while I was riding. I mean, I know you want to get away from the math sometimes, but it was actually a good thing to do for me.

PP: Oh, yeah, no, I totally agree. I was really into running for a while. And I loved that sort of time to decompress and add space to your brain, right? So with climbing, I don't have that sort of flow, I don't reach flow quite as quickly as I do in other exercise. But it's very good for getting out the work stress, I’ve got to say. You just, like, work really, really, really hard physically. And it is very rewarding. I think it's the type of thing where, just like math, in my opinion, you know, “natural ability” is not actually the thing I think that determines how well somebody does in math over time. I think it has a lot to do with how hard you work, right? And if you're in training, taking care of your body, learning things, watching climbers, being very observant, you tend to pick it up pretty quickly. And there's a lot of big, burly guys in the gym who struggle, and then they're sort of surprised when, like, petite women get up and sort of just crush the problem, which I mean, I don't mind that. You know, I think it's a good lesson for everybody. So yeah, I also think, you know, for me, climbing and math are both very dominated by certain genders and races, it's a very white dominated sport, it's a very white male dominated sport, often like math is. Math spaces are very dominated in those same ways. And a lot of my work in math has been to promote diversity and equity and justice, really, in my math communities wherever I exist. And it turns out that that extends to my climbing communities as well, because I co-founded a group called Color the Wasatch, which is an affinity group for people of color climbers in the Wasatch Valley. So it's been really great, and I've learned a lot from that. And it's very similar. When you have people around you that have similar experiences to you, it can be so enriching, and it can be such a relief to just sort of feel yourself relax, and feel just comfortable in your own skin wherever you might exist. And that's a big one that I sort of learned very recently. I’ve sort of always been into sort of activism in the math community. But this was a real first big thing I did in my community outside of my work as a university professor, and it's incredibly rewarding. And it's kind of taken off. It's been great.

KK: Cool.

EL: And we also like to give our guests a chance to talk about anything, you know, you'd like to plug. And actually, I think I just saw on Twitter, some people tweeting about the Roots of Unity conference that is happening. Is that — I don't know if conference is the right word for it — happening this summer. And I think this episode will probably be published in time for people who learned about it to apply,

KK: I think next week, this will be out.

PP: Yeah, that’s great. Because the application deadline for the Roots of Unity workshop is actually February 15, which is great. As you said, it's going to be out in time, the podcast will be out in time. So the Roots of Unity workshop, I am actually co-organizing with phenomenal women in math who I really looked up to, actually. And we designed this workshop to sort of support people who might not see people that look like them at their home institutions early on in their graduate career. So there's amazing programs out there, like EDGE, the EDGE network for people going into grad school, there are amazing research-focused conferences, like the Women in Numbers group, we actually have a Women in Geometry, Groups and Dynamics group now. And we felt that there was a sort of gap in between. And it's very hard, speaking from personal experiences, to be the only woman of color in your department or in your graduate cohort. And so we're really aiming to support anybody who would benefit from this kind of kind of program, but especially gearing it towards people who don't get that same training or preparation or encouragement at their home institutions, especially women of color. So it's going to be a professional development and research development workshop. And one of the things that we're doing is we're sort of helping grad students learn how to read papers, because like, gosh, we just are given papers and said, Go read this, right? And there is a skill to reading a paper well, I think. And so that's one of the big things that we're focusing on, is that transition between early coursework that feels very much like an extension of the undergraduate curriculum, and then into this whole new world that really requires a big pivot mentally, of reading papers and coming up with research problems and having a good network of support and collaborators as you do that. So that's occurring in June of this year at the IMA, so the program is you know in the process, the schedule is in the process of being set. But yeah, applications are open, and we would love to see lots and lots of applications.

KK: Excellent. All right. Where can we find you elsewhere online?

PP: So I am on Twitter. What is my handle? Evelyn, do you know what my handle is?

EL: Um…

PP: I think it might be priyam886. [Editor’s note: It is!]

EL: Maybe?

PP: That might be it. But you know, y'all can post it if you want on the website, eventually. My website actually has a link to the Roots of unity workshop. So that's patelp.com. And I also have a little page actually about Color the Wasatch there as well. So if people are interested in the climbing aspect of things also, that's that's all on my website.

KK: Oh, cool. All right. Well, this has been a lot of fun. I always like learning new things about the Brouwer fixed-point theorem. You know, I've taught complex analysis, and so I've thought about — I think I've shown my students sort of how linear fractional transformations, which are these isometries, are acting on the upper half plane, but I never thought about it in terms of isometries. So this is good for the next time I do it.

PP: Yeah, absolutely. So those linear fractional transformations are like the basic ingredient you need, right? Once you know those, you say which ones are actually going to map the upper half plane to itself, and then which ones are going to be distance-preserving, and everything falls out from there, so it's really nice.

EL: Yeah, this is a lot of fun.

KK: Yeah. Thanks.

[outro]

On this episode of My Favorite Theorem, we're revisiting the popular Brouwer fixed-point theorem with Priyam Patel of the University of Utah. Below are some links you might enjoy after you listen.
Patel's website and Twitter profile
Our previous episodes about the Brouwer fixed point theorem with Francis Su and Holly Krieger
A pdf of Allen Hatcher's algebraic topology book (available, legally, for free!)
The Lefschetz fixed-point theorem
Douglas Dunham's page about Escher and hyperbolic geometry
A blog post Evelyn wrote about putting pictures into the hyperbolic plane
Information about the Roots of Unity workshop (application deadline: February 15, 2022; if you're listening to this in later years, poke around and see if it's happening again!)

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I'm joined today by my fabulous co-host.

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City where we are preparing for another snowstorm this week after we had one last week, which is great because we are so low on water right now and we need every bit of precipitation. So even though I'm from Texas, and I don't naturally love shoveling snow or being below 50 degrees, I am thrilled that we're supposed to get snow tonight.

KK: So when I lived in Michigan — you know, I grew up in North Carolina, so snow was a thing, but we didn't shovel it. We just sort of lived with it — and I had a neighbor across the street who was in his 70s. And he had a snowblower and he let me use it and I thought this is amazing. So if you and John haven't invested in a snowblower yet, you know, maybe it's time.

EL: But we'll see. Climate change means that we might have to do less and less snow shoveling.

KK: Well, it's true. Actually, I remember growing up, you know skiing was a thing in North Carolina and I think you might still be able to but like the natural snow ski resorts, kind of they have to manufacture all their snow now. It's, it's things have changed even in my lifetime, but it's not real. As we're told, it's not real. Sorry to editorialize. Anyway, let's talk math. Today, we are pleased to welcome Courtney Gibbons. Why don't you introduce yourself?

Courtney Gibbons: Hi, I'm Courtney Gibbons. I will see far more snow up here in Clinton, New York, then either of you, I think.

EL: Definitely.

CG: I just sent in my plowing contract for the year. So that's awesome. Make sure that I don't have to shovel or snowblow my own driveway, which is not long. I'm a professor. I'm an associate professor of mathematics at Hamilton College up here in beautiful Clinton, New York. There is a Hamilton, New York, but that's where Colgate is. So don't get them confused.

KK: Oh, right.

CG: Yeah. It’s weird. It's a strange thing.

EL: Yeah. At least it’s not the whole Indiana University of Pennsylvania thing because that is not okay.

CG: Yeah, no, no. I think there was an incident on one of our campuses where, like, Albany sent some kind of emergency response squad to the wrong Hamilton. But they're only 20 minutes away, so it was a quick thing. Yeah.

KK: All right. Well, welcome. So, I mean, well, maybe we just get into it.

EL: Well, I will say that we have talked to Courtney before on the podcast, although extremely briefly, when we, I guess this must have been the joint meetings that 2019 a decade ago. (Hahaha.)

KK: Yeah.

EL: We had people give us, you know, little, like, minute or two sound bites of their favorite theorems. And she did talk about a theorem, although I understand it's not the theorem that she's going to talk about today, which, I mean, you don't have one and only theorem in your life? Come on, Courtney.

CG: I am a lover of many theorems. I think back then I mentioned Hilbert’s Nullstellensatz, which is the beautiful zero point theorem that links roots of polynomials to factors of polynomials over algebraically closed fields. It's beautiful. It's a really nice theorem. I initially thought I was going to talk about Hilbert’s syzygy theorem today, which by the way, syzygy, excellent hangman word.

EL: Yeah.

CG: Unless you're playing with people. You've used that word on before, in which case, their first guess will be Y. But I decided today I wanted to talk about Emmy Noether’s isomorphism theorems, in part because they're usually just called the isomorphism theorems. And Emmy Noether’s attribution gets lost somehow. So I wanted to talk about those today because I'm a huge Noether fan. I mean, I'm also a huge Hilbert fan. You kind of have to be a big fan of both. But these theorems are super cool. They're theorems you could see in your first course in abstract algebra, and that's actually where I first saw them. I'm a commutative algebraist and I do a lot of homological algebra. So I love arrows. I love kernels. I love cokernels. I love images. I love anything you can set up in an exact sequence, and I think this was my first exposure to a theorem that was best explained with a diagram. And I remember at that moment being like, “This is what I want to do! I want to draw these arrows.” And I'm lucky because I got to grow up to do what I want to do.

EL: That’s kind of funny because I loved abstract algebra when I took it in undergrad, and I think as it got more to, like, you know, having all these kernels and cokernels and arrows, that was when I was like, “I just can't do this,” and ended up more in geometry and topology. So, you know, different, different things for different people. That's fine. So yeah, let's get into it. So what are these theorems?

CG: Excellent, well, they are often numbered. I grabbed a couple books off my shelf, and it wasn't consistent, but Rotman and Dummit and Foote, kind of numbered them the same way. So the first one, which is usually the first one that you see, it's true for rings and groups and modules. I most often use it for modules, but I'll state it for groups. And it says that if you've got a homomorphism F from a group G to a group H, then the kernel of that homomorphism is a normal subgroup of your group G. Or if you're working with rings, it's an ideal of your ring, or you know, a submodule of your module. And you can mod out by it. So you take G mod the kernel, and it's going to be isomorphic to the image of your homomorphism. And so if you've got a surjection from G to H, and you are like, “I kind of want to build something isomorphic to H, but built out of the parts of G,” you're like, “Cool, I can just take the kernel and mod out by that and look at the group of cosets of of that normal subgroup.” And you've got this — you're done! You've built this cool isomorphism. And I advertise it to my students as, like, a work-saving thing. Because usually to build an isomorphism, you've got to show one-to-one/injective and onto/surjective. And this is like, well, take the thing you want to be isomorphic to, try to imagine it as a better group, a nicer group, mod the kernel of something, and build that homomorphism. Make it surjective and then you get one-to-one for free from this theorem. So I love this theorem. It's a really, it's a nice theorem. I actually use it. I don't reference it, but I think when you do the first step of finding a free resolution, which is which is what I do, it’s like my bread and butter, I love doing this. If you're calculating it by hand, you take a module, and you surject onto it with a free module. You look at the kernel of that thing, and then you build a map whose image is that kernel. And the big deal here is that your module M is isomorphic to the cokernel of that image map, which is the same thing as what you get from the isomorphism theorem. It's that first free module mod the kernel. So this gives you a nice presentation for a module. You can do this in certain nice cases. And I always sort of give a little thanks to Emmy when I start building free resolutions. I'm like, “I know that this is your theorem in disguise.”

EL: Yeah.

KK: I don’t think I actually knew that attribution, that Emmy Noether was the first person to explicitly notice these. But you know, she's the one who figured out that, you know, homology is a group, right?

CG: Yeah, exactly. So she was thinking about rings and groups. And, you know, a lot of the terminology is thanks to her and Hilbert. When you think about integral domains, I think that was what Hilbert initially called rings. I dug this up at one point for my students, they're like, where don't come from?

EL: That makes sense, right? Because I think of a ring as something that's like the integers.

CG: Yeah, it is really. Yeah. And like when people started generalizing to super bonkers weird examples, like, you know, the ring of quaternions and stuff, you're like, okay, so everything isn't quite like the integers. But we've got these integral domains, integral being the “like the integers” adjective and domain, I think of like, where stuff lives. The stuff that's like the integers lives here.

EL: Yes.

CG: Which is nice. But yeah, these these are, these are attributed to me. And it really bugs me to see them without her name attached since we have Hilbert’s syzygy theorem and Hilbert’s Nullstellensatz and it's like, but what about Emmy?

KK: Yeah, she she, she doesn't get the recognition she deserves. I mean, everybody knows like, she's like a mathematics mathematician. But yeah, she she definitely doesn't get credit a lot of the times.

EL: Yeah, I will say I am so, so tired of the headline, “The most important mathematician you've never heard of,” and I read this, and I know that I know more mathematicians than your average person. But like, if it's Emmy Noether, I’m just like, come on. You can't say you’ve never heard of her.

KK: Even the physicists.

EL: A lot of people haven’t heard of her. But yeah.

CG: But it's also interesting the way she's talked about. Because at the time, of course, it was difficult to be a woman in math even though she somehow was able to be a professor, although unpaid. But you know, you look at the way people described her and there's that one guy, I forget, who was like, “I can testify that she was a mathematician, but I can't testify to the fact that she was a woman.” You know. And it's like she was accepted because she was so unfeminine. And it was not threatening to the status quo of men doing math to have this, in their words, coarse, rough, simple soul wearing men’s shoes, blah, blah, blah, great heft, among them, because it's like, well, she's basically a man. So that always bothered me, too. I'm not a particularly feminine person, but occasionally, I do feel like I get a little bit of a brush off because they're like, “Oh, you're so cute.” I'm not cute! I'm a big strong mathematician!

EL: Yeah.

CG: That’s okay. I'm aging and rapidly getting less cute.

EL: You’ll become invisible soon.

CG: Yeah, I know. I'm really I'm excited for that. Yeah, I think I'll enjoy that better than the like, “Oh, aren't you a cute little mathematician?”

EL: Yeah. All right. So…

KK: So there's a first isomorphism theorem. So there must be a second.

CG: There’s a second, there's a third, there's a fourth, there's probably a fifth, then at some point, you just start calling them nth. And I mean, I think I've only seen four. I'm only going to talk about the first three, because I actually don't really know what the fourth one says. There's no quiz at the end. But there's homework. The fourth, look up the fourth. Okay, so the second isomorphism theorem is kind of a special case of the first one. You can prove it using the first one. It says that if you've got a group G, and you've got a normal subgroup, and you've got — we’ll call it N for normal — and you have another subgroup doesn't have to be normal, so we'll just call it S for subgroup, or Su or whatever. This says a couple of things. It says that the product of S and N is a group (and you form the product the way you would expect; you just multiply pairwise elements together, things from S with things from N). The intersection S intersect N is a normal subgroup of S, which is a really nice exercise. And you've got an isomorphism between the product SN mod N, and the quotient group S mod S intersect N. So this is a way of sort of saying like you've got this one group, maybe you don't understand SN really well, and you're working with SN mod N. Instead, replace it with a group you might understand better, which was your original group S mod S intersect N, which is a nice, normal subgroup. So it gives you some opportunity to work with a nicer group to understand the one you don't know. And the proof of this is you build a surjection from S to SN mod N: you’re going to just sort of send S to its coset representative over there and argue that it's onto. And then you calculate that the kernel is S intersect N and then you're like, boom, first isomorphism theorem, take it away. Which is, which is really cool.

KK: It’s kind of a corollary.

CG: Yeah, it's kind of a corollary, but the diagram is super cool. It looks like a diamond because you've got S and S intersect N and then SN and and then you've got all these things all over, the arrows. I love the arrows! And you can put the group G at the top and you put the identity at the bottom, and then you've got like a nice lattice. So I think it even is sometimes maybe called the lattice theorem. But that may. Yeah, I think so. But I, I wouldn't bet your final exam on that if you need to say something.

EL: Yeah. Well, and lattices are a different thing also. So I would be confused if I called it the lattice theorem. I’d think it was about like lattices in the plane or something.

CG: Right. Yeah. I think the idea is it looks like like a lattice. And most algebraists are like, “Oh, cool. We can call that a lattice.”

KK: It is a lattice though, right? The set of subgroups is a lattice. I mean, it’s a partially ordered set with meets and joins and blah, blah, blah.

CG: It is. Yeah, and what's funny is that the person I learned abstract algebra from in undergrad studied lattice-ordered stuff. I forget exactly. It was algebraic stuff. Now I'm of course blanking on it. That was Marlow Anderson, and I am the retirement replacement for Bob Redfield at Hamilton, who studied lattice-ordered subgroups. So I inhabit this space where lattices surround me. It’s kind of exciting.

KK: Yeah. But you're right. This one's a little more obscure. I can't remember ever using this in life except for, like doing homework?

CG: Yeah, I don't recognize a place in my life where I've used this one. If I have I was ignorant of the fact that that was what was making things work, which is not unusual. This is why I need to have collaborators. So they can be like, “Ah, Gibbons.” Hope only gets you so far.

KK: Right? All right. All right, number three.

CG: So number three, I think this one is low-key my favorite even though again, this is not one — actually no, I did use this. We used this in my modern algebra class this semester. This says let's say you've got a group G, and you've got a normal subgroup K in G. And you've got a normal subgroup N in K, so you're normal all the way up. This says that if you take G mod N mod K mod N, which kind of looks like — if you think of it as fractions, like if you're taking 1/2 then dividing by 3/2, this basically says you can “cancel” the denominators, and you're going to get G mod K. Which is really nice, because if you think about what G mod N mod K mod N is, G mod N is a group of cosets and K mod N is a subgroup of cosets. And now you're making cosets out of cosets. And that's a level of abstraction that you don't really want to work with directly, and so having this isomorphism that's like, that’s the same thing as G mod K, just work over there, that’s it's a really powerful thing. And in my class, we're doing a bunch of Galois stuff. Like, our first course in algebra follows that the story of why the quintic isn't solvable by radicals, basically, why there's no nice quintic formula that looks like the quadratic formula. And when we get into the solvable groups thing, and you're building these chains of subgroups, and you want to do stuff with them, we got on this sort of sidetrack of like, well, what if you wanted to mod out by one of the modded out things? Yeah, there’s an isomorphism theorem. This is so cool.

EL: So yeah, I remember seeing this and possibly having it like a homework question or something in abstract algebra, probably in grad school. And I remember feeling like I was getting away with something. When you could just like, like, cancel the denominators, which is not what this actually is.

CG: It’s one of those things where like, I'd be like, sure, that should work. And that's the hope I'm talking about, like, you’ve got to check the details. But again, this one is one you prove with the first isomorphism theorem. You set up your surjection from G mod N to G mod K, and you show all the stuff that's going to go to zero there was the stuff in K mod N. So this is another nice one, but it just looks so cool. You're like, “I can cancel the denominators. Yes, life is good.” You know, some people worry that mathematics is fundamentally flawed, and we don't know where the error is. And I'm like, it can't be. Look, this worked out. We can cancel denominators. That’s got to be on solid ground! It worked out exactly how we think it should.

EL: I love that. That's a great attitude to go through life with.

CG: W could worry about the fundamental flaws, or we could enjoy the wins where denominators cancel. I'd say we've done pretty well for ourselves. We've built this in a sensible way.

KK: I sort of feel like even if we do find the place where it's bad, it doesn't mean that bridges are going to fall down. Right? I mean, most of what we’ve built it seems okay. I mean, yeah, yeah.

EL: It will probably only break something like what Kameryn talked about with us last month.

KK: Some weird set theory.

EL: Yeah, who cares about that?

CG: Let it burn.

EL: Just kidding, Kameryn.

CG: No, I mean, isn’t there, I don't follow this very much. But I thought there was a modern push to sort of replace the foundations of math, the set theory stuff with category theory instead.

EL: Homotopy?

CG: Yeah, homotopy type theory, which in my mind is just like, you know, an evil cousin of category theory. But I, I don't know. I like homotopy. But I don't know what the homotopy type theory is.

EL: I know. I’m afraid to say it.

CG: It is a little scary to say.

EL: Yeah, I have not been able to figure out the correspondence between what I think of as a homotopy, which is like, “Look, you could drag the little loop around on the doughnut,” and homotopy type theory. Don't tell anyone.

CG: Well, and as someone who studies homology, I'm like, why would we do homotopies? They're kind of yucky. Like homology makes it nicer, right? Homology is the abelian version.

KK: It’s abelian, right.

CG: I mean, I guess, you know, if you actually want to describe stuff in the real world, he can't just live in commutative land, but who wants to live in the real world? That’s what the physicists do.

EL: Yeah, that's why we do math, to not have to do that.

CG: Exactly. Exactly. The real world is full of snow. And climate change deniers.

KK: Sure. Yeah. So it seems like you've had a long love for these theorems, right? This goes way back, right?

CG: This goes way back. For me, I was not actually intending to be a math major. I failed seventh grade math. My elementary school report cards were like, “She's very creative. Not so great at math.” “Asks too many questions,” I think was one of them. My first existential crisis was like, what happens if we keep adding forever, and we run out of names for the numbers, and I was like, I can't go to school. They asked me to add as high as I can, and I got to 100. And I was like, I'm going to stop here because at some point, I know what to do, but I don't know what to say. And that was pretty stressful. For me, I hadn't worked out like the time calculation, like how long it would take me to get to the point where I didn't know the names of the numbers anymore. But that was at a time where like, most of my classmates thought 31 was the biggest number because we only ever counted on the calendar.

So I was really surprised in college. I mean, I was also a college dropout who went back. I was surprised in college to get caught up in math and actually find that I could ask these questions. And my professors weren't like, “Oh, my god, shut up.” They were like, “What a cool question!” And they would actually talk to me, and I think it was the relationships more than the math that drew me in at first. By the time I transferred I was starting in multivariable calculus, having finished my first year elsewhere. And I was like, it's cool. We're gonna parameterize the way a leaf falls from a tree, mathematical poetry, all that, but like, it didn't set my soul on fire. And then I take differential equations, which really didn't set my soul on fire, although it was fun. I had fun time in it. But when I got to algebra, I felt like the language I was trying to speak my whole life was finally available to me. It was all about relationships and these like, very picky, not like picky in the sense of real analysis, but you could talk about really subtle differences in things. Like the difference between an ideal and a subring to me, was this really compelling thing, because they're so similar, but they're just a little bit different. And that little difference makes such a huge difference when you start talking about quotient objects and cosets and things like that. I really loved it. And then when I saw the arrows come out, I was like, I'm hooked. I want to build the isomorphism theorem of friendship. This is the way I want to describe everything in my world. And I just got worse from there. I had a brief flirtation with topology in grad school, and then I realized that you could be a commutative homological algebra person and steal all the cool diagrams, like the Meyer-Vietoris sequence, and all of that, and you could do with polynomials! And, and I was like, all right, that's, that's for me. That's what I'm gonna do.

KK: Very cool. I mean, I'm a pretty algebraic topologist myself. I mean, not not so much anymore. But earlier in my career, I was doing topology, but I was doing, like, homology of groups. So there's some geometry there, but not really, right. You’re really just thinking about algebra all day long.

CG: Yeah, I love that. I love that the tools of homology. I was tempted a little bit by geometric group theory, because it's so pretty and so fun. And I think some of the best, most fun conversations I had in grad school before starting my research work were with my friends who were taking that class too. We could just draw something on the board and argue about it for hours. And that’s one of the things I love most. I think I'm a mathematician — some people are mathematicians because they want to uncover these deep, beautiful, abstract truths. And I'm a mathematician because I like talking to people about math. So all my research work, my job is really like, if I didn't do research, I wouldn't have math to talk to people about, so I've got to keep doing the research and ideally doing with my friends, so we can talk about math. I love talking to my students about math. I love talking to you guys about math. I love talking my family about math, and they're like, “Oh my god. Please don't explain again what an algebraic geophysicist does.” Which is at one point what my mom decided. I said, “I do commutative algebra, which is close to algebraic geometry.” And in her mind, it became algebraic geophysicist.

KK: That sounds cool though.

CG: Which, that sounds like a cool gig. Yeah, like I would do it if I knew what it was. Yeah, like homology of rocks. Yeah, yeah.

EL: Figure out when the next earthquake is gonna be using commutative diagrams.

CG: Yeah. Maybe that's the the new topological data analysis stuff where you're doing persistent homology.

KK: That’s right. Well, you can use that to try to understand the structure of various things like you're looking for cavities inside of these. Yeah. So I mean, there are people who've done these analyses. It's real.

CG: All right. Maybe that's my mid-career pivot. Yeah. Algebraic geophysicist. Okay, joint appointment in the geology department. I get to go on their cool trips where they go hiking. I’m in.

KK: Right. Okay, so part two of this podcast. So well, we don't have a part three. Maybe there is a part three? Anyway, we’ve got three things here. We ask our guests to pair their theorem with something. So what have you chosen to pair the isomorphism theorems with?

CG: Well, I decided the best way to do this would be a clickbait-y Buzzfeed listicle kind of thing. So I'm going to start with theorem number three. Theorem number three was the theorem where you are cancelling denominators. I think it's a really fun and satisfying theorem, and so what I recommend you pair this with is a small batch craft beverage, alcoholic or non, of your choosing because when you use this, you should indulge all your senses and just be overwhelmed with joy that this works out and you could just cross out the denominators. For me, that would be a nice local, unfiltered wheat beer, just sit back, cross those little denominators out and just be like, “Yes, life is good. We have built a good thing.”

EL: That sounds great.

KK: Actually, one of the things I might be looking forward to the most about the joint meetings being in Seattle. [Transcriber’s note: Whoops! The 2021 Joint Mathematics Meetings were canceled/postponed after all.] And Evelyn knows this. My favorite distillery happens to be in Bainbridge Island. And so I think I'm going to make a little side trip to procure their excellent whiskey that I cannot get here. I can only get it there. So this is good. This is a good pairing.

CG: Oh, excellent. Are you taking orders?

KK: Well, I don't know. For you? I guess I could ship you something.

CG: Yeah. I’m not going to make it in person this year. [Narrator: She wasn’t the only one.] But I do love a good whiskey.

KK: Their whiskey is spectacular.

CG: We’re going to have to have an offline conversation about this and compare tasting notes. All right. Theorem two. It's about multiplication and intersections. And what goes better with multiplication than bunnies?

EL: Okay.

CG: And did you know there's a huge intersection of math people with bunnies, especially on Twitter?

KK: Yes.

CG: I think you've got to pair theorem number two with bunnies, you've got to it. It's not a theorem you use super often, but if you do use it, you have to immediately go out and acquire a pet bunny and join the math bunny Twitter crowd. It's just, I don't think there's any way around that. I think that's one of the laws of the universe.

KK: Okay.

CG: I should apologize to the math bunny Twitter folks, because they can infer that this is not the best thing, right? Number one is always the best thing you can you can have. But I hope they'll forgive me when they hear that my pairing for theorem number one, the OG, the original isomorphism theorem, is friends. When you use this theorem, you should think about your friends, you should write a note to your friends, you should talk to your friends. I've got friends on the brain this week because I was just thinking about a couple of friends of mine who passed away pretty young one was in his late 30s and one was in his mid-40s. And you know, I thought about Emmy's life and how she had a pretty tumultuous life, right, trying to make it onto the faculty at a university in the first place and then being shipped out of the faculty to Bryn Mawr’s gain. She ended up at Bryn Mawr.

EL: Basically dodging Nazis.

CG: Yeah, dodging Nazis. We talked about “math is apolitical,” but is it? Bryn Mawr lucked out, I guess? Because they got — I mean, America in general lucked out because we got all these amazing mathematicians who were fleeing Nazi Germany. But that sort of seems to me, and I'm not a math historian, but it seems to me that that must sort of mark the point where mathematics in Germany really took a backseat and why English has become the predominant language of mathematics.

EL: Yeah, I mean, the American mathematicians, the great, the big names from the early 20th century studied in Germany. And now it does very much go the other way. There's even a quote about, like, the center of gravity of research crossed the Atlantic because of the Nazis.

CG: Yeah, so I mean, that must have been such an awful thing to experience, being expelled and the uprooting of all of that. And then she died fairly young. She died at 53. I think it was complications from surgery, a cancer surgery. And, you know, it just it makes me wonder what did she have planned? What were her mathematical plans? How was she about to revolutionize physics again? Her most famous theorem probably is the theorem that talks about conservation of stuff under different types of symmetries. Like when you do something, something is conserved, which is, I guess, super important to physicists. I've never used that one in my own work. Right. But it seems like 53, she must have had so much more stuff planned for herself mathematically, physics-y stuff, personal stuff.

EL: Well, and how would Bryn Mawr have been different? And how would American — you know, would she have stayed at Bryn Mawr? Would she have ended up at a different university? Because I think she was only in the US for a couple of years before that happened. Yeah, she barely even got to teach anyone. I mean, I guess it should really be reversed. People in the US barely got to learn from her. And yeah, it's really tragic when you read about it. I think it was maybe an ovarian cancer surgery or something.

CG: I think that's right.

EL: I'm trying to remember if she's one of the people who seemed to be recovering fine, and then just kind of dropped dead a couple days later, or something like that. I know that I’ve read about it. I don't know for sure.

CG: I read up this morning, actually, to make sure that I had her age right. And it seemed it seems like the internet consensus at least is it was a viral infection. So the surgery went fine. But she had incurred some infection after the surgery and just dropped dead. You think of Maryam Mirzakhani. Also someone who was taken really young with so much math on the horizon. But it's sort of hopeful, right? Like, you know, Emmy really didn't get the recognition she deserved. But Maryam Mirzakhani, of course, got at least some of the recognition she deserved, which is, I guess we're making some progress.

This weekend, my friend and collaborator Nick Baeth passed away from a really rapid pancreatic cancer. I think he got the diagnosis maybe six weeks before he died.

EL: Gosh.

KK: That’s awful.

CG: And I wish I had finished writing the note to him that I was saying to him that it was just such a — he and I are math siblings. He was my math “older brother,” although we were in grad school at different times. But we met through our advisor, Roger Wiegand, who — his students become basically family. I know all my “siblings,” pretty much now and we hang out and they're great. But that's how I started collaborating with Nick. And we started working on some semigroup stuff, which is a little outside of my wheelhouse, but related to some questions that I want to answer in my research life. And it was just so much fun to work with him. We'd started up a new collaboration, and we paused it, obviously, six weeks ago. But I thought I had enough time to tell him just what a wonderful surprise it was to find such a like-minded person to collaborate with, somebody who was super generous with ideas, super generous with coauthorship. It really felt like if you'd had a good conversation with Nick about something, he'd be like, “Do you want to be a coauthor? Can you prove this thing? Let’s actually keep doing it.” And I know that's not always the norm in math, and people are very protective of their ideas, not without reason. But it was just such a joy to work with somebody who actually felt like it was more fun to do math with people, and that was why you did math, than you have to go chasing down these big, outstanding results. Not that what he worked on wasn't important. I mean, he had a Fulbright in Austria. It was pretty cool. And I've just been thinking about that. And you know, this other, my graduate office-mate who died, like, a year and a half ago, who was also really young, and just thinking about for someone like me, who does math, to talk math with friends, I need the reminder that I should tell my friends that I enjoy talking math with them, and I enjoy being friends more often. And so for me every time I sit down and calculate a free resolution, and I do my little “Thanks, Emmy,” I'm going to also jot a note to somebody, just a short note like, “Hey, I was using Emmy Noether’s first isomorphism theorem today, and I thought I'd say hi.”

EL: That’s great. I love that.

KK: So you know, maybe there was a third part of this podcast. We always like to give our guests a chance to plug anything, if they like. Where can we find you on the internets? Or do you have something you really want to promote?

CG: Oh boy. You can find me on Twitter, I am addicted to Twitter. I am @virtualcourtney. And that is my online presence as opposed to my corporeal presence, which is at corporeal Courtney, although I don't know if that's on Twitter? Anyway, I tweet a lot there. My pinned tweet, there is a 10-minute pep talk that I recorded for my students during pandemic times. This is a pep talk I give pretty frequently to students who are maybe in their first math course where it's not just calculus 1-20. We're starting to develop proof techniques, or we're dealing with things that are more abstract than things they've thought about before. Or we're encountering the isomorphism theorems and they're like, “Oh my God, what do these arrows mean?” And they're having this crisis of confidence, like “am I cut out to be a mathematician?” And my little 10-minute pep talk goes into my own bumpy route to becoming a mathematician. I'm plugging it a little bit self-consciously. I put it as an email attachment to my students. And I had put it on YouTube and meant to make it unpublished but clicked the wrong button. And then suddenly it had been shared, and I was getting emails about it and thousands of views. And I was like, I guess I'll leave it up. So I'll plug that for anybody who's feeling a little bit, I don't know, shaky in their mathematician identity, or, you know, not sure that they're allowed to have a mathematician identity. I certainly feel like the last person who was “supposed” to become a mathematician. My initial plan was to be a French major. And I did take my graduate language exams in French. So it's good that I studied enough French to know the subjunctive. But yeah, so I'll plug that and then I'll say one more thing. I am working on writing an open-source, remixable, free WebWork-enabled algebra textbook that takes my predecessor Bob's idea of using Galois theory for the first course in modern algebra, and makes it into an active pedagogy kind of thing, sort of in the model of Active Calculus.

KK: Oh, cool.

EL: Oh, cool.

CG: So I am on sabbatical next academic year, and that's my main project. And if anybody wants to, test WebWork problems, or help out when I'm like, “Okay, why did I think I could write a textbook?” I'm happy to talk math with old friends, new friends, math book friends, WebWork friends, anybody who wants to help make that thing actually happen.

EL: Yeah, and selfishly keep me in the loop about it, I'd love to know about it when it comes out here. You're actually convincing me, and several of our guests are always like, working on me. Like, I should really learn algebra a little better and feel a little more comfortable with some of these ideas.

CG: Sure. I’ll keep you in mind as a beta tester. Over the years, I've developed all these active learning worksheets that are the basis for the book. They look fun. My students are — well, they used to smile. Now they’re masked, so I just assumed they're smiling all the time. But they would smile and argue when they were working on them, which is what I had hoped would happen.

EL: Oh, that’s great. Yeah.

CG: You should rope some friends in to learn algebra with.

EL: Yeah!

CG: Because I think that's the most important thing for sticking with it is having pals.

KK: Well, good luck. You know, as somebody who, I’ve written a couple of books, you always reach this point where it's like, “Why am I doing this? What did I get myself into?” But once you push through that, it's fine.

CG: I’m in the planning stage where I have a big wiki essentially, like a big table with, like, here’s chapter one’s outline, here’s all the resources for everything in here and now, it's a matter of actually connecting the things. I mean, PreTeXt is its own special thing, learning learning PreTeXt. But yeah, I think it will be a good experience. I feel a little bad because you know, both my algebra Professor Marlow Anderson and my predecessor Bob Redfield have these great books. And I'm like, “No, I'm not going to use those. I’m going to write my own. Stick it to you guys.” But I think what I haven't seen is a mixture of the active pedagogy with the Galois theory as the thing that links rings, groups, and fields. So that's what I want to create and put out there.

KK: Cool. All right. Well, this has been a lot of fun, Courtney. Thanks for joining us today.

CG: Yeah, absolutely. I have sort of been like, “When are they going to ask me?” for a little while, not thinking like, maybe I could just reach out and be like, “Hey, ask me.” So I was really excited when you did. And I've loved listening to the podcast. And I appreciate you both for for making it happen. It’s awesome.

EL: Well thank you.

KK: Yeah. All right. Well, take care.

CG: Bye.

[outro]

On this episode of My Favorite Theorem, we were delighted to talk with Courtney Gibbons, a mathematician at Hamilton College, about Emmy Noether's isomorphism theorems. Below are some related links you might find useful.
Courtney Gibbons's website and Twitter account
The Wikipedia article on Noether's isomorphism theorems, which includes a helpful chart describing differences in labeling the theorems
An article about Emmy Noether by astrophysicist Katie Mack and her biography on the MacTutor History of Mathematics Archive
Evelyn's 2017 article in Undark about the effect of Nazism on German mathematics in the 1930s
Our episode with Kameryn Williams
Active Calculus, a free, open-source resource for teaching calculus

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn lamb, a freelance math and science writer in Salt Lake City, Utah, which is very chilly, and I'm trying to warm up from my bike ride just now that I probably should have worn better gloves on. The downhill!

KK: Yeah, yeah, that sweater you're wearing those should help this. Yes.

EL: Yes, it is. I'm showing all of our listeners that this is a wonderfully rainbow chunky sweater that I recently inherited from my grandmother. Probably purchased in 1973 or something.

KK: Yeah. Well, so today's actually, it's November 11. It’s Veterans Day and the University of Florida is closed and so I've done wonderful things, like I went to brunch with my wife this morning. We sat outside. It's not cold here, of course. We sat outside, and then I fixed her, the light in her closet, it was one of those pull cord deals, you know, the core just broke. I mean, inside the lamp, so I had to actually go to Ace Hardware, which I highly recommend. Your local Ace is much better than the big box places and and get the stuff to fix it. So that's been my day.

EL: Thanks to our sponsors, Ace Hardware. No. They care so much about math podcasts, I’m sure.

KK: I’m sure. All right. Well, today we are pleased to welcome Kameryn Williams. Kameryn, why don’t you introduce yourself?

Kameryn Williams: Hi, I’m Kameryn Williams. I'm a mathematician at Sam Houston State University about an hour north of Houston, Texas. It's good to be on.

EL: Yeah, and I bet it's nice and warm there where you are.

KW: It’s in the 70s today, you might think that's warm, being in Salt Lake.

EL: Yeah, I went to grad school in Houston. So I do — November, February, March. Great months there.

KW: Oh, yeah, definitely. Well, like I have family in Idaho, and they've been complaining they got snow recently. And I'm like, come on, it’s not that bad. Quit complaining.

EL: Did you grow up in a warm place? Or?

KW: I spent some time in California growing up but moved around a fair bet.

EL: Okay. Yeah.

KK: All right. Well, so what sort of math are you interested in?

KW: I’m a logician, mostly specializing in set theory.

KK: Oh, okay. Well, that’s, that’s —

EL: I guess that’s okay.

KW: You’ve had a few logicians on, I’ve seen.

KK: We have.

EL: We just had your advisor on a couple of months ago.

KW: That’s true, Joel [David Hamkins] was on. So he took my favorite theorem, so I had to come up with a new one.

EL: So rude.

KK: Oh, well, okay. Well, that's the perfect segue. So what is it? What is your what do you want to talk about today?

KW: Okay. Yeah, so I guess maybe I should hedge? It's hard for me to pick favorites. It's like your children. They're all my favorite.

KK: Sure. Sure.

KW: But definitely among my favorites, one of them is Gödel’s condensation lemma.

KK: Okay, so I don't know this one. I mean, everybody knows his famous incompleteness theorems, right, but —

KW: Yeah, so this comes, I don't know, what, about 10 years after his incompleteness theorems. Okay, so maybe I should step back a bit and not talk about Gödel for a second and kind of set the stage for the theorem.

KK: Okay, that would be great.

EL: That would be fantastic.

KW: By itself, you kind of need to know why it's there to understand why anyone would love it. So set theory, like the study of well-foundedness, we kind of care about these transfinite iterative constructions. So here's a good example: Let’s generate all the sets. You can start with whatever non-sets you have — numbers, whatever — and then you just look at sets of those, sets of sets of those, sets of sets of sets of those, and so on transfinitely. So you just iterate the power set operator, along the ordinals transfinitely out to infinity, you get all the sets that way.

Okay, so it looks like if we want to do this, we need all these sorts really large, weird infinite objects going on. But 1922, Thoralf Skolem made an interesting theorem now known as Skolem’s paradox — whether you think it's a paradox is on you — and he noticed that you could just have a single small countable collection of things that had all the properties what it looks like you needed all these really large sets for. So there's this small little countable thing, it’s got what it thinks are all the real numbers. You know, externally you can see it's countable, but internally, it doesn't know that. It thinks there's uncountably many real numbers. You can do everything with that.

EL: Whoa. Sorry, so you can tell I've not done much with this kind of stuff here, so you’re blowing my mind here.

KK: This is good because that’s kind of how it was when I first learned this. Like, if you would have brought me on as an undergrad, I would have said downward Löwenheim-Skolem, that’s my favorite theorem. Given any structure, you can find a countable elementary substructure, a countable thing that has all the same sort of intrinsic properties.

EL: Okay.

KK: Okay, so from the outside the reals look countable, but inside they don’t?

KW: Yeah, well, so what does it mean that the reals are—

EL: That seems way too powerful.

KW: What does it mean that something’s countable? It means there exists a bijection with the natural numbers. Well, if you don't have all the objects, you just have a small piece, you might be missing that bijection.

KK: Okay.

KW: So you might think it's uncountable, but it's just because your model was too small to see all the bijections.

KK: Okay.

KW: Okay, so Skolem, he called it a paradox. He thought this, you know, revealed all the set theory people were doing was nonsense. Like come on, you can’t even nail down what it means for something to be uncountable. And I mean, the other people kind of had your reaction, you know, it’s blowing my mind. That’s kind of where I fell. But it was really not treated as anything more than a curiosity. So maybe you thought it was a paradox, maybe you thought it was curiosity, but no one really had any use for it beyond that.

And then, late 1930s, about 15 to 20 years after Skolem did this, Gödel was interested, you know, he was fresh off the incompleteness theorems. Everyone loves those. He wants to go onto something a bit more difficult. And so he's concerned about Hilbert’s first problem. Can you prove Cantor's continuum hypothesis? Can you prove that there's no cardinality intermediate between the natural numbers and the real numbers?

KK: Right.

EL: Right.

KW: Okay. Skolem says, you know, maybe it's hard to say what that means, but let's set that aside. Okay, so how is he going to do this is he wants to do something like this construction I mentioned, iterate the power set, but you want to have a bit more control. The problem is if you take the power of sort of the natural numbers, that's going to contain basically every countable object, and how do you get any sort of handle and what this says? So Gödel’s idea was to restrict this to as small as possible to just the intrinsic things. So rather than taking all the sets of natural numbers, you just want to take the ones that you can define just by quantifying over natural numbers. So you'd have, like, the set of prime numbers, because you can say what it means for something to be prime, you can have the set of even numbers, you can say that, but you wouldn't have a weird thing, for example, like these, you know, weird countable models of set theory, you can't really define that just by quantifying over natural numbers, okay, or isomorphic copies of them, however you want to say it.

KK: Okay.

KW: So he iterates this process where you just take the definable sets. And then if you iterate that transfinitely instead of the full power set operator, you get some other hierarchy, some other transfinite hierarchy of construction. So this is called the L hierarchy. I looked this up because I didn't know: L stands for law. Not entirely sure why he chose law.

EL: Like, rules, that kind of law?

KW: Yeah, I'm not entirely sure what he thought with it, but that's what it was. Okay. So just like with this, like, iterating the power set hierarchy you can do like a sort of Skolem collapse there. You can do the same thing with Gödel’s L hierarchy. So you can build it up to a certain stage. And then you can say, okay, I built it up to some uncountable stage. I'm going to take just a countable piece that looks like this. It's got all the same elementary, or to use a bit of jargon, first-order properties.

KK: Okay.

KW: Okay, so Gödel’s condensation lemma just says that if you do this Skolem collapse process, take this elementary sub-model, it's just going to be isomorphic to an initial segment of the construction. So maybe you built up to stage alpha, you take this countable piece, it's like building up to stage beta, some stage beta below of alpha.

KK: Okay.

KW: Okay. And looking at your faces, it's not clear why this is interesting.

EL: Right. Well, I do have a question that — sometimes it's really fun to be very naive and ask ridiculous questions. So like, okay, if we're trying to do this process for the real numbers, say, and find this countable thing underneath it, like, is there a canonical way to say, okay, these are the real numbers that are in this countable piece of it. Like, can I say is pi in here is, is 1/4 in here? Could I ask you that? And you could say yes or no.

KW: I can say yes to both pi and 1/4. Other numbers might be harder to say. But those are easy. Yeah.

EL: All right.

KW: This is the point, is you have a very firm control over what's going on. This is kind of the opposite of taking the power set construction where, what is the power set of the natural numbers? That's kind of the vague question that the continuum hypothesis is trying to get at. This is we're going to get at it very firmly. But this is interesting because Skolem’s paradox gives us this condensation lemma when you apply it in Gödel’s constructible universe. So this paradox becomes a theorem. And this theorem then is kind of the key lemma to prove the continuum hypothesis for L. So you want to now prove that you can't have too many reals. Well, any real you have, it shows up at some countable stage. So if every real shows up at a countable stage, there's aleph-one many countable stages. Aleph-one is the smallest uncountable cardinal. That means there's aleph-one many reals, the smallest uncountable cardinal, nothing in between.

EL: I feel like you're pulling one over on us.

KK: I don't think that's a proof of the continuum hypothesis because, you know —

KW: It’s a proof of the continuum hypothesis in Gödel’s model he built.

KK: Okay.

KW: So this is how he showed, then, that it's, you can't disprove it from this.

EL: Right. It's not a contradiction.

KW: Yeah. So it's consistently true because it's true here. Now, this isn't necessarily true in the full universe of sets. You might miss out on a lot of things in Gödel’s construction, and then later Paul Cohen showed, okay, you can't prove it for the full universe of sets.

KK: Right.

KW: But yes, this is just one step, this thing that was a paradox became a key lemma in order to kind of do half of Hilbert first problem. And I think it's kind of interesting, something that’s first is introduced, maybe it's a paradox, maybe it's a cute little curiosity, and then Gödel, you know, 15, 20 years later or so, realizes, oh, no, this is actually — a version of this is a really important step in this proof.

KK: Hm. Okay, this is completely foreign to me.

EL: Yeah. Well, I'm a little confused on the timeline here. Had Gödel already done the incompleteness theorems, but then this is something else that he did later?

KW: This is after. Yeah, yes. The stuff with L, this is about 1940. Incompleteness theorems are about 1930. So it's about 10 years later.

EL: Okay. But he kind of figured out that this work fits in with the incompleteness theorems.

KW: Well, I mean, the incompleteness theorems are one of the first forays into a larger program. Like, okay, you know, they established, you can't write down a computable list of axioms that decides all the things you want. Well, can you then find natural questions that you can decide? Well, here's one that was a candidate: the continuum hypothesis. So it's easy to say, okay, here's these weird diagonalization tricks to get something. How do you show that a “natural statement” is undecidable? That takes more work. It’s a more difficult problem.

KK: Okay.

EL: Yeah. And so we — I emailed you to be on the show, because I saw a tweet you had done about how maybe you felt it was a little unfair that Gödel is so known for the incompleteness theorems, whereas you feel like this work is actually more impressive, and people should be more, I guess, should like it more than the incompleteness theorems.

KW: Ah, that was maybe a bit of a hot take. But I think it's it's uncontroversially mathematically more impressive. The proof of the incompleteness theorems is basically he noticed the same sort of similar sort of weird thing people had kind of dismissed, and he realized this could be important, this diagonalization trick. If you can somehow code proofs and formulas into number theory, they can refer to themselves and you get this self-reference. So, you know, cute little curiosity, and he takes this cute little curiosity and he turns into a theorem. So he did the same thing here, but it was a much more difficult theorem, much harder to do. And not to pooh-pooh the incompleteness theorems, but if you take a senior undergraduate-level logic class, you'll prove the incompleteness theorems by the end. They’re not that difficult of results.

KK: Well, that's probably why they get more love, right?

KW: Yeah. I mean, that's a big part of it. And so that's maybe where a bit of a hot take came from.

KK: Right, right. Yeah, this is all very Kafkaesque to me, right? So, you know, Kafka has these — he uses infinity a lot in his in his writing, you know. It reminds me of — so the L standing for law made me think of Kafka’s story “Before the Law,” which is part of The Trial.

KW: I’m not familiar with that one.

KK: It’s part of The Trial, but it's often pulled out as a separate story. So the deal is that the man goes to see the law, and there's a gatekeeper, and he can't get through because the guy won't let him through. But he says even if you do, there's another door inside. It’s sort of like these infinite gatekeepers that you can never, so you can never get to the law. And then the man who came to see the law, eventually he just stays because he needs to get in there, but he can never get in. And he keeps asking the guard, and the guard says, eh, maybe tomorrow. And then finally he just dies. You know, it's very, it is a very Kafka sort of story. But in a way, sort of these hierarchies are embedded in all of that. Okay. So I mean—

KW: I mean, what he should have done is he should have taken an elementary sub-model.

EL: Yeah.

KW: Don't wait till the very end. Just get it down to a smaller piece.

EL: Exactly. We're about to publish “Mathematicians Rewrite Kafka.” Mathematicians rewrite all 20th century literature, solve any problems anyone ran into, just take care of it.

KK: So you've loved this theorem for how long? Is this a love at first sight kind of thing?

KW: Ah, I mean, I'll be honest, it took me a good amount of time to really appreciate it. And you kind of see the statement in a textbook, and it's this abstract technical thing, and it's really, I mean, okay, if I'm being honest, Kanamori, Aki Kanamori, has some really nice papers that go through the history of this, and reading those made me appreciate a lot more, seeing where it came fromm some of the intellectual development, rather than just after the fact, here is this lemma in the middle of some technical exposition.

EL: Yeah, well, and often, when you're seeing theorems in a textbook, it's like, they don't have a flag next to the one that like you should really pay attention to or love. I remember, when I first took, like, my first algebraic topology class, there was — this wasn't a theorem, it was a definition — called the fundamental group. And it wasn't clear to me for a while that like, this was a particularly important object. And then a couple of weeks later, I was like, man, two weeks later, we're still talking about this fundamental group? And sure, it has the name fundamental, so that part is on me that I didn't really notice that. But yeah, sometimes when you're reading, you know, the first time you encounter this, you're just deluged with all this information, and you’re making your own hierarchy of which things you know, and how the information fits together, and which things are really central, is difficult.

KW: Well, I mean, if it was easy, they wouldn't pay us to do it.

EL: Right.

KK: That’s true. So the other thing we do on this podcast is we ask our guests to pair their theorems with something. So what pairs well with Gödel’s condensation lemma.

KW: Ah, I would say probably a nice brie with some green apples and some honey.

EL: Okay, that does sound delightful.

KW: I mean, I don't think it has particularly anything to do with the theorem, but I just think brie is nice.

KK: You just want to eat that anytime, right? Oh, yeah. I'm there with you.

EL: I mean, yeah, that sounds good. While you're reading Kafka also?

KW: Maybe a nice wine too. Okay. All right. Yeah.

EL: Well, that does sound just lovely.

KW: So we always like to give our guests a chance to, to plug something. Where can we find you on the worldwide intertubes?

KW: My website is just my name, KamerynJW.net. Okay, you can spell it because it will be linked in the episode description, presumably.

KK: We’ll link straight there.

Yeah, the thing I would like to plug as I'm on the job market. Please hire me. I’ll just be honest. That's what I would like to plug.

KK: Okay. Great.

EL: Excellent.

KK: Everyone needs work.

EL: Yeah. Any hiring committees that ask us for references, we'll send in this episode.

KW: Yeah. Just I would like a tenure track job.

KK: Okay. How long have you been at Sam Houston?

KW: This is my first year here. Previously, I was at University of Hawaii for a postdoc.

KK: Okay. That's quite the switch.

KW: A little bit, but you know, they offered me a position there. Like, I’m not going to say no to a postdoc in Hawaii.

KK: No, no, no.

KW: If I don't like it, I just don't stay. It’s a postdoc.

KK: And Sam Houston is in Huntsville, correct?

KW: Yeah.

KK: Okay. So yeah, different vibe.

KW: A little bit different, yeah.

EL: Slightly. Yeah. Do you live in Huntsville or in Houston?

KW: I live in town. It's a bit too far to commute every day.

EL: Yeah. I know, actually, a friend of mine, who is one of your colleagues I know lives in Houston because her spouse has a job in Houston. And so they drive opposite directions in the morning, but that sounds hard.

KK: Yeah, well and Houston is so big. You could still live in Houston and have like a two hour commute, right?

KW: Oh, yeah. Right. It's an hour to Houston, but that's just the outskirts of Houston and depending on where you want to go, it might be two or more.

EL: Great city, though. I miss living in Houston, even though people like to trash it for being such urban sprawl, and muggy and humid and full of mosquitoes and everything, but I liked it.

KW: My problem is I did grad school in New York. So I see any other large city and I'm like, Here are the ways this is not like New York. I don't like that.

KK: Well, New York's dense, right?

KW: Yeah, exactly.

KK: I once knew a mathematician who got a job in LA and and he complained that it wasn't big enough for him. He wanted to live in Tokyo or someplace like that. I just thought, well, okay, more power to you. Here. I am in my little, you know, 100,000 person town in Florida.

KW: I mean, that's larger than where I am.

KK: Sure. All right. Well, this has been this has been great fun, Kameryn. I learned something, I think.

EL: Yeah.

KK: I have to turn it over. Yeah.

EL: Yeah. One thing I love about writing the transcripts for the episodes is I get to, you know, experience it again. And then I often make a lot of connections. Like, oh, it would have been really smart of me to ask a question about this. But in the moment, I didn't think of it. So you know, everyone could listen to it again and get something else out of it.

KK: Right.

EL: Thanks for joining us.

KW: Yeah. Thank you for having me.

[outro]

On this episode of My Favorite Theorem, we had the pleasure of talking with Kameryn Williams from Sam Houston State University about Gödel's condensation lemma. Here are some links you might find interesting after you listen to the episode.
Their website and Twitter account
The Skolem paradox in the Stanford Encyclopedia of Philosophy
Gödel's incompleteness theorems in the Stanford Encyclopedia of Philosophy
Akihiro Kanamori's paper about Gödel and set theory
A shorter and more accessible paper by Kanamori and Juliet Floyd on the same topic
Before the Law by Franz Kafka

Evelyn Lamb: Hello and welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Evelyn Lamb, a freelance math and science writer based in Salt Lake City but currently podcasting from my parents’ house in Dallas, which is actually not any warmer than Salt Lake City right now, unfortunately. This is your other host.

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. I'm in my faculty office. I'm usually — I'm the chair of the department. But I'm hiding out in the faculty office today. Actually, I was looking for better wireless. And it seems to be working a little better in here. But it's so weird because I have nothing in this office, like nothing. It's very strange. So anyway, how are things going for you?

EL: Not too bad. Yeah, seeing my family, which is nice, and very excited about today's episode. So let's get right to it! We're happy today — both of us are music lovers, and we're very happy to introduce our guest, Emily Howard, a composer. Emily, do you want to tell us a little bit about yourself?

Emily Howard: Yeah. So I'm based in the UK, in Manchester. I'm originally from Liverpool. And I'm a composer. I love writing for large ensembles, large acoustic ensembles, such as the orchestra. I also write vocal music, choral music, and also chamber music. So a lots of different areas. And I suppose probably the reason that you've got me on here is that I've got a real interest in mathematics. And actually, I have a degree in mathematics and computer science, my undergraduate is in mathematics and computer science. And I suppose that, you know, definitely it's one of the key influences on my work.

EL: Yeah, I was listening to — it might have been the BBC Proms a few years ago — I was listening and saw this piece that I think was called Torus. And I thought, “You don't accidentally name something Torus.” So I decided to try to find out more about this person who had named her composition Torus. And so yeah, I found out that you had a math background, and thought it would be just really fun to talk to you on the podcast. So yeah, can you talk a little bit about the — I know, you've done some collaborations with mathematicians, you know, written pieces, like kind of in conversation with mathematicians in the composition process, and I would love to hear about that.

EH: Yeah, so I mean, I suppose actually, in 2015, I had I think it was a Leverhulme fellowship at the University of Liverpool, working within the mathematics department. I had been invited by Lasse Rempe-Gillen. He's a professor in dynamical systems. And I think he'd been in touch because he had himself played the violin in an amateur orchestra in Liverpool, and they had performed a piece of mine called Mesmerism. Actually, it was after Ada Lovelace. Ada Lovelace used to dabble in all types of things, including mesmerism, sort of a form of early hypnosis. And the piece, I mean, that piece was a piece for solo piano and orchestra, and he was playing in it. And I think he thought it will be great to invite me as someone with a mathematics background back — after 15 years away in the music world— actually back into the maths department. And, I'm so glad that this happened, because going back in and speaking to lots of different mathematicians in a different way, rather than, I don't know, having to take exams and study, I was more being an observer, having amazing conversations about people's research. Regularly I'd speak to someone in one area of mathematics and another, and I suppose I felt — what I realized was that mathematicians often don't understand what each other are speaking about. So I didn't feel so bad about it, that I could sort of dip in and I suppose it helped me to take a more global approach, or to take in more general ideas, because I think that's one thing, perhaps, you lose if you don't practice mathematics regularly, you know, you lose this very detailed approach, and that's kind of annoying. It’s really annoying in many ways, but I also find it annoying when you can’t completely understand something because you'd have to spend quite a few years really thinking about it in depth. But then something is also gained from sitting back and taking a look at everything and absorbing it in a very different way.

So anyway, we worked together on a set of chamber pieces. You can find them on my website. One’s called Leviathan, another is called, well, these are all pieces, exploring ideas from lattice research in dynamical systems. And I suppose they're all based on thinking about perturbations, and thinking about — I’ve got some etudes as well, they’re called Etudes in Dynamical Systems — just trying out very particular things with datasets, but also conceptual ideas about reaching stasis and loops from something. I also spoke a lot with colleagues in mathematical biology and also in topology. And I remember, I think this is where, kind of, talking about Torus, it was kind of born, the idea, because this period of time gave me more confidence to actually do something with mathematics. Just to go back back a bit, after my undergraduate in mathematics, I'd spent a couple of years doing a Masters at the Royal Northern College of Music in composition, and then more years at Manchester University doing a PhD in composition as well. And I suppose, when I was doing this, I really wanted to use these ideas or concepts from mathematics, but I didn't really have a musical technique to do that. I suppose that grew. And at this point, it sort of all merged, and I was able to perhaps achieve something that I wanted to achieve with these mathematical ideas through music. And so yeah, just to go on, you mentioned Torus. And I'm immensely proud of Torus. It's a big orchestral work, it's like 23 minutes long, based on the mathematical shape, a torus, a doughnut.

And this is when I had actually just met a mathematician, Marcus du Sautoy, who I’ve worked with quite quite a lot. He's based at Oxford, and we met because of this thing. And when we spoke first, actually, he was writing about, no, he’d written a play about a torus. And I was sort of saying, “Well, actually my piece is about a torus,” so we really bonded on this. And one of the things that interested me most was, you know, the, the BBC Proms are held at the Royal Albert Hall in London. And the first thing he said was, “Well, that is a torus.” And I just thought, How funny. It really is, because you can run round and round it, you know, it's just just really nice to, you know, hear this.

KK: So I'm especially fascinated here, because my son is a composer. Well, he just finished his undergraduate degree, and now he's in graduate school for composition. And so I'm always — first, I'm encouraged that composers can actually make a living, right, because whenever I would tell a mathematician, one of my friends, “Oh, my son's studying composition,” they say, “Well, how’s he going to live?” And I say, “Well, I don't know,” but clearly it’s possible to figure it out. So I'm super encouraged. Yeah, so Evelyn, you had a real question, though.

EL: Oh, well, I was just saying this is an existence proof then for your son.

KK: Yeah, that's right. That's right.

EL: Yeah, well, I was — you know, music is a very non representational art form, unlike a canvas where you could — I mean, it's still hard to represent mathematical ideas on a painting. But can you say anything about how you do use the form of music to represent mathematical ideas?

EH: I suppose — it's really interesting that you say that, because I completely agree with you actually, that it is very difficult to do that. And I wonder if, even if I, especially in abstract music, if we take aside — you know, if it has text in it, that's a very different thing — but in an abstract form, actually, I wonder if I ever tried to absolutely represent something from mathematics within music, whether in fact, anyone would know. And I suspect, actually, there are cases when I've worked very closely with people and they really know what I'm doing. I think they can certainly tell, but actually, I don't think that that's necessarily happens. And also, it's definitely not what I'm trying to do. Absolutely not. So I suppose my aim is not to represent mathematics. My aim is to — I mean, I love hearing about mathematics, and I'm completely inspired by processes and systems and patterns. And I suppose what I'm doing is taking them, and that’s a catalyst for my creative process. And so I do think something comes through, but I think it's more that I couldn't make what I'm making without doing this. And it's more that something new is occurring that came from thinking about these things. But it's definitely not a representational thing. I mean, that's definitely the case for maybe the more large-scale pieces we could talk about, like Torus.

But I think also there have been a couple of pieces when I have tried my very best to represent ideas, and one of them would be this set of dynamical system etudes. That was one of them. And then another one would be the music of Proof, which is, I wrote a string quartet because Marcus and I were discussing proof and different ways you can prove things: proof by contradiction, by induction. He wanted to, to put those to me, and then I would create responses to different kinds of proofs. And so I suppose that’s as representational as I've been, and actually, it was really useful, because in doing that, then I gained a whole set of like, I thought about — I've never tried to write music before as though I'm solving a mathematical proof. But actually, in doing that, that led me to new places. So that's kind of what I'm trying to do, find new ways to do things and make new sounds.

EL: Yeah, maybe respond more than represent you?

EH: Definitely.

KK: I mean, certainly, some music is sort of, you can tell, it's not deliberately mathematical, I don't think. I think of like, like Steve Reich's minimalism percussion pieces, right?

EH: Clapping Music.

KK: Yeah, they’re so cyclical. And you know, you can kind of, if you think about it as a mathematician, you can kind of imagine, well, if I saw this on the page, it would almost look like — our listeners can't see my hands doing this — but you can imagine sort of, you know, intersecting sine waves and things like that. So I can see how you could do that.

EH: And to take that further, I wonder if all music in some way can be — well, I say reduced, and I don't like to reduce — but you could certainly represent ideas in very complex music, I think mathematically probably. Not that anyone necessarily has. But perhaps the secret to lots of things is in really complex music represented. I mean, I don't know.

KK: Well, there’s a whole journal of mathematics and music. I mean, you could you could certainly, I sat on a PhD committee for composer on campus here, who really was trying to do these very strange time signatures that were sort of approximations of pi and things like that.

EH: Wow.

KK: And I put the question to him. I said, “Okay, I mean, you can make a machine do these. But, you know, can a human do this?” And he said, “Well, no, not really.” But it was interesting.

EL: Can humans do math anyway?

KK: This is not a philosophy podcast. I don't know.

EL: Yeah. Thankfully,.

EH: Just to say on that on that subject. I mean, I've got certain things — there are certain things you do put in notation that are absolutely beautiful mathematically, you know, but in fact, yes, they're not really possible. But they they do make a performer think in a certain way, and it will give you certain results. So I mean, I think there can be, I think they're beautiful, and they can be there. But you just can't expect perfection, perhaps.

EL: Yeah. All right. So we always like to ask our guests, what is your favorite theorem?

EH: I mean, we've kind of been speaking about it. That's, that's the main problem with this. I mean, actually, I would say, rather than a theorem, it's definitely these shapes. So I mean, let's take the torus, but I've got this fascination, I'd say, with thinking about mathematical shapes and thinking about, you know, them being far away, and also thinking about being on them. And I suppose, I mean, so you've got a torus. I mean, so, if you think about the difference between, say, flat geometry and a sphere with the spherical geometry, and then, I mean, there's a pseudosphere, and then I would call, like a negatively curved geometry, I've got a piece actually called Antisphere because I've worked out that an antisphere, there's a word antisphere, which is the same as the pseudosphere, and the word antisphere is a lovely word, and I don't like the word pseudosphere. So I called the piece Antisphere. And I suppose I've just got this — because, you know, my music is usually notated. So I find it very interesting as a process to be in my mind wandering around these shapes. And I suppose, I also think it’s very useful for music because they don't necessarily need to be 3d, I mean music, potentially, you could perhaps hear really high-dimensional mathematics. Because, you know, you could be traveling around and you're not necessarily visualizing it.

So my answer to your question is that I like these mathematical shapes. And I've been thinking about them and really studying them and using them to influence these large-scale orchestral work. So that's Torus, Sphere, and Antisphere. And going forward, I'm now looking at, well, I'm trying to look at Thurston’s eight geometries and pushing myself in this direction with the aid of a number of very kind mathematical friends, because they're very difficult. And yeah, so that's kind of the direction where I’m going.

And I find, you know, I suppose that's what we were saying earlier about the representation thing. So there are a number of stages. So I'm thinking about this tours, I'm thinking about traveling around it. Shall I give an example of the compositional process for writing Torus?

KK: Sure, please.

EL: Yeah.

EH: Okay so Torus is a piece, yes, it was for the BBC Proms. It was performed first in 2016, by the Royal Liverpool Philharmonic Orchestra with Vasily Petrenko. Now I’d say, I wrote it over at least a year or so. And I'd also been thinking a lot about this just previously, thinking about traveling on this torus. And what I was thinking about when I wrote it was the idea that you're traveling round and round one direction of the torus, you know, and I took these very consonant chords, see, I've got major sixths. If you hear it, they’re a very constant chord. And I start with this major sixth, and I travel up one and down one and up another — they're going up in semitones, and down, and they reach a point and they come back again. So that the harmony is very much also shaped like a torus all the way through. And it's skewed, when you hear it, it's more complex than just listening to it timed because I've got a real combined fascination with exponential functions and the idea of really big things becoming absolutely minuscule, but also kind of being related. So, you know, this kind of journey around the torus, the first loop in this orchestral work is about five minutes long. And if you do listen, you'll hear it in the strings. And to my mind, I've got this sort of ever-expanding enclosing torus idea in the strings, but, but maybe on the surface of it, as you go round, you hear different things each time you go round. And that's perhaps in the wind and in the percussion, and in the brass. So it’s almost like, you're on a landscape and it's changing.

Now, the piece works by this — suppose the radius of the doughnut-shaped Torus, sort of shrinking. So you go around it, and it gets quicker and quicker. And you'll hear — I think it's around about maybe 17 minutes in, there’s a viola solo. Because it's really big, and it sort of shrinks down, I think there are seven, sort of around this this way of the torus, and you get this viola solo that encapsulates these major sixth idea because you can just play all of that on one solo viola. And then there's this sort of huge shift in thinking in the piece. And, again, that almost came from thinking about dynamical systems and just completely changing the goalposts, and then you're traveling very fast. And in my mind, we've flipped to that we’re now thinking about going on the other direction of a torus forever and ever. Do you know what I mean?

EL: Okay.

EH: You’ll hear that. It’s a huge moment. And that music seems to completely change. And I don't think anyone listening to it will hear a torus shape, because you can’t, because you're hearing my ideas about journeying around it. Do you see what I mean?

EL: Right.

EH: But actually, it really helped me to think about this thing and you know, it is absolutely about this sort of journeying around on it.

EL: Yeah, I don't know, I'm almost imagining like you could, maybe — I wish I had like a, some kind of inner tube or something. But like, maybe the first loop around is on the top where it's further apart, and then you're kind of almost falling into the middle where the You know, the two sides of the torus, or the hole of the tours is kind of small. And then you you kind of flip the other way and start. The next time I listen to it, I'll have to imagine that kind of journey.

EH: The thing is, I’m sure now I've told you that, you probably will hear it, you know? And I think as well, I mean, it's not, it’s never so obvious, because I'm also doing a couple of other things as well. So each journey around this torus — so I said the first one's about five minutes — you’ll also hear these major sixths alternate in the string, so it’s almost like there's one side and there's another side. So you hear this sort of, they appear to travel like this, and then on the other way, they're sort of all, everything is together, so it's rhythmic unison. And it and there's a sort of written-out rall, so it gets slower and slower. And then you'll hear, it comes around again. So I mean, there’s that and there, and then sometimes as well, one side of the torus. That is going on in the background in my mind, and that's definitely happening. But in fact, there's something else on the surface, like there are these very loud things going on sometimes. And they obliterate this thing, but it's always there. And a bit like, say, a painter might have a layer of something to start the piece, this is my layer, this sort of journey. And then it builds up.

KK: And you totally, you had Evelyn at viola solo.

EL: Yeah, yeah, I do play viola.

EH: Great. You’ll hear that.

EL: Yeah, you're pushing all our buttons. Kevin’s son’s composer thing, my viola thing. You’re just excellent. Yeah. What I was wondering is, do you know, why were you drawn to the torus specifically? Do you know?

EH: It’s a really good question. No! I’m not sure I do.

EL: Well, you sort of mentioned the Sphere and Antisphere and Torus are kind of this set. And those can be the three different two-dimensional geometries. But I'm wondering whether you kind of already liked the torus and just were excited that you could use it here, or if you kind of sought it out because it is this model of flat geometry.

EH: That's very interesting, because I think I was trying — when I was writing Torus, I was already thinking about writing Sphere as well. And, you know, in many ways, there's something too perfect about a sphere.

EL: Yeah. It constrains you. It’s very limiting.

EH: It's a problem. And the other thing is, I was having to think of a title. And actually, the title Torus is just the most beautiful word. And there, you know, I think it's the fact that the sphere was too perfect. And there was a way to sort of have two very different things on this torus. But I think it's also true to say that I hadn't thought until later on to do the set. It emerged, you know. So I was sort of playing around and suddenly, I thought, when I'd written Torus, I thought, “Well, actually, that's quite a really wonderful way to think about things.” And then then came Sphere. And that's a shorter piece. It’s a five-minute piece for chamber orchestra. I suppose, I was thinking more about spherical geometry. As I said, I found that one more difficult because it had to come straight after I was writing Torus. However, a couple of years later, that's when I decided to write this Antisphere. And that's for the London Symphony Orchestra with Simon Rattle. And it was just performed a few months before lockdown. So I was very lucky to have that happen. I'm really grateful. And I'd say that, I mean, it's interesting, because somehow, I feel that my link to the maths in it is stronger with Antisphere. And I think that's just because I've gained experience and maybe confidence. I suppose with Antisphere, I was thinking about harmony. I like using quarter tones. They’re present in a lot of my music over the last, say, 10 years. We’re used to maybe the 12 semitones of the scale, and the quarter tones are just in the middle. And I suppose I was thinking about, you know, the classic angles in a triangle, which add up to 180. And then on the sphere it’s more, and then on the negative curves, it’s less. And actually in Antisphere, I use that so that you get these chords that we might all recognize, like a major chord, but actually it's been shrunken in some way.

EL: Oh!

EH: And so it's weird. And it sounds weird. And also, there's a section in there, it’s a very fast section of a kind of circle of fifths idea. But actually, I think it's a circle of 4 3/4. And they're great, because they last longer. Obviously, you know, these players are amazing, because they can do this in an orchestra ensemble.

EL: I was going to say how did you get — I think I would struggle to play quarter tones on purpose. I’m sure I’ve played some by accident.

EH: I mean, I was really blessed with this string section, but it is an amazing experience. And I think you do feel because you do feel this — I mean, the association of a major chord, I mean, I don’t think I’m using major chords, but whatever I’m using — you can feel it, but then you can certainly feel that something's happened to it. And I really liked that. And another thing I did was the rhythmic thing. We've mentioned Steve Reich, and I like the thought that you've got a regular pulse in our world. But if you're somewhere else, or looking in on a different kind of — well, the thought is that you're the pulse might, if it was like 1-1-1. But in fact, it could go 1-2-4-8. So actually, therefore our perception of this piece of music is not one of a regular pulse. In fact, I have created it as though it is, but it isn't. So we hear a short section and a longer one. And then and by the time you get to here, it's so long, you perceive it as something completely different. And I love that as a compositional process. So I did that. In Antisphere, I've actually taken a chord sequence that I used in Mesmerism, this this early Piano Concerto I wrote in 2011. And that’s, quite regularly, you know, there are a few chords, at the opening, you hear solo piano playing these chords, and they're quite regularly spaced. But in Antisphere, what I've done is I've taken them as the basis, these piano chords, as the basis of these sort of chords that start off like a bit and actually end up as three minute sections. And loads of weird things going on in between you like this with the percussion, there's loads of weird metals and sort of resonant sounds. And so, yeah, as I said, it gives a completely different sort of perception of what's going on.

EL: Yeah, I don't think I have listened to Antisphere yet. But I am now going to seek it out because I really want to hear this — especially if I've got in my mind this idea of, like, hyperbolic triangles with their, you know, curved in — or they look to me like they're curved in. If I actually believed enough, they would look like straight lines to me, because I would really embody the hyperbolic metric.

EH: But and this so this is happening in pitch, but it's also happening in sort of rhythmically and timbre in lots of ways as well. I did write an article, Orchestral Geometries, which I will post, you know, we could put there. And actually, I've been lucky enough to have wonderful recordings of the three pieces, and also the scores as well alongside so yeah.

EL: Oh, wonderful. So yeah, another thing we do in this podcast, which we kind of already done, is we ask our guests to pair their theorem, or mathematical object, with something. And so yes, I kind of assume that you would be pairing it with your compositions based on these, on the torus and these other shapes. But yeah, do you want to add anything else about that, or any other pairings? You know, if it's just a nice cup of tea or something?

EH: Or a doughnut? I was going to pair them with that. And actually, I was going to pair them perhaps with, you know, we could we could play a little clip from one of them. So this clip is a little bit from the orchestral work Torus, and I think, it’s sort of about two thirds through where everything changes. So before then you've been rotating round, the kind of doughnut shape of the sphere, no sorry, the doughnut shape of this torus. And then you go into this viola solo and then you hear the huge perturbation, and that's when you change as though you're rotating. At least when I was writing, I was thinking about rotating around the other part of the torus.

[music clip]

KK: So I really, now I'm sort of curious to know how you’re — I mean, I know you haven't thought about it yet. But pursuing the sort of 3D geometries, that's going to be weird. Have you seen these, people have tried to visualize these things using VR. Have you seen any of these explorers?

EH: Yes, yeah, I have. And actually, if you have any more, I would love to have them because it's very helpful to see them. We've been looking online, and there are quite a few. You know, there are some amazing websites and people, actually programs for doing it. And we were kind of exploring it. But I am interested. And it's a new pursuit, you know, so I'm really interested to, well, think about it more. I think it'll take some time as well.

I'm currently writing a piece. And I was thinking that it will be very nice, now having written these orchestral geometries, to embody sort of a process of moving between the different geometries a bit. So that you could, I mean, no one need necessarily know this, but musically, there might be a difference between this spherical and the — but they could, so I'm really interested in that. So I'll do that. But I also I'm interested in this, is it H3 [hyperbolic 3-space]?

EL: Yeah.

EH: So yeah, I'm interested in that. And it's kind of frustrating, because I'd really like to understand why. And it's quite difficult, I think. I'm not sure. I think it's probably well beyond me. But, you know, the thought of this dodecahedral space and moving around and the twist, I'm really interested in the fact that it would you know, that the spherical that these twists, change what dimension, it’s really interesting.

EL: Yeah, yeah. And I'm wondering how you can use like, the specific — what am I trying to say, like, the opportunities that an orchestra gives you to sort of, you know, can you, like, pass off ideas from one section to another, can that give you a twist? Or, you know, something like that, how you could, I guess, use the tools that you have at hand to kind of explore it in a different way than you might if you were trying to draw it on a piece of paper?

EH: Yeah, I think, I mean, it’s probably better doing that than drawing on a piece of paper actually. Like, I feel there must be something about the way I think that I like — these thoughts, and these kinds of systems and shapes immediately present musical ideas to me. So, absolutely, you know, what I would be interested in is a process that you could audibly hear becomes something different when it's spherically curved, and becomes something different when it's, you know, hyperbolic, and making that more and more extreme. So the math underlies that, and perhaps, maybe it's a process of me imagining going through this, these these dodecahedron and things. But in fact, that will be a layer, and then after that, perhaps then, I do something more musical, as in this has given me this data, but I accentuate it in ways, and perhaps also go against it and things. Because I do think as well with art, usually, you're asking a question rather than answering anything, which is a real difference, I think. Because mathematicians, you're so interested in truth, or mathematical truth, and you’re really bound by it; it’s really important. But I always feel that I'm more interested in what your process is in these wonders, and then I'm just using that to sort of leap somewhere unknown.

KK: Well, so are we. We're leaping into the unknown all the time, we just, then want we want the answer once we get there, right?

EL: I do think there's a similarity in that, you know, the way mathematicians approach things a lot is, you kind of set up these axioms, and this is the rule system I'm going to be working within. And I think that forms of art do that as well, you know, say, this is the aesthetic system that I'm working in, or this is, and maybe they're not as rule-bound as mathematics is, because when you say like, these are the axioms, or these are, you know, whatever, I'm doing, then you're just really stuck with them. And with art, a lot of the times it's about breaking the rules. But I do think, you know, you kind of set up these, sometimes composers can set up compositional rules where, like, Okay, well, I'm writing a fugue, which means that I have, you know, I have this kind of structure. And the allowed things to do are like transposing it or flipping it backwards, or things like this, and say, like, well, I'm working within this form, in this way. So, I mean, I did a lot of music in college, and I was kind of torn between going into math and music. And I think the way that I thought about them kind of tickled the same part of my brain, that’s why I was interested in both things, and ended up, you know, in math instead of music professionally, but I do think we've made these aesthetic or form kind of rules in music or art. And now we're going to work within them, just the way mathematicians do with axioms.

EH: Yeah, I mean, I completely agree with you on that. And I also would say as well that I agree, and actually working closely with professional mathematicians has really kind of opened my eyes to how much they are going, because I don't think you learn that, you know, these are the answers, actually it is a lot of guesswork. And it's really, yeah, leaping, as you say.

EL: Yeah, well, this, this has been a lot of fun to talk with you about this. I really hope our listeners will go find your pieces. And we'll definitely link to your website and the article you talked about and everything. Is there anything else that you want to share, you know, things you want to suggest that they look into or read or any concerts coming up that people could actually attend or livestream?

EH: Yes. So actually, next week, the BBC Philharmonic is playing sphere in Manchester. And that should be on Radio 3, as well. I’m based at the Royal Northern College of Music in Manchester. I’m a professor of composition, and I run, I direct, the PRiSM lab. PRiSM is a research center for Practice and Research in Science and Music. And I'm lucky enough to work there also with colleagues. Marcus du Sautoy joins us and David De Roure, as well. And I suppose we're interested in mathematics meets music meets science meets AI, and there are lots of different types of composers. We have a blog, and that's where this orchestral geometries blog will be. And lots of very exciting, very different things going on there. So I just wanted to mention that and maybe I could also give you a link for that as well.

KK: Sure.

EL: Yeah, that would be great.

KK: We will include it. All right. This has been fantastic. Thanks for joining us, Emily. It's really been great.

EH: Thank you so much for having me. It's been a pleasure.

[outro]

On this episode of the podcast, we were delighted to talk to composer Emily Howard, who uses her mathematics background in her compositions, about her favorite mathematical object, the torus, and the orchestral work it inspired. Below are some links you may enjoy after you listen to (or read) the episode.
Emily Howard's website
Her page about the composition Torus, including a recording by the BBC Radio Orchestra
Her article Orchestra Geometries
The November 11, 2021 BBC Radio concert featuring Howard's composition Sphere
PRiSM, the Royal Northern College of Music Centre for Practice and Research in Science and Musicthat Howard directs
A website visualizing the eight Thurston geometries for 3-dimensional space
An article by Evelyn about the pseudosphere (or antisphere)
Our episode with Emily Riehl, who is relevant to this episode because she is both an Emily and a violist

Evelyn Lamb: Hello, and welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Evelyn Lamb, one of your hosts. I'm a freelance math and science writer in Salt Lake City, Utah, and this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. I like your background, Evelyn. Our listeners can't can't see it. But it looks like you had a nice camping trip somewhere in Utah.

EL: Yes, that's actually a couple years ago — we’re going to describe this in great detail for all the listeners who can't see it, no — this was a camping trip in Dinosaur National Monument on the Utah-Colorado border, which is a really cool place to visit.

KK: Excellent. It’s beautiful over there. Yeah. So how's things?

EL: Oh, not too bad. A bit smoky here, we're getting a lot of wildfire smoke from the west coast. And it, you know, makes some of those outdoor activities that are so fun, a little less fun. So I hope it clears out.

KK: Right. Yeah, well, my big adventure lately was getting my son settled into his new apartment in Vancouver. And it was my first time on an airplane in a year and a half. That was weird. And then of course, Vancouver, such a lovely city, though. We had a good time. So he's all set up and starting grad school and his nice new adventure.

EL: Yeah. And he's kind of, almost as far as he can be in a populated place in North America from you.

KK: That’s correct. That's, like, almost 3000 miles. That's far.

EL: Yeah.

KK: It’s great. Anyway, enough about that.

EL: Yeah. Well, today, we're very happy to have Joel David Hamkins join us. Joel, would you like to introduce yourself and say a little bit about yourself?

Joel David Hamkins: Yeah, sure. I am a mathematician and philosopher at the University of Oxford. Actually it’s something of an identity crisis that I have, whether I'm a mathematician or a philosopher, because my original training was in mathematics. My PhD was in mathematics. And for many, many years, I counted myself as a mathematician. But somehow, over the years, my work became increasingly concerned with philosophical issues, and I managed somehow to turn myself into a philosopher. And so here in Oxford, my main appointment is in the philosophy faculty, although I'm also affiliated with the mathematics department. And so I don't really know what I am, whether a mathematician or a philosopher. I do work in mathematical logic and philosophical logic, really all parts of logic, and especially connected with the mathematics and philosophy of the infinite.

KK: Well, math is just kind of, it's just applied philosophy anyway, right?

EL: This might be a dangerous question. But what are — like, do you feel a big cultural difference between being in a math department and a philosophy department?

JDH: Oh, there's huge cultural differences between math and philosophy, I mean, on many different issues, and they come up again and again, when I'm teaching especially, or interacting with colleagues and so on. I mean, for example, there's a completely different attitude about reading original works. In philosophy, this is very important to read the original authors, but in mathematics, we tend to read the newer accounts, even of old theorems, and maybe for good reason, because oftentimes, those newer accounts, I think, become improved with greater understanding or more connections with other work and so on. Okay, but one can certainly understand the value of reading the original authors. And there's many other issues like that, cultural differences between math and philosophy.

EL: Yeah, well, I, when I was at the University of Utah, I taught math history a couple times. And I was trying to use some original sources there. And it is very difficult to read original sources in math. I don't know if part of that is just because we aren't used to it. But part of it, I do feel like the language, and the way we talk about things changes a lot really quickly. It makes it so that even reading papers from the 1920s or something, you sometimes feel like, “What are they talking about?” And then you find out Oh, they're just talking about degree four polynomials, but they're using terms that you just don't use anymore?

JDH: Yeah, I think that's absolutely right. And oftentimes, though, it's really interesting when there's an old work that uses what we consider to be modern notation. Like if you look at Cantor's original writings on the ordinals, say, it's completely contemporary. His notation, he writes things that contemporary set theorists would be able to understand easily, and even even to the point of using the same Greek letters, alpha and beta to represent ordinals, and so on, which is what we still do today. And so it's quite remarkable when the original authors’ notation survives. That's amazing, I think.

EL: Yeah. So we invited you here to share your favorite theorem. What have you decided to share with us today?

JDH: Well, I found it really difficult to decide what my favorite theorem is. But I want to tell you about one of my most favorite theorems, which is the fundamental theorem of finite games. So this is the theorem, it was proved by Zermelo in 1913. And it's the theorem that asserts that in any two player finite game of perfect information, one of the players has a winning strategy, or else, both players have drawing strategies if it's a game that allows for draws. So that's the theorem.

EL: So what is the definition of a game? This is, like, the most mathematical thing to set, you know, like, yeah, okay, games, we've all played them starting from when we were, you know, two years old or something. But now we need to sit down and define it.

JDH: Yeah. Well, that's one of the things that really excites me about this theorem, because it forces you to grapple with exactly that question. What is a game? What is a finite game? What does it mean? And it's not an easy question to answer. I mean, the theorem itself, I think, is something that you might think is obvious. For example, if you think about the game of chess, well, maybe you would think it's obvious that, look, either one of the players has a winning strategy, or they both have drawing strategies. But when it comes to actually proving that fact, then maybe it's not so obvious, even for such a game as chess. And then you're forced to grapple with exactly the question that Evelyn just asked, you know, what is a finite game? What is a strategy? What is a winning strategy? What does it mean to have a winning strategy, and so on?

So there's a wonderful paradox that surrounds the issue of finite games called the hypergame paradox. So if you have a naive account of what a finite game is, maybe you think a finite game is a game so that all plays of the game finish in finitely many moves or something. Okay, so that seems like a kind of reasonable definition of a finite game. And then there's this game called hypergame. And the way that you play hypergame is, say, if you and I are going to play hyper game, then the first player, maybe you go first, you choose a finite game. And then we play that game. And that's how you play hypergame. So the first player gets to pick which finite game you're going to play. And then you play that game. And if we said every finite game is a game, so that all plays end in finitely many moves, then it seems like this hypergame would be a finite game, because you picked a finite game, and then we play it, and then the game would have to end in finitely many moves. So it seems like hyper game itself is a finite game. But then, the paradox is that if hypergame is a finite game, then you could pick hypergame as your first move.

KK: Right.

JDH: Okay, but then we play hypergame. But we just said when playing hypergame, it's allowed to play hypergame as the first move. So then I would pick hypergame as my first move in that game. And then the next move would be to start playing hypergame. And then you could say, hypergame again. And then I could say hypergame, and so on, and we could all just say hypergame, all day long, forever. But that would be an infinite play. And so what's going on? Because it seems contradictory. We proved first, that every play of hyper game ends in finitely many moves, but then we exhibited a play that didn't end. So that's a kind of paradox that results from being naive about the answer to Evelyn's question, what is a finite game? If you're not clear on what a finite game is, then you're going to end up in this kind of Russell paradox situation.

KK: Exactly. Yeah. That's what I was thinking of, the sets that don't contain themselves. Right.

JDH: Right, exactly. It's actually a bit closer to the what's called the Burali-Forti paradox, which is the paradox of the class of all ordinals is well ordered, but it's bigger than any given ordinal. And so it's something like that. So if you think a lot about what a finite game is, then you're going to be led to the concept of a game tree. And a game tree is the sort of tree of all possible positions that you might get to. So there's the initial position at the start of the game, and the first player has some options that they can move to, and those are the sort of the child nodes of that root node. And then those nodes lead to further nodes for the choices of the second player, and so on. And so you get this game tree. And the thing about a finite game is that — well, one reasonable definition of finite game is that the whole game tree should be finite. So that in finite infinitely many moves, you're going to end up at a leaf node of the tree, and every leaf node should be labeled as a win for one of the players or the other, or as a draw, or whatever the outcomes are. And so if you have the finite tree conception of what it means to be a finite game, then hypergame is not a finite game, because at the first move, there are infinitely many different games you could choose, and so the game tree of hyper game won't be a finite tree, it will be an infinite tree. It will be infinitely branching at its first node.

And so if you have this game tree conception, then a finite game can mean a game with a finite game tree. And then we can understand the fundamental theorem of finite games. So my favorite theorem is the assertion that in any finite game like that, with a finite game tree, then one of the players has a winning strategy, or both players have drawing strategies. And what does it mean to be a strategy? I mean, what is a strategy? If you think about chess strategies or strategies as they're talked about conventionally, then oftentimes, people just mean I kind of heuristic, you know, the strategy of control the center or something, but in mathematics, we want a more precise notion of strategy. So controlling the center isn't really a strategy; that’s just a heuristic. It doesn't tell you actually what to do. So a strategy is a function on the game tree that tells you exactly which moves to make whenever it's your turn. And then a play of the game is basically a branch through the game tree. And it conforms with this strategy if whenever it was your turn at a node in the game tree, then it did, what the strategy was telling you to do at that node. So a strategy is winning if all the plays that conform with that strategy end up in a win for that player.

KK: Okay, so is this the point of view Zermelo took when he proved this?

JDH: So yes, Zermelo didn't have, he didn't quite have it all together. And this is maybe related to the fact that we don't actually read Zermelo’s original paper now when we want to prove the fundamental theorem of finite games because we have a much richer understanding, I think, of this theorem now. I mean, for example, in my in my proof-writing book, I gave three different proofs of this theorem. And Zermelo didn't have the concept of a game tree, or even a finite game in terms of game trees, like I just described. Rather, he was focused specifically on the game of chess, and he was thinking about positions in chess as pictures of the board with the pieces and where they are. But nowadays, we don't really think of positions like that, and there's a kind of problem with thinking about positions like that. Because if you think about a position in chess, like a photograph of the board, then you don't even know whose turn it is, really, because the same position can arise, and it could be different players’ turns, and you don't know, for example, whether the king has moved yet or not, but that's a very important thing, because if the king has moved already, then castling is no longer an option for that player.

KK: Right.

JDH: Or, or you need to know also what the previous move was, in order to apply the en passant rule correctly, and so on. So if you just have the board, you don't know whether en passant is allowed or not. I mean, en passant is one of these finicky rules with the pawn captures where you can take it if the previous player had moved two steps, and your pawn is in a certain situation, then you can capture anyway. So I mean, it's a technical thing, it doesn't matter too much. But the point is that just knowing the photograph of the board doesn't tell you doesn't tell you whose turn it is, and it doesn't tell you all the information that you need to know in order to know what the valid moves are. So we think of now a position in a game is a node in the game tree, and that has all the all of the information that you need.

So Zermelo’s proof was concerned with games that had the property that there were only finitely many possible configurations, like chess. There are only finitely many situations to be in, in chess. And he argued on the basis of that, but really, it amounts to arguing with the finiteness of the game tree in the end.

KK: Yeah. So is this at all related to Nash's equilibrium theorem? So I was scrolling Twitter the other day, as one does when one has nothing else to do, and I saw a tweet that had, it was a screenshot. And it was the entire paper that Nash published in the Proceedings of the National Academy proving his equilibrium theorem. I mean, it's a remarkable thing that it's just a few paragraphs. Is there any connection here?

JDH: Right, so actually, this kind of nomenclature, there are three different subjects connected with game theory. There’s game theory, which, Nash equilibrium and so on is usually considered part of game theory. And then there's another subject standing next to it, which is often called combinatorial game theory, or it's sometimes also called the the theory of games. And this is the the study of actual games like Nim, or chess, or Go, and so on, or the the Conway game values are part of this subject. And then there's a third subject, which I call the the logic of games, which is a study of things like the fundamental theorem of finite games, and the sort of the logical properties. So to my way of thinking, the Nash equilibrium is not directly connected with the theory of games, but rather is a core concept of game theory, which is studying things like the stability of probabilistic strategies, and so on, whereas combinatorial game theory isn't usually about those combinatorial strategies, but about sort of logically perfect strategies, and optimal play, and so on. And that isn't so much connected to my way of thinking with the Nash equilibrium.

KK: So fine, I hand you a finite game. It has a winning strategy. Can you ever hope to find it?

JDH: Oh, I see. Is it computable? Well, yeah, this is the big issue. Even in chess, for example, I mean, chess has a finite game tree. And so in principle, we can, in the sense of computability theory, there is a computable strategy you can prove, because it's a finite game. So the strategy is a finite function, and every finite function like that is definitely computable. And we can even say more about how to find the strategy. I mean, one of the proofs is this backpropagation proof, you look at the game tree, and you propagate. You know that the leaf nodes, the terminal nodes are labeled as a win, and you can propagate that information up the tree. But it turns out that the game tree of chess is so enormous…

KK: Right.

JDH: That it wouldn't fit in the universe, even if you used every single atom to represent a node of the tree. And so in that sense, you can never write a computer program that would compute the perfect chess play. It's just too big, the strategy. If you're really talking about the strategy on the whole game tree, then the game tree is just too enormous to fit in the universe. And so it's not a practical matter. But theoretically, like in terms of Turing computability or something, then of course it's computable. There are some other issues. For example, I've done a lot of work with infinitary games. I mean, this connects my interest with infinity. And it turns out that there are some, some positions, say, in infinite chess, which I've studied, we identified computable positions in infinite chess, for which White has a winning strategy, but but there's no computable winning strategy. So if the players play computably, in other words, according to a computable procedure, then it will be a draw. So it's very interesting, this interaction between optimal play, and computable optimal plays not always, they don't always align.

EL: So I've got to ask you to back up a little bit. What is infinite chess?

KK: Yeah.

JDH: Oh, I see infinite chess. So imagine a chessboard without any border. It just goes forever in all four directions.

EL: So you start with the same, you know, don't start with infinite number — or, yeah, I’ll let you keep going.

JDH: So of course, infinite chess. I mean, we can imagine playing infinite chess, you know, at a café or something, but it's not a game that that you sit down in a café to play. It's a game that mathematicians think, “What would it be like to play if this if the board looked like this, or if it looked like that?” And so there's no standard starting position. You present as a starting position, and you say, “Well, in this position, it has a very interesting property.” So Richard Stanley asked the question, for example, on mathoverflow. And that sparked my interest in this question. Well, one of the things that we proved was that you can have positions in infinite chess that White has a winning strategy, so it's going to win in finitely many moves, it’s going to make checkmate in finitely many moves. But it's not made in n for any n, for any finite n. In other words, Black can make it take as long as he wants, but it's hopeless. In other words, Black can say, “Oh, this time, I know you're going to win this time, but it's going to take you 100 moves.” Or he could say, “This time, it's going to take you a million moves.” For any number, he can delay it that long, but still White is going to win infinitely many moves following the winning strategy. So these are called games with game value omega, and, and then we produce positions with higher game, higher ordinal game values, omega squared, omega cube, and so on, the current record is omega to the fourth. So these transfinite game values come in. And so it's really quite fascinating how that happens.

KK: Well, knowing that you like logic and philosophy so much, now I see why you like these games, right? This is kind of a logic puzzle, in some sense. Yeah. Yeah.

EL: Well, and what made you choose this theorem as your favorite theorem?

JDH: Well, I mean, there's a lot of things about it to like. First of all, what I mentioned already, it forces you to get clear on the definitions, which I find to be interesting. And also it has many different proofs. I mean, as I mentioned, there are already three different proofs, three different elementary proofs. But some of those proofs lead immediately to to stronger theorems. For example, there's a slightly more relaxed notion of finite game, where you have a game tree, so that all plays are finite, but the tree itself doesn't have to be finite. So this would be what's called a well-founded tree. And then these would be called the clopen games, because in the in the product topology, the winning condition, amounts to a clopen set in that case. And then the Gale-Stewart theorem proved in the ‘50s, is that infinitely long games whose winning condition is an open set, those are also determined: one of the players has a winning strategy, open determinacy. And then Tony Martin generalized that to Borel determinacy. So Borel games also have this property. So infinite games whose winning condition is a Borel set in the product space are determined, one of the players has a winning strategy. And then if you ask, Well, maybe all games are determined, in the sense that one of the players has a winning strategy, you whether or not the winning condition is Borel or not. And this is called the axiom of determinacy. And it's refutable from the axiom of choice for games on omega, say, you can refute it. But it turns out that if you drop the axiom of choice, then the consistency strength of that axiom has enormous strength. It has large cardinal strength. In large cardinal set theory, the strength of the axiom of determinacy is infinitely many wooden cardinals, if you've ever heard of these large cardinals. And for example, under AD, the axiom of determinacy, it follows that every set of reals is Lebesgue measurable, and every set of reals has the property of Baire. And so there's all these amazing regularity set theoretic consequences from that axiom, which is just about playing these games.

And so what I view as the whole topic, the fundamental theorem of finite games just leads you on this walkway to these extremely deep ideas that come along much later. And I just find that whole thing so fascinating. That's why I like it so much.

EL: So I guess I want to pull you out of these extremely deep things into something much more shallow, which is, are there games like, games that kids play, or that like, people play, that are infinite games by their nature?

JDH: Oh, I see. I mean, I've studied quite a number of different infinite games. For example, well, I have a master's student now he just wrote his dissertation, his master's dissertation on infinite checkers, but we're in the UK, so we call it infinite draughts. But I guess ordinary checkers is usually just on an eight by eight board and so that doesn't count. So, I've done some work on Infinite Connect Four, I mean, infinitary versions of Connect Four and infinite Sudoku and infinite Go. And there are infinite analogs of many of these games that are quite interesting.

EL: I guess what I'm wondering is, so, in my mind, maybe there's a game where you can, like, go back and forth forever. And there's no rule in the game that says you can't just like walk towards the opponent and then walk backwards. And then the game tree might not be finite, or maybe I don't quite understand how the game tree would work.

JDH: No, no, you're absolutely right about that. Yeah, in that situation, the game tree would be infinite. I mean, it's related — for example, in chess, there's this threefold repetition rule. If you repeat the situation three times, then it's a draw. But the actual rule isn't that it's automatically a draw, but that either player is allowed to call it a draw. Right? And that's a difference. Because if you don't insist, if both players choose not to call the draw, then actually the game tree of chess would be infinite then because you could just keep moving back and fourth forever. And there's also another rule, which is not so well known, unless you're playing a lot of chess tournament play, which is the 50 move rule. And this is the rule in chess tournaments that they use, where if there's 50 moves without a pawn movement or a capture, then it's a draw. And the reason for that is — I mean, of course, I view both of those rules as kind of practical rules just to have an end of the game so that it doesn't just go on forever in the way that you described. But when we were deciding on the rules for infinite chess, we just got rid of those rules, and we thought, look, if you if you want to play forever, that's a draw. So any infinite play is a draw is the the real rule, to my way of thinking. And and the reason why we have the threefold repetition rule, and the 50 move rule is just those are proxies for the real rule, which is that infinite play is a draw. But it doesn't quite answer your question. I'm sorry, I don't know any children's games that are just naturally infinite already. Name the biggest number.

EL: Yeah, well I’ve also been sitting here wondering, like, can you make a game — like other children's games that aren't maybe on a board? Tic tac toe, of course, is the first thing I think, but then what about Duck Duck Goose [for the Minnesotans: Duck Duck Grey Duck] or Simon Says, or these other children's games that aren't really board games? You know, can you even work those into the framework of these finite games, or infinite games, or not? I don’t know.

JDH: There is this game of Nim where you play with stacks of coins and you remove coins from one stack or another. It's a beautiful game with a really nice resolution to it in terms of the strategy, but that game has infinitary versions where the stacks are allowed to have infinite ordinal heights. And basically, the classic proof of the Nim strategy works just as well for ordinals. And so if you if you think about the sort of the balancing strategy, where you look at the things base two, and so on, well, ordinals have a base two representation in ordinal arithmetic also, and you can still carry out the balancing strategy even when the stacks have infinite height.

EL: It never occurred to me to think about infinite ordinal height Nim.

KK: Well, you never have that many toothpicks, right?

EL: Yeah. Yeah. limited by my environment, I guess. I must say that I've spent a lot of my mathematical career sort of avoiding things with with too much infinity in them, because they're very intimidating to me. So maybe we have kind of different mathematical outlooks.

KK: Right. Yeah. So the other thing we like to do on this podcast is invite our guests to pair their theorem with something. So what pairs well, with the fundamental theorem of finite games?

JDH: Well, there's only one possible answer to this, and I worry that maybe I'm cheating by saying, of course, I have to pair it with the game of chess.

KK: Sure.

JDH: Because Zermelo’s theorem was really focused on chess, and he proved that, look, in chess, either White or Black has a winning strategy, or else both of them have drawn strategies. And I never played a lot of chess when I was a child, but when I had kids, they got involved in the scholastic chess scene in New York, which is quite hyperactive, and fascinating. And so my kids were playing a lot of chess, and I went to hundreds of chess tournaments and so on, and so I started playing chess and I learned a huge amount of chess from, they had such great coaches at their schools and so on. But actually, I'm a pretty mediocre chess player even after having played now for so many years. And one of my coauthors on the infinite chess papers that I wrote, is quite talented chess player. He's a national master, Cory Evans. He was a philosophy graduate student when I met him at the City University of New York, which is where my appointment was at the time. And so I got to meet a lot of a lot of really talented chess players, and it was really great working with him on that infinite chess stuff, because I realized that that actual chess knowledge is really focused on the 8 by 8 board, and that once you go to these much bigger boards, the the chess grandmasters even become a little bit at sea. And so I would know what I'm trying to do mathematically to create these positions with high game values, and I would show them this crazy position with, you know, 20 bishops and hundreds of rooks and so on. And I would talk a little about, and he would say, hang on, this pawn is hanging here, it's totally unprotected. And it would completely ruin my position. So their chess ability, their chess reading ability was such that they could look at these crazy infinite positions and point out flaws with the position. And that was really something that was important for our collaboration. These chess positions are so finicky, these huge, infinite ones. And so many details are running on whether the things are protected properly, and whether — because oftentimes, you have to argue that the play has to proceed according to this main line. And if you want to prove the theorem, you have to really prove that. And if there's some little upset that means that the flow of play isn't exactly like what you thought, then the whole argument is basically falling apart. And so it really was depending on on all of that. So I really had a great time interacting with a lot of these talented chess players. It was really fantastic.

KK: I’m a lousy chess player.

EL: A lot of interest among chess players in the mathematical study of the game of chess? Even leaving aside the infinite versions, but, you know, the finite version. I assume there are theorems being proved about regular chess. Do players care about them much?

JDH: Well some of them definitely do. And actually, there's a huge overlap, of course, between chess players and mathematician

EL: Oh, yeah, that's true.

JDH: And so maybe maybe a lot of the interest is coming from that overlap. But, for example, there was a problem that I had asked, I think I asked it on Mathoverflow. Take chess pieces. On an empty board, take a full set of chess pieces and just throw them at the board. So you you get some position. What's the chance that it's a legal position? So in other words, a random assignment of pieces. And you can you can make some calculations and prove some interesting things about the likelihood that it's a legal position. In other words, a legal position, meaning one that could in principle arise in a game, in a legal game, right. And

EL: Do you happen to recall any ballpark, you know, is this, like a 1% chance?

JDH: It’s way less than 1%. It’s exceedingly unlikely.

EL: Okay.

JDH: If you allow, if you insist on all the pieces, because then there haven't been any captures. So the pawns have to be sort of perfect. There has to be one column and opposing. And already just because of that, that already makes it extremely unlikely to happen if you have all the pieces. And then some other people answered on Mathoverflow, I think giving better bounds when you don't have all the pieces and so on. But it wasn't quite open. But I think the general conclusion was that it's extremely unlikely that you get a legal position.

KK: Well, that makes sense. Given the complexity of the moves, it would be pretty remarkable if a random placement would would actually work.

JDH: There are some amazing — there’s a book by Raymond Smullyan, about the “chess detective,” and he has these many instances. It’s sort of like he gives you a chess position, and and you have to deduce, what was the previous move? Because these positions are often extremely strange. Like you think, “How could that possibly arise?” So there's sort of logic. I mean, he's a logician. And so there are sort of chess logic puzzles to figure out what the previous move was. And there's often a story associated with the game that, you know, so it must have been Black, who was the murderer because… It’s really some fascinating work that way. I really like that.

KK: Well, this has been informative. I've certainly learned a lot today.

EL: Yeah.

KK: So we like to give our guests a chance to advertise. Where can we find you online? And if there's anything you want to promote, we're happy to let you do it.

JDH: Oh, I see. Well, you can find me online, I have a blog, jdh.hamkins.org. And also, I'm on Twitter, and also on Mathoverflow. And I just published a number of books. So one of them I mentioned already, it's called Proof and the Art of Mathematics. And this is a book for aspiring mathematicians to learn how to write proofs, Proof and the Art of Mathematics with MIT Press. And I have another book, a philosophy book called Lectures on the Philosophy of Mathematics, also with MIT Press. And that is a book that I use for my lectures on the philosophy of mathematics here in Oxford. And it's, I would say, a kind of, grounded in mathematics perspective on issues in the philosophy of mathematics.

EL: Yeah, and if I can praise you a little bit, I will say something that I have enjoyed ever since I've been following you is that you — some of some of the things you write are about, like very technical, you know, deep mathematical things. But you've also had some really neat, like, puzzles that you've shared with children and stuff like that. I remember I was working on a Math Circle project one time about paper folding and cutting and you had a fun, I think it was like you show someone a configuration of holes in a piece of paper and say, can you fold the paper so that you just have to punch one hole in this folded paper to get the holes looking like this? Or something like that. And so it kind of spans a big range of mathematical sophistication, and what level you want to jump into something. So I think that's something fun and other people who might be looking for activities like that might enjoy it.

JDH: Thank you so much. I'm so glad to hear you mention that project. Those projects are all available on my blog if you click on the math for kids link, which is one of the buttons on my blog. And they all arose because I was going into my daughter's school every year, or a couple times a year, with these different projects, including that one and a number of other ones. So have about a dozen or more math for kids projects on my blog.

KK: Very cool.

EL: Well, thanks for joining us. I enjoyed talking about chess, a game that I have probably played, you know, 10 times in my life.

JDH: Well, it's a pleasure to be here. Thank you so much for having me.

KK: Yeah. Thanks.

[outro]

In this episode of the podcast, we were happy to talk with Joel David Hamkins, a mathematician and philosopher (or is that philosopher and mathematician?) at the University of Oxford, about the fundamental theorem of finite games. Here are some links you might enjoy perusing after you listen to the episode.

His website, Twitter, and Mathoverflow pages
On his website, check out Math for Kids for some fun activities for all ages
His books Proof and the Art of Mathematics and Lectures on the Philosophy of Mathematics
The Wikipedia page about the fundamental theorem of finite games
The PBS Infinite Series episode on infinite chess
The Mathoverflow question and answers about legal chess board positions

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, where there is a family of quail that live outside my window. And they don't know that I'm here so I can watch them scurrying around in the bushes. There are at least five young ones right now.

KK: Cool.

EL: They are so cute. Oh, it's just like, sometimes — they're not here right now, which is good. Because otherwise, I would just be like, staring out my window. Looking at these cute little quail.

KK: Oh, see, so in Florida, we're in actually the boring birds season because you know, it's just, this is the locals. So I see my cardinals and the titmice and all of that, but it's still fun. I still feed them. I'm out there every day. They’re eating me out of house at home. It's true. It's a good thing. All right, well, so today we are very pleased to welcome Ranthony Edmonds. Why don't you introduce yourself, please?

Ranthony Edmonds: Hi. Yes. So I'm Ranthony Edmonds. I'm a postdoctoral researcher at The Ohio State University.

KK: The Ohio State.

RE: The “the” is very important in Columbus. We take this very seriously. And I've actually become one of those people who corrects and makes sure that they add the “the” in conferences and notes and things like this. It’s very obnoxious. Yeah. So I'm a postdoctoral researcher at The Ohio State University. I study commutative ring theory, classically, specifically factorization theory. And I'm in the midst of this sort of interesting transition, where I am looking into applications of algebraic topology. So I spent the last year learning a bit about topological data analysis, and I’m specifically interested in applying that to redistricting. So I, you know, I my interests are kind of broad, but specifically, to kind of give you some keywords, commutative ring theory, topological data analysis, redistricting. And, of course, you know, my general mission is to increase access to mathematics for Black Americans and members of other traditionally underrepresented groups in the mathematical sciences. And I'm trying to do that through a combination of inclusive pedagogy. academic research, and community engaged scholarship.

EL: Nice, and I was perusing your website before this to familiarize myself a little bit, and I saw that you're working on a kind of a history-related project about Black mathematicians at The Ohio State University historically?

RE: Yes. I think a lot of people had a lot of different reactions to what happened last summer with with George Floyd and the protests that swept over the country. And one thing that I sort of questioned is this idea of, well, if we're going to try to improve access to mathematics for Black Americans, for other traditionally underrepresented groups, how do we really begin to do impactful work if we're not really aware of what's happened historically? And I've always been interested in history. I think a lot of this comes from a previous project I'm still doing with the Hidden Figures story and sort of using that to center discussions about diversity and equity in the discipline. But yeah, I just love math history. I have a team that's very interdisciplinary, and we're looking at the history of the math department, specifically at Ohio State this summer, and then on into the fall. So there are really two things that we want to do. One is, you know, a lot of the narratives of these the pioneers who graduate from the department with PhDs, with master's degrees, they're kind of just hidden. There’s not a lot of recognition about the work that they've done. But we have discovered that there are seven Black PhDs who have graduated with a doctorate degree in math from Ohio State. And we’ve got two former university presidents among that midst. We've got lawyers, authors, you know, program officers in the NSF, just people who have gone on to do really prolific things, and yet are still somewhat kind of unacknowledged by the university themselves, and then just in the wider math community, and I think that there are a lot of hidden stories out there. And I think when I reflect on the Hidden Figures story, this is what made that so impactful is because people didn't know. So I think that there's a lot of work out there that's being done by wonderful people that people just don't know about. And so what we're trying to do is to highlight and amplify those stories, one. And then two, examine and contextualize their experiences at the university. So what was happening when they were students here? What influenced their trajectories after graduation, where they went to work, if they went to industry or academia? I think if we think about trying to get more people in graduate school or get more people at the Faculty level, well, we should start by thinking about how we're serving our undergrads who are in that actual population and how we've done that historically.

So we're doing a lot of things. I'm working with some people in strategic communication, some people in our Office of Diversity and Inclusion here at Ohio State. Also we have a connection with the National math Alliance. So they are an organization that I'm very intimately familiar with from my time in graduate school at the University of Iowa, but they are really focused on trying to increase the number of minorities entering PhD programs in the mathematical sciences. And this is pretty broad, right? It's not just math. It’s statistics, it’s economics, it’s something that requires quantitative training as an undergraduate. So we're working with them. And we're also working with a local museum, the Ohio History Connection, and there is a specific branch which is, they call themselves Afro-Am, but the official title is the National Afro American Museum and Cultural Center. And they're located in Wilberforce, Ohio. We're working with them with some of our archival research, and also our community programming. So we've learned a lot of really interesting things. We’ve sort of broken up the history of the department starting at 1963, when the first Black male PhD, his name was William McWorter, graduated from Ohio State up to the present and just identifying individuals who earned degrees during that time period and interviewing them, as well as sort of contextualizing what are the big things? How did selective admissions affect Black student enrollment in general and specifically in the math department? How did you know the protest of the ‘60s and ‘70s impact the campus environment? We have students who are wonderful who are helping us look at these different questions. And then there's actually like a lot of cool math. For instance, the the first Black male PhD from Ohio State, his name was William McWorter. And he was part of this camp along with like Axler and others that was like “death to determinants.” We don't need them, why are we teaching them to students? He felt like it was a very crippling tool pedagogically, in that students just used them for computations and had no idea what they were. And which makes sense, because I think I've experienced that on the student end of things.

EL: I’d say guilty as charged there.

RE: So he wrote a couple of papers for that were published in Math Magazine about determinant-free methods in linear algebra. And specifically he competed came up with an algorithm for computing the characteristic polynomial of a matrix and computing eigenvalues and eigenvectors of a matrix without using the determinant. And I say, “a matrix,” there are obviously conditions imposed upon it, but it was really cool. So I'm working with a student this summer, and we're reading through this paper, and then we want to create a lesson plan related to that algorithm. Because it mainly focuses on dependency relationships, like do you understand the difference between, like, given a list of vectors, can you determine if they're linearly independent or dependent? And then it requires doing that via elementary row operations. So it's just sort of hitting some of the high points from introductory linear algebra, without getting into the weeds of what the determinant really is. So we're working to create a lesson plan from that, and then hopefully, that'll be incorporated back into the honors track here. It was when he taught here. And disseminating that. Our main goal is learn it and then disseminate it, you know, so other people can can learn from what we're figuring out. So there's this history component, and then there's a lot of math that we're uncovering from the history that's just really interesting in its own right, that we hope to, over time turn into lesson plans that other people can use for their classrooms.

EL: That’s really cool. Like, so I've done a little bit of dabbling in math history and stuff. And it's always really interesting to me how much the language has changed. And you'll see an abstract for a paper written it decades ago and realize, like, we just talk about things differently now, and it's kind of hard to dig down and figure out, Okay, how would I think about what they're doing here? You know, they have these names for different curves that aren't names I use anymore, and like, how do I translate it? It's like almost a translation project. Even going back just to the ‘60s, maybe.

RE: Yeah.

EL: So that must be really interesting. And I think that's a great project for students to do. So yeah, that sounds so cool.

KK: It is. You're very busy. And you know, your list of mathematical interests is super interesting to me too. I mean, I'm not a ring theorist, but the whole TDA and redistricting.

RE: Yeah. We’ll have to talk a bit after the podcast. But yeah, it is really interesting. And I think, you know, it's part of this whole approach of just trying to humanize mathematics. We're studying it, and we're getting into the nitty details, but we're also thinking about how people came to be mathematicians, and how this has actually been affected historically, especially for Black Americans, by policies. You know, a lot of the PhDs that we're studying about were supported by NSF fellowships, and this is a direct response to the space race that was happening. And they saw the influx of federal funding. And so it's all really interesting. I feel like I'm learning a lot about — even though it's focused on Ohio State — I feel like I'm learning a lot about the math community. And one thing that is really cool about us is sort of how we do our lineage. Right? And so the math genealogy websites are really cool, because you can sort of track back, very easily, Oh, this person who would who would they have been working with? You know, I think in another discipline, if you are trying to figure some information out about the person that you want to know, who their academic “siblings” were, that might be actually difficult to discern, but we have the Math Genealogy site where we can get that information easily.

KK: Yeah. All right. So this podcast, though, is called My Favorite Theorem, so we asked you on here for a reason. So, Ranthony, what is your favorite theorem?

RE: Yeah, so I am taking it back, back, back, back. So I actually would say that my favorite theorem is the fundamental theorem of arithmetic.

KK: Okay.

RE: It’s very classic. And the reason that I like it is because it's sort of the first introduction to really meeting math in disguise, because I think a lot of people are at least aware of the concept of it in grade school, even if maybe we don't get into the implications. And so, you know, the fundamental theorem of arithmetic, it states that, given an integer, so positive whole numbers greater than one. So yeah, greater than one, excluding zero, you know, it can be written uniquely as the product of prime numbers. And that this decomposition into primes is unique except for the order. So in practice, it means give me a number, like any number that's an integer, and I can factor it uniquely into small pieces called primes. And that’s it. That's the only way I can factor this number. And it gives it a unique signature. And it's telling us that in the same way that atoms are the building blocks of ordinary matter, these prime numbers build up the integers. And I love it, because there are a lot of implications in the work that I do in factorization theory that can all sort of be traced down to this fundamental idea. And I also love it because when I talk to younger students about ring theory or things like this, I always start with the fundamental theorem of arithmetic. And I tell them when they're drawing factor trees — at least that's how I learned, I'm not sure how you guys — is that what you remember? You had the number and then you do the branches?

KK: Yeah.

EL: I loved doing that when I was a kid. I don't know if you two were also like that. That it was kind of a soothing little exercise. Like, write down a big number — not too big; I wasn't super ambitious — but like, write write down a number and just do a little tree figuring out, you know, yeah, that kind of thing. I don't know. I thought it was fun.

RE: Yeah, I usually start off with when I’m talking with — I don't want to say little kids, right — with general audiences. I'll start off by asking people to pick their favorite three-digit number. So I guess maybe, do you guys have a phone or calculator handy?

KK: Sure.

RE: So this may not pack the same sort of punch, but I ask people to pick their favorite three-digit number. And then I ask them to create a six-digit number by taking that three-digit number and repeating it twice. So I usually use 314 because it's an approximation for pi. I was also married on Pi Day. And so 314 is is my number, and then I create a six-digit number, so that's going to be 314,314. Yeah. Okay.

KK: I chose 312.

RE: Okay, all right. And so I want you to take your six-digit number and divide it by 11.

KK: Okay.

RE: Okay. I have 28,574 right now. And then I want you to take that number and divide it by 13.

KK: It’s amazing that you're getting integers here.

RE: Yeah.

EL: Or is it?

RE: So now I have 2198. Okay, and so now I want you to take this number and divide it by your original three-digit number.

KK: Yep.

RE: And did everyone get seven?

KK: Yes.

EL: Yay!

RE: So yeah, so it's like this really cool thing where if you take a number and you multiply it by 1001, it has the effect of creating a new six-digit number. That's your original original number repeated twice. And so essentially, because we know that 1001 factors uniquely into primes, which is guaranteed to us by the fundamental theorem of arithmetic, you know, 1001 is 7×11×13. And so if you divide away 11, then divide away 13, if you divide away that original number, no matter what it was, you're going to be left with 7. And so it's really exploiting this property of the integers. This is really cool. And so I don't know, I just really love the theorem.

So why, I guess maybe, do I care about it? In terms of the mathematical sense, besides the fact that it's cool? It’s because there's a lot of deeper underlying mathematics here. So it's like, we have this statement that tells us, given any integer, we can decompose it uniquely into the product of prime numbers. And so like I mentioned before, these prime numbers are acting kind of like the atoms of the integers. And so in factorization theory, this is sort of the name of the game. We're really interested in, how do we decompose a mathematical object into its smallest pieces? And this is our very first introduction to this idea. It's in elementary when we're breaking numbers into primes. And then typically, when we kind of level up, the next thing we try to break down are polynomials, right? And so in algebra, whatever level in which you had it, you have a polynomial and you want to break it down too. And so it's like, okay, we want to factor it. And the question is, how do you know when you're done factoring? So you know, with a prime number, you circle it, and it's like, we have our, you know, but with polynomials, it’s a little bit more hazy. There's not a list of, well, there are, but a list of just all the irreducible polynomials that ever are. And so the question is, is there some sort of fundamental theorem that exists for the set of polynomials over the reals? So if we had something like x4−1, I remember in algebra that this was a difference of squares, and so there was a pattern. So I could break this into x2+1 and x2−1. And then this was always a tricky one, because it was like, aha, another another square, x2−1. So you can keep going. But then the question is, you know, do you circle x2+1 or not? Is it irreducible? And the question depends on the setting. Like, it depends on if we're working over the reals, or if we're allowing complex numbers, because if we allow complex numbers, then we can suddenly say that x2+1 is (x+i)(x−i). But if not, then, you know, maybe we're done.

So feasibly, it's like, well, we don't want to have to come up with a fundamental theorem for every single set of polynomials that exist. That's not very efficient. So we kind of generalize this idea of the integers into something called a commutative ring. And we generalize this idea of primes into irreducible elements. AndI think that living that abstraction is what I've spent most of my mathematical career looking at, like how things decompose, but I think tracing it back down to, you know, what we're really trying to do here is to come up with really nice notions that generalize the fundamental theorem of arithmetic. So this is probably why it's my favorite theorem, because I feel like if you keep going down to just the bare bones of what it is we're trying to do, the best example I think is there in that theorem, and also the best things are the things you can talk about.

KK: Yeah, and it's kind of like the first real theorem you learn.

RE: Yeah.

KK: Because, you know, I mean, you start learning mathematics in elementary school, and you learn how to add and subtract, but there aren’t really — well, there are theorems there, or definitions, maybe, but this one, you learn how to do it somewhere like fifth grade, maybe?

RE: Yeah, you’re really young, but I don't know that it was given the name.

EL: Yeah, I didn’t know the name of it, I think until I was probably in grad school, maybe college?

RE: Yeah.

EL: Still, but you learn it. Maybe you don't learn it as a theorem.

KK: Yeah. You learn an algorithm, right? Essentially.

EL: Yeah.

KK: Yeah. How do you do it? I mean, what do they teach you to do? Like, start dividing by primes, maybe?

RE: I guess I felt like at that point, well, this was when I was still just using a lot of memorized facts. And that was math to me. And so I guess I had my list of things that I thought were prime. And then maybe if they threw in a big number, I'd have to think about it. Like if they threw in like a 37. It's like, Oh, wait, what's happening? But 2, 11, 13? You know, we were pretty good to know. But yeah, I think I had no idea that it was a theorem. I do remember learning it, though. And so my favorite things are when I'm learning something, especially in a more advanced mathematical setting, and it takes me back to a very young me who just didn't know that there was a lot more to this when I was first exposed to it.

KK: Mm hmm. Yep. And hopefully didn't fall to the Grothendieck trap of thinking that 57 was prime, right?

RE: No. So basically, I've done a lot of work looking at unique factorization. And so because I work with zero divisors, which I don't know that I need to get into the nuts and bolts, but I thought a lot about what makes a unique factorization domain tick. Because I think a lot about settings where we don't have those nice properties. And so a unique factorization domain is the is the exact generalization of the fundamental theorem of arithmetic. So the fundamental theorem of arithmetic, you know, we've got a setting, the integers, where everything factors uniquely into primes, and in a unique factorization domain, it’s commutative ring, which is a generalization of the integers and the nice properties that they have. And it's a commutative ring, where everything factors uniquely into atoms, so we're generalizing primes now into atoms. And so there are some really nice results related to polynomial rings, where if you have a ring that has unique factorization, then the polynomial ring extension also has that same property, and vice versa, too. And so in the world that I live in, there are a lot of times where these factorization properties don't extend. And so I spend time thinking about what can we do to try to make them extend. So yeah, I think a lot about factorization theory and commutative ring theory. And so a lot of this is sort of based on this very gold-star standard of a factorization setting, which is a unique factorization domain. It's the nicest place that you can live, where factorization is just really well-behaved. You don't have to distinguish between primes and irreducibles, it's just a beautiful place to be. And so I call this a utopia and it's really mimicking or generalizing the fundamental theorem of arithmetic and the results there.

KK: So another thing we like to do on our podcast is ask our guests to pair their theorem with something. So we hear you might have multiple pairings. You’re only obligated for one.

RE: Okay, so originally, my first thought with a pairing was was alcohol, and I don't really drink that much, but I do love mead. So there is a meadery here. Okay, so mead is like it's like when they fermented grapes to make the wine, they ferment honey. So it's sweeter. And so there's a meadery here in Columbus called Brothers Drake. And I believe that they have a cousin, or a brother? You know, another brother meadery that's in California. But that's really broad. I'm not sure which part in California, but they have an apple pie mead. And it's my happy place when I do you know, partake in a little bit of something. So, I think the apple pie mead, and just any mead in general, if you would like to try it, especially when we get into history and stuff. I feel like this is like a very historical drink. Yeah.

EL: Yeah. Like, I don't know, I think of, like, dank castles and that kind of thing. Probably a lot of like, Disney fantasy, you know, people coming from battles and drinking their mead or something.

KK: Right.

RE: Yeah. I think a lot about Thor because I just am a big Marvel Universe person. And so I feel like Thor and Loki would just be having some mead, you know, catching up. But okay, so I was trying to think of what else would go with my theorem that wasn't alcohol. And so because I feel like the fundamental theorem of arithmetic is a very classic thing. So I would just like to pair my theorem with two things. One, sleep. So this is like a shameless plug for everyone to attempt to get eight hours of sleep. This is something that I tried really hard to do last year. And it was really crazy how much I fought against. Like, “I don't have time for this because XYZ,” but it took me maybe a whole semester, and I finally am now sleeping eight hours a night no matter what. And so this is a nice pairing with math, is sleep because I think that it's really good to do math when you're rested and your head is clear. So that's one. And then the second would just be like nice long walks. I love nature. I love cycling and strength training, but you just can't beat a good walk. And so for those who are able and mobile, I just think taking the time to go on quick walks during the day, even if it's just, like, 10 minutes, in between a meeting or something it’s a really great thing to do. So those are, I guess, like, my classic pairings. So what I say is apple pie mead, eight hours of sleep, and a walk.

EL: A nice long walk. This sounds like a great day.

KK: This is amazing. And you’ve got your bike there in the background.

RE: Yeah. Oh, my gosh, we're getting to know each other.

KK: Yeah. Well, when I was a postdoc, I was a very serious cyclist. I mean, I spent a lot of time, it was good therapy for me to get on the bike. Like if I was stuck on my math, I went for a ride.

RE: Yeah.

KK: But I lived in Chicago at the time, so you can't really ride in the winter.

RE: Yeah. That's an interesting city. I guess what like, did you go maybe to like suburbs and ride out there?

KK: Yeah, I was in Evanston. So I was at Northwestern. So that was good, because you could just head north, and then you're out in the country pretty quick. But you know, I had a group I rode with and all that, but just very good therapy all the way around. And then I had a kid and moved to Detroit. And those two things will just kill your cycling.

RE: What about what about you, Evelyn? Do you have — because you were talking about birds in the beginning, and I see that your background is very scenic.

EL: Yes, this is a cold day at Bryce Canyon National Park down in southern Utah. It's extremely hot in all of Utah right now. So this was kind of nostalgic, like, bringing some cold weather into it. But yeah, I love biking and taking walks and stuff. I'm really lucky in the neighborhood I live in. Basically, if you go north from my house, which is also uphill, you end up in less than a mile going into this extensive trail network that can get you all over the place if you're willing to go for a long walk. And it's like, I live less than two miles away from the state capitol building in downtown Salt Lake, but the fact that you can get up into nature so quickly is amazing.

KK: Well, it's right up against it.

EL: We’re built into the foothills here. And it's great. So yeah, I love taking walks in nature and I I've never tried that kind of mead, but there's actually a local like fruit wine place here that has a whole mead series in addition to fruit wines. And it's really cool because they have some that are sweeter and then some that are less, where they fermented, like, all of the sugar and it's amazing some of these, they almost taste like a Chardonnay or something. Yeah. Because you think of mead, honey wine, it's going to be super goopy and sweet, and depending on how much you ferment it and stuff, it actually has all sorts of different flavors so yeah, it's a cool place. I think they've got some like apple and honey cider mix things so I should check those out. Yeah,

RE: You definitely should, but I do agree that there are so many different like flavor profiles. The meadery here, you can go and do samples and they have like, music nights, pre-COVID, I think they're starting to resume this, and, like, empanada nights, which was a very personal weakness of mine. But yeah, I love — like, some of it just is too strong for me, right? Because I was I was leaning towards mead because I was like, okay, I don't know if I'm a hard alcohol drinker. But it's not all just sweet. It's not all just like juice with alcohol. I really like it. And so the last time that we tried was called Purple Rain. And I believe that the guy said that he, and it depends also on the barrel on which you're aging the the the mead, but he did something, like it was like using some sort of like blackberry something, and they accidentally like made too much and it was like overflowing the barrel when they came to check in on it. And so he called it Purple Rain. So I thought that was was pretty cool. Back to the cycling comment, my bike was actually stolen out of my garage at the beginning of COVID, and I was so upset. So at the beginning of COVID and work from home, when I realized that we'd actually be here for a while, I started nesting. I did a lot of things to my office space. So you see this black peel-and-stick wallpaper that I put up and actually turned out really nice.

EL: And a beautiful — I was hoping to see the rest of that picture that you’re tilting up now because yeah, I was thinking that looks really cool.

RE: Yeah, I got art. So this is um, I Gosh, I want to say it's just an Israeli painter named Itay Magen, and I just really love. It does a lot of really vital prints, colorful art. And so this came as a canvas. And then I realized when you buy canvas prints, you actually have to go get them mounted, which can be a little pricey. And then I ordered this Blackboard that you see, but the point that relates to the bike is that I also put together some shells, there's a landing gear, so you're kind of blinded, but I was painting and staining the shelves in my garage, and I left it open for a little bit, just to let the air sort of, you know, vent because of the paint fumes, and my bike was stolen. I got a little too trusting living downtown, you know, moving here from Iowa, you know, I kind of learned my lesson in the big city, I guess. So I got a new one. And I'm still getting to know this one a little bit better. But I took it out for the first time last week, and I have new clipped-in pedals. I got a different pedal than I had last time. I've been practicing just getting clicked in and out just at home because it's a little bit more challenging. So yeah, but I love doing outdoor things. And I think it's really nice to get fresh air, just for balance. And then also it does help, I think, with math. There's a tendency, I think, to try to double down, like, “no, I got to get this result. And then like sleep can happen or life can happen.” But I found that, you know, actually taking the breaks is really helpful.

EL: Yeah, the number of times where, you know, you're stuck on something, and then you actually let yourself sleep and wake up and realize, Oh, I can approach this in some different way — I wish I learned better from that rather than continuing to torture myself sometimes.

KK: Yeah, yeah. All right. Well, this has been great fun. So where can our listeners find you online?

RE: Yeah, so you can find me online on Twitter. My handle is @RanthonyEdmonds, let's see, with regards to the OSU Black math history project I mentioned, we will have a website, probably by September. But in the meantime, you can contact us at [email protected] if you're interested in telling a story related to your time, you know, at Ohio State or affiliated, or just you just want to tell a math story. You know, that's the place to go. And then I think lastly, I'll start posting a lot soon on Twitter about another project that I'm working on just by the end of the summer related to redistricting and communities of interest, and sort of synthesizing community input so that when the redistricting process happens at the end of this year, we are taking into account communities of interest, which is this sort of traditional redistricting principle that says that communities with shared interests should be kept together in the mapping process. But what are those communities look like? Where are their boundaries? What are their key characteristics? We're working with the MGGG Redistricting Lab along with Ohio Organizing Collaborative and their independent citizens redistricting commission to really collect a lot of public input related to communities of interest. And so I'm focused on what's happening here in Ohio. But this is an effort happening over 10 states this summer, as we prepare for redistricting in the fall and all that's going to happen with the release of the census data. So I guess just stay tuned. Some good places that aren't my Twitter profile will be Common Cause, Ballotpedia, and of course, here in Ohio, the let's see, don't let me lie, ohredistrict.org. And so this is where you can find the Ohio citizens redistricting commission information. And so this is an independent commission that is sort of focusing on modeling good redistricting practices. And we're working closely with them this summer. But like I said, this is definitely happening in over 10 states. And so I'll start posting about this soon. But in terms of not me, specifically, just, you know, look some things up about what's happening in redistricting. Try to get involved and make your voice heard, because it affects all of us and it's really important, but I don't want to go on a separate tangent. This is supposed to be like a closing plug. So follow me on Twitter @RanthonyEdmonds. Email me if you're interested in telling your story related to Black math history at [email protected]. And then just you know, ohredistrict.org and Common Cause are really great resources for learning more about redistricting that's happening this year.

EL: Excellent. That is a fantastic set of resources. Thanks so much for joining us. This was a lot of fun.

KK: It’s really great.

RE: Thanks for having me.

[outro]

On this episode of My Favorite Theorem, we had the pleasure of talking to Ranthony Edmonds from The Ohio State University about the fundamental theorem of arithmetic. Here are some links you might enjoy after you listen to the episode:
Edmonds' website and Twitter account
An interview with NPR about her Hidden Figures-based course about mathematics and society
Math Alliance, a program that supports mentorship for early-career mathematicians from underrepresented groups
Ohio History Connection and the National Afro-American Museum and Cultural Center
An article by Evelyn about why 1 isn't a prime number, which mentions the distinction between prime and irreducible
The Metric Geometry and Gerrymandering Group (MGGG)
Ohio Organizing Collaborative
Ohio Citizens Redistricting Commission
Common Cause

Ballotpedia

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Evelyn Lamb, one of your hosts. I'm a freelance math and science writer in Salt Lake City, Utah, and this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. How are you doing, Evelyn?

EL: I’m doing okay. I was trying to think of something interesting to talk to you about at the beginning of this and life isn't very interesting, but in a good way. So that's good.

KK: Yeah. Yeah. You know, monotony is underrated, isn't it?

EL: Yeah.

KK: We just had our 29th wedding anniversary on Sunday.

EL: Congratulations.

KK: Thank you. That's our big news. We went out and sat at a restaurant for the first time in nearly a year. So we were very excited

EL: Excellent. I hope that, you know, this podcast is going to go on for decades and decades, and someone is going to be catching up on old episodes, you know, in 15 years and say, “Why do they keep talking about all these very boring things that they're doing for the first time in a year?” So that'll be great. You know, last time it was haircuts, this time it’s restaurants.

KK: That’s right. That's right. Yeah.

EL: But whether they are saying that or not, they will be very excited that we are talking today to Rekha Thomas, and Rehka, would you like to introduce yourself?

Rekha Thomas: Hi, thanks for having me on the show. So I am a professor of math at the University of Washington in Seattle. And I come originally from optimization. My PhD is actually in operations research. But I have worked in a math department ever since I graduated. And my work lies somewhere at the intersection of optimization, applied algebraic geometry and combinatorics. And I very much like problems that have an applied background, or a motivation. Not necessarily because I work in applied things, but because I very much like the problem to be motivated by something real. And in the last 10 years or so I've been working quite a bit in the mathematics behind questions that come from computer vision. And this has been especially fun, but optimization in general does have that applied side to it. So I like problems like that.

KK: Right. You know, I talked to my colleagues over an industrial and systems engineering, where they do a lot of that work all the time. And I think, “You guys are basically mathematicians. Why don't you come over here?” Heavy overlap there. Yeah.

EL: Well, and I just want to say, how we came to invite Rehka onto the podcast.

KK: Yes.

EL: I think it's pretty cool. So one of her students sent in an email to our little submission form on Kevin's website, saying that he thought that his linear algebra teacher was great and would be a great guest for My Favorite Theorem. And this is especially remarkable because this is in the semester, we're all teaching has been online and stuff. And I just think to be connecting with students enough when you're doing Zoom teaching, that they actually reach out to a podcast to get you as a guest on a podcast, just you must have really made an impression. So I think that's very cool.

RT: Thank you. I was very honored by that piece of news. I thought, “Okay. We finally made contact.” That was great. But especially for an undergraduate to take that initiative to write to you was very touching.

EL: Yeah. So you know, not to build things up too much, but I'm sure that this is going to be a great episode.

KK: Yeah.

EL: So with that introduction done, what is your favorite theorem?

RT: Yeah, so I announced my favorite theorem in that class and told them about your podcast. So when you invited me, I thought, “Oops, is that really my favorite theorem? Is that what I really want to speak about?” And I think yes, so I'm going to stick with that theorem. And this is a theorem from linear algebra. It was a linear algebra class. And I think it's one that's not very well known to pure mathematicians. So it goes by the name of Eckart-Young theorem or the Eckart-Young-Mirsky theorem, but apparently the history is more complicated. I've been reading about this so I can tell you about it if we get to that. So the theorem is basically says the following. So if you have a matrix, say a real matrix of size N by N, and we know its singular value decomposition [SVD] — which is a very special decomposition of the matrix as a sum of rank-one matrices — then the closest rank-K matrix to the given matrix is just made up of summing the first K rank-ones in that decomposition. So if I wanted the closest rank-one matrix, I just take the first rank-one in the singular value decomposition of A. If I wanted the closest rank-two matrix, I take the first plus the second rank-one matrices in the singular value decomposition, and so on. So there is this neatly arranged set of rank-one matrices that add up to give you A, and if you truncate that sum anywhere at the K-th spot, you get precisely the closest rank-K matrix to the given matrix. This is the theorem.

KK: Closest in what sense? What’s the metric?

RT: Yeah, so that's a very good question. So closest in either Frobenius norm or spectral norm. So these are standard norms on matrices. Frobenius norm is just, you think of your matrix as a long vector, where you just take every row, let's say, and concatenate it to make make a long vector, and then take the usual Euclidean norm. Okay. So the sum of Ai,j squared, square root. [Maybe a little easier to read in math notation: (Σ Ai,j2 )1/2.] Spectral norm is the largest singular value, so it is the biggest stretch that the matrix can make on a unit vector.

KK: Okay.

RT: So in either norm this works.

KK: That’s pretty remarkable.

EL: Yeah. So just doing a little bit of mathematical free association, is this, can this in any way be thought of as like a Taylor's Theorem, like, you're kind of saying like, Okay, if you want your approximation to be this good, you go out this far, if you want it to be this good, you go out this far, maybe that’s — I don't I don't know if that's a good analogy.

RT: No, I think it is. So I think there are many constructs in mathematics like this, right? Like Fourier series is another one, where you have a break down into pieces, and then you take whichever part you want. Taylor series is similar. The SVD is special in the sense that the breakdown, this breakdown into rank-one matrices is actually tailored to the matrix itself. Like, for example, as opposed to, say, in Fourier series, where the the basic functions that we are trying to write the function as a combination of, they are fixed, right? It is always cos(θ) + i sin(θ), or cos(Nθ) + i sin(Nθ). So it's not particularly tailored to the function. It's just a fixed set of bases, and you're trying to write any function as a combination of those basis functions. But in the SVD, the basis that you construct, the factorization that you get, is tailored to the actual data that's sitting inside the matrix. So it's very, very nice and is incredibly powerful. So it's similar, and yet, I think, slightly different.

KK: Right. In other words, yeah, that's a good explanation. Because, as you say, with Taylor series, you're choosing a basis for a subspace of the space of all smooth functions or whatever. Whereas here, you're taking a particular matrix. Does it matter what — so if you change the basis of your vector space, do you get a different SVD?

RT: No. So what the SVD is — there is a very nice geometric way to think of the SVD, which may answer that question better. So the SVD is sort of a reflection of how the matrix works as an operator. So what it's telling you is if you take the unit sphere in the domain, so let's say we have an N by M matrix, so the domain is Rn, take the unit sphere, under the map A, the sphere goes to an ellipsoid.

KK: Right.

RT: In general, a hyperellipsoid, right? And this ellipsoid has semi-axes. The singular values are the lengths of those semi-axes. Oh, so it tells you, yeah, the length of the semi axes, and the unit vectors in the directions of those semi-axes are one set of basis vectors. So that's one set of singular vectors, and the pre-images in the domain are the other set of singular vectors. So if you're willing to change basis to the special bases, one in the domain, one in the codomain, then your matrix essentially behaves just as diagonal matrix. It just scales coordinates by these singular values. So it's a generalization of diagonalization in some ways, but one that works for any matrix. You don't need any special condition. So they're very canonical bases that are tailored to the action of the matrix.

KK: I’m learning a lot today. So I have to say, so before we started recording, I was mentioning that I think students need to take more linear algebra. But the truth is, I need to take more linear algebra.

RT: I think we all do.

EL: Yeah, I mean, I am really realizing how long it's been since I thought seriously about linear algebra. So this is fun. And it goes really well with our episode from a few months ago. I believe that was Tai-Danae Bradley who chose a singular value decomposition as her favorite theorem. So we've got, you know, a little chaser that you can have after this episode, if you want to catch up on that. So Rekha, you said that there was a bit of a complicated history with this theorem. So do you want to talk a little bit about that?

RT: Yeah, sure, I'd be happy to. So I always knew this theorem as the Eckart-Young theorem. I only recently learned that it had Mirsky attached to it. But then in trying to prepare for this podcast, I started looking at some of the history of the singular value decomposition. And there's a very nice SIAM review article by Stewart, written in 1993, about the history of the singular value decomposition. And according to him, singular value decomposition should be attributed, or the first version of it, is due to Beltrami and Jordan from 1873 — so the Eckart Yang theorem is from 1936, so almost 60 years before — and they were looking at more special cases, they were looking at square real matrices that are non singular, perhaps. So they, you know, people were interested in that special case. Then there were several other people, like Sylvester walked on it. Schmidt from Gram-Schmidt, he worked on it, Hermann Weyl worked on it. So from 1873 to 1912, this went on. And this article says that this approximation theorem that I mentioned, the Eckart-Young theorem, is really due to Schmidt from Gram-Schmidt fame. And he was interested not so much in matrices, but he was studying integral equations, where you have both symmetric and asymmetric kernels, non-symmetric kernels. And he wrote down this approximation theorem. So really, Stuart claims that that this theorem should actually be attributed to Schmidt. And then in 1936, Eckart and Young actually wrote down the SVD for general rectangular matrices. So that is in that paper, for sure. And they seem to have rediscovered this approximation theorem. So, that is my understanding of the history of how it is, but I did not know this till two days ago. And I'm not really a bonafide historian in any way. So, but this is what I've understood. It's an interesting story.

EL: Yeah, that sounds like an interesting article. I mean, I guess in the history of math, there are an uncountable number of places where unravelling back to where the idea first appeared is more complicated than you think.

RT: Right.

KK: And also, this sort of gets at your interests more generally, how you like things to sort of come from an actual application. Well, if this really came from integral equations, right, that's really an application.

RT: Absolutely.

KK: So it's working on lots of levels for you.

RT: That’s right. So he, of course, did this in infinite-dimensional vector spaces. And approximation will allow you to approximate an operator as opposed to a matrix, right? And apparently, that really elevated this whole theory from just a theoretical tool to something that's actually widely used. It became much more of a practical tool. And I guess, in modern day, the SVD, and versions of the SVD in exists in all kinds of mathematical sciences. So in signal processing, and fluid dynamics, all sorts of places. So it's, in some sense, one of our biggest exports from the math world, and yet we don't quite teach it normally, to math people. So yeah.

KK: Right. So was this like a love at first sight thereom? Was this the sort of thing that came up a lot in your work, and that's why you're now so enamored?

RT: So I did learn of at first in the process of writing a paper. I did not know about this theorem before, maybe about 10 years ago. But I think this theorem sort of perfectly fits me, which is why I love this theorem. So for different reasons, right? So first of all, it's an optimization problem, it's about minimizing distance from a matrix to the set of rank-whatever matrices. So it's an optimization problem. The space that you're trying to minimize to, which is the space of rank at most k matrices, that is an algebraic variety. So it can be written as the set of solutions to polynomial equations. So there's the applied algebraic geometry side, or at least the algebraic geometry side. And it's not a very simple variety. It's actually a complicated variety. So it's an interesting one. And lastly, this problem is sort of a prototypical problem in many, many applications. So a lot of statistical estimation problems are of this flavor. You have a model, which, let's say is our rank-K variety, so the rank being some measure of complexity. And then you have an observation that you have in the field with instruments, and it tends to be noisy, so it's not on the model. So that's your observed matrix. And now you're trying to find the maximum likelihood estimate, or the closest true object, that fits the observation. So this is a very standard problem that comes up in many, many applications. So in some sense, I feel it really lives at this intersection of optimization, algebraic geometry and applications, which is sort of what I do.

KK: That’s you.

RT: Yeah. So that's one reason that I think this theorem is so cool. And the other thing is, I think it's a very rare instance of an optimization problem where the object, the observed matrix, knows the answer in its DNA. It doesn't need to know the the landscape that it's trying to get to, which is your space of matrices of rank at most K. it doesn't need to know anything about that variety. Just inside its own DNA, it has this SVD. And from the SVD, it knows exactly where to go. So this is completely unusual in an optimization problem, right? Like even if you're minimizing, say, a univariate function over the interval [0,1], I really need to know the interval [0,1], to figure out what the minimum value is. The function doesn't know it. But this is, I think, kind of a gem in that sense. You don't need to know the constraint set. And then lastly, it appears all over the place in applications. So in things like image compression, you know, the Netflix problem is sort of a version of this, distance realization, you know, things that would come up in areas like molecular modeling, or protein folding, and so on. It's all — many of these problems can be thought of as low rank approximation to a given matrix.

KK: Very cool. Yeah, I'm now I'm thinking about that variety. It's not a Grassmannian. But it's sort of like, is it stratified by Grassmannians? Let's not go down this path.

RT: It’s just — yeah, it's a set of solutions to the equations that you get by setting all the rank, whatever minus to zero.

KK: Right.

RT: So the matrix cannot be rank more than K. So you set all the K+1 by K+1-minus to zero.

KK: That’s complicated.

RT: Yeah, yeah. It's a complicated variety with singularities and so on. Right.

KK: So another thing we like to do on this podcast is ask our guests to pair their theorem with something. So what pairs well with this theorem?

RT: So I thought about this. This, to me was a very interesting challenge that you posed. So one thing that I always think about when I teach this theorem or teach the SVD in general, is there’s sort of a layer cake analogy. Okay, so I have always drawn this layer cake picture in my class. But when I started thinking about what I should tell you on the podcast, I thought, “Okay, it's not quite a layer cake, like you would buy in a store.” But it's sort of like a layer cake. So there is a layer cake analogy going on here. And that is simply we can think of a matrix. So if you have, say, an M by N matrix, let's start by just thinking of it as a rectangular chessboard lying on the floor. And then every entry of the matrix is creating, let's say, a building on each square. So you have, you know, the buildings have different heights, depending on the entry there. And then in that sense, what we're doing is, if you think of what a rank-one matrix is in picture that I'm trying to draw, then what it is, would be this sort of city block with skyscrapers that we've constructed, where when you look from East Avenue or West Avenue, you see one skyline. And then you see, like, various up and down versions of the same skyline as you look across. And then similarly, if you stand on North Avenue or South Avenue, you see one skyline. And then you see these up and down versions of that skyline as you go in the other direction. So that's a rank-one matrix. And then the whole matrix is built of these puzzle pieces, if you like. They’re all rank-one matrices. And what we're doing is sort of, take different puzzle pieces, you know, the first puzzle piece captures the most amount of energy in the matrix, then the first and second, the next amount, and so on. So that's sort of one geometric thing. And so thinking of a pairing, that’s just a geometric thing, not not exactly a pairing. But in my mind, another way to think about the whole thing is you could think of your matrices as, say, living in a universe. We have this M by N, universe. And then each of these landscapes, these, you know, matrices of rank, at most K, they form landscapes inside this universe. They're nested one inside the other. And you could almost think of your matrix as sort of a flying object. And if it needs to make an emergency landing on one of these landscapes, it knows exactly where to land. It doesn't need any, you know, radio control from the ground, right? There's no air traffic control on the ground who needs to tell this matrix where to land. So that's sort of my geometric vision of what is happening. I love geometry. So I always try to make pictures like this. But the closest physical phenomenon that I was thinking that maybe we could match with this is with the way migratory birds work, right? Like these migratory birds, they have sort of an inherent genetic compass in their head that tells you where they should land. So Florida being one of the biggest places.

KK: Yes.

RT: And that's exactly before they fly over the Gulf, right, where there's a long stretch of water so they know exactly where the end of the land is, or where the beginning of the land is when they fly in the other direction. So I think that's some amount of this sort of DNA information that's in their head. So there’s either genetic information — of course, they also use celestial signals like the sun and the stars and so on. But yeah, so that that, that to me was the best pairing I could come up with, just thinking of matrices as having this inbuilt computer inside them.

EL: I love that!

KK: I do too. As an avid birdwatcher, I'm really into this pairing a lot. This is really nice. Yeah, and luckily for the matrices, it doesn't get messed up. You know, I mean, I get various — I subscribe to Audubon and things like that. And I just read an article recently about how light pollution is really a problem for migratory birds. Especially, you know, they fly over New York City. You think they're not there, but they are, and you can catch it on radar data and all of that. And it's really become a problem for them. And especially with climate change, they're getting messed up on all these things — not to get off into birdwatching, but this is a really excellent pairing. I love it.

EL: Yeah, well, I know Kevin is quite the birdwatcher. I have only recently been getting a little more into it. And so I will think about flying matrices the next time I go and look at some birds. I recently discovered a new birdwatching place not too far away from where we live. And there have been some great blue herons nesting there. They're probably leaving soon, but they were there for the spring. And so that was cool as a very beginning birdwatcher to suddenly have your first serious like, “I'm going to watch birds at this place” have these nesting great blue herons at them. It really raises the bar for subsequent birding outings.

KK: They’re impressive birds, too. I mean, I've seen them. We have lots of them here, of course, I mean, they're walking around campus sometimes. But yeah, they'll catch up like a big fish swallow it whole. And it's really pretty remarkable. So yeah. All right. Well, Rekha, so we always like to give our guests a chance to plug anything they're working on. Where can we find you online?

RT: Oh, so I have a basically just by webpage. I'm not a social media person at all.

KK: Good for you.

RT: That’s basically where you can find me.

KK: You're part of a wonderful department. I've visited there several times.

RT: Okay.

KK: Great department, great city. And we'll be sure to like the your homepage. Anyway, thanks for joining us. I learned a lot today. This has been great.

EL: Yeah. And can I just, I don't do a little bit of tooting our own horn was saying that I love this podcast because we get to do things like now every time I decompose these matrices I’m going to think about migratory birds. And, you know, it's just, like, building all these little connections. I love it. Thanks for joining us.

RT: Thank you.

[outro]

On this episode of the podcast, we were excited to talk to Rekha Thomas, a mathematician at the University of Washington, about the Eckart-Young-Mirsky theorem from linear algebra. Here are some links you might find interesting after you listen to the show:
Thomas's website
Our episode with Tai-Danae Bradley, whose favorite theorem is related to Thomas's
Stewart's article about the history of singular value decomposition

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast with no quiz at the end. I'm Kevin Knutson, professor of mathematics at the University of Florida. I am joined today by your fabulous and glorious other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah. And I assume the “fabulous and glorious” means my new haircut, which I actually got professionally cut for the first time in quite a while recently.

KK: Good for you.

EL: I had been playing around with, you know, scissors — not official hair cutting scissors, just you know, the scissors that are lying in the kitchen drawer — and, like, my clippers and stuff for a while, but I went in and I look very sleek and chic right now.

KK: You do. You look very sleek. I’m getting a haircut tomorrow, I look less sleek.

EL: Not too bad, though.

KK: You know, I still have the plague beard though. I don't know, I can't decide what to do about that. And my hair has gotten longer, but you know, thinning, and I don't know what to do anymore.

EL: My husband actually decided to try for the ponytail. So he had been very excited about, like, two weeks after the second shot going and like getting a real haircut for the first time in a while. And then by that time, it was long enough that it's almost ponytailable. So he's like, “I've never had a ponytail. I think I want to try this.” So so now we're like at opposite, you know, going in opposite directions.

KK: I have a ponytail and my wedding pictures and then that’s the last time.

EL: Okay.

KK: Anyway, today we are pleased to welcome our good friend Liz Munch. Liz, you want introduce yourself?

Liz Munch: Hi. Yep, so I'm Liz Munch. So I am an assistant professor at Michigan State University. I'm in the departments of Computational Mathematics, Science, and Engineering, which is a mouthful, but we just say CMSE. And I'm also in the Department of Mathematics.

EL: All right, and what is your recent hair story?

LM: It’s the kind of, like, what can I do rolling out of the shower to like, make it do anything? And it's kind of like, wherever it lands. I did get a haircut, though, recently. It was very exciting. I had — it was getting so long that it was going out of control. And usually what happens is I I go get a haircut and I do like extreme versions. I'm like, “I hate my hair!” And then I chop it off to my ears. And then I grow it out again. And then “I hate my hair!” and I chop it off to my ears. This time, she convinced me not to chop it off all the way, but I'm due for one. So we're gonna work on getting that one soon.

KK: Looks good. Looks good. Yeah. So CMSE, that's interesting. You have this joint appointment. How's that? Yeah.

LM: I like it. It's different. So at least for me. So I do very interdisciplinary math, very applied math, but not your usual applied math. And so for me, it's a it's a really nice setting to be in. So CMSE is — so the faculty in CMSE are all sorts of different backgrounds. So there are mathematicians, but there are also statisticians and biostatisticians and biologists and plant biologists and geologists and physicists and engineers, and I'm sure I'm forgetting people. But it's basically rigged to be interdisciplinary. And so it's made a lot of fun, because you can find these interesting projects that are sort of in the intersection of complicated math and interesting stuff to do there with applied projects I would never have thought of before. So at least for me, that's been a really good fit for trying to do interesting applied math research explicitly outside of the usual academic silos.

KK: Yeah. Well, that's nice, because I think a lot of us in math departments often want to engage in these activities, but you know, the physics department’s in another building, and it's more effort, right? You know, it's true, there’s this idea of these collision spaces. So you know, if you have colleagues in all these different disciplines just right down the hall, it might lead to more interesting stuff. So that's a nice model. I wish we could do more of that. Anyway. So you have a favorite theorem, I hear.

LM: I do have a favorite theorem.

KK: You going to tell us what it is?

LM: I was going to talk about max flow min cut.

KK: Okay.

LM: Which is sort of my, I don't know — so the reason I like max flow min cut, is partially because, again, I do interdisciplinary applied mathematics, and I tend to fall into the theoretical computer science land a lot, just based on a lot of things we want to do. Because I work in TDA, topological data analysis research, and so a lot of things come down to here's a complicated thing I want to compute, and I’ve got to go figure out how to do it in a computer in a reasonable amount of time. So this is an example of that.

So max flow min cut. So here's the game. So you get yourself a directed graph. And this graph has a source and a sink. So you've got your S vertex source and your T vertex sink. And you essentially have capacities on each of the edges. So my directed edges, I now have an amount of stuff I can push across the edge. So I like to think about this with, like, water tube flows kind of thing, right? So I've got how much capacity each of the edges can take. And so the game is, try to see how much stuff you can push along from the source to the sink, basically playing nice with those capacities, so flowing from one side to the other. So if you're at any vertex, all of your inward flow amounts should be equal to all of your outward flow amounts, because you can't have anything hanging out at the vertex. And so your other restriction is your flow values — this is essentially like a second choice of weighting on all your edges — your flow values have to be less than your capacity values, right? So you can't flow more across an edge than the capacity allows.

KK: Sure.

LM: And so the trick is, what's the most amount of flow you can get across? I hand you one of these graph setup problems, what's the most flow you can get across these things? So the max flow min cut, as the title would suggest, is that you also need to know something about the minimum cut. So what's what's a minimum cut? So same starting input information. What you can do is you can try to divide your collection of vertices into two piles. And the rule is that you have to have your source in your sink in different piles. And the cost of whatever choice of piles has to do with, essentially the cost of the capacities that go from one pile to the other, right? So you sort of add up those values, there's some negatives if things are backwards, etc. But that's the cost of a cut.

KK: Okay.

LM: And so the game with that part of the problem is you want to make that as small as possible, right? Can I reorganize these vertices in such a way to make that cut cost low? And so the max flow min cut theorem says the capacity of the highest possible flow value is equal to the cut, the minimum possible cut value you could get. And so this is cool, because this means that you could answer your problem either by going and hunting for a maximum flow value or by hunting for a minimum cut value. And so it gives you two very different ways of looking at this problem to try to solve something. And these show up in all sorts of different application settings, right? So you can imagine, like, railway networks, where you're trying to move stuff around, you could try to do things with electric grids, and you can do things with water flow. There are lots of places where this thing would show up. And so having access to two different ways of looking at the problem is super useful.

EL: Yeah. So I have never thought about this before. So I'll just kind of — I’m not even sure if I can formulate the right question. But I'm trying to think, my mind immediately goes to extremes. So like, if you put everything except your source in one pile, or everything except your sink in one pile, is it obvious that usually that's not going to be good?

LM: I guess it kind of depends on the sort of setup you have, right? So I guess the simplest version would be like, okay, let's start with a graph that's just a path, right? And so in that case, I can stick a bunch of random numbers as capacities on my path, and so the maximum flow I can get across this path is going to be something like the minimum weight capacity that I've got, right?

KK: Yes.

LM: That minimum weight capacity is also going to be if I chop my path graph in half at that particular edge, that's as low as my cost for my path can go.

EL: Sorry, just to slow down a little bit for me. So that capacity is like, this one edge can only hold, you know, whatever amount of whatever stuff?

LM: Yeah.

EL: Okay. Great.

LM: This is kind of like the bottleneck in the system, right? Like I could have pushed, like, 30 gallons back here. But if this particular tube only allows for five gallons, it doesn't matter how much I can push in the backend, right? It's got to be, it's limited by this one, right?

EL: Yeah.

LM: And so then you basically take that, that path problem and scale it up. So then you've got all these other possible paths where you could push stuff through. You could imagine a vertex where you have, like, a capacity of 10 coming in, and maybe five edges coming out. And so your flow could be 10 in and then it got split up and there's all sorts of things. And so the way you prove this, so this is the Ford–Fulkerson algorithm from the 1950s, I think? It basically comes from looking for paths like that. So the whole problem comes down to basically that path example I was giving you. So you essentially rig up this modified version of your graph, and then look for paths from your start to your sink, where you try to see places where I could push some amount of flow across that line, across that path. And what you do is you essentially update some sort of flow that you're trying to build as big as possible. So if I've got a path where the bottleneck value was five, I now push value of five across each of the edges in that path. And then I update this other graph I can use to keep track of things. This is called a residual graph. Basically, then I update this flow, and then I go look for a new path. And then I update this flow, and then I look for a new path. And it all comes down to just finding these paths that have some sort of bottleneck in them and updating accordingly.

KK: That seems slow.

LM: Oh, yeah. Oh, yeah. This is not nice.

KK: So there must be better algorithms for doing this. But that’s the easiest to explain algorithm?

LM: Yeah, right, easiest to explain, and the one you get the proof off of. So that's the the Ford–Fulkerson one. And of course, I did not check this beforehand, but I believe the running time has something to do with the value of your maximum flow. And there are some caveats in here. So if you have integer-valued flows, so I'm only allowed to put integers on each of my edges, that process will terminate, which is a good sign, right? It will terminate at the best possible flow, which is a good thing, right? And worst-case scenario, you basically updated by one every time, and so the the order of the number of times you have to do that update procedure has to do with essentially the maximum flow value.

KK: Okay.

LM: So yeah, it's not pretty, but it works. This is another one of these examples where in practice, it tends to not be that bad. But of course, you can construct the nasty examples.

KK: Sure. Right. So yeah, the theory is bad, but the practice is good. So that's fine.

LM: Yeah.

KK: So do you use this much? Does it come up for you?

LM: I haven’t in a long time. The reason I was thinking of this was a while ago, I was teaching this summer program for bright high school kids, where we were doing a topology class for high school kids that basically have seen calculus. And so one of the reasons I like this was I was remembering I had rigged up this thing where I created a graph on the ground. And I put capacities on it involved in the — sort of like, I don't know, I think it was squares of paper or something like that — and messing around with getting them to actually manually push stuff around on this graph and start thinking about flows and how you can update information. Yeah, it hasn't come up in my research recently. But I think in general, just the messing around with problems where you can have multiple ways to find a solution, that's been really interesting lately, because trying to rig, trying to look at a problem from a different angle where you might have an easier ability to go solve something than something else, right? So like, if you told me go find the minimum cut in this particular graph, the knee jerk reaction is like, “Okay, let's test all possible ways of splitting the two piles!” which you really don't want to do, right? Bad idea. Never do that, right? And I don't have any good intuition for how — if you had to be that problem, I would start going at any sort of algorithm that would get me anywhere. But because we've got this duality idea, right, I can go back and do this pushing flows through a network thing that gives you access to a way to solve a problem that wasn't there before.

EL: Nice. And so do you remember, like really loving this theorem when you first saw it? Or is it something where your appreciation has grown over time?

LM: Ah, there's probably — now I;ve got to remember where I saw it. I'm going to guess I saw it in an algorithms class in my PhD. Yeah, I don't know. I don't know that I appreciated it at the time. I feel like a lot of the beginning of my math career was like, “Okay, this theorem exists, but I'm not sure why I care.” Right? It's taken me much longer to start getting to the like, “Oh, the point is the duality. The point is accessing a problem from different sorts of viewpoints.” And having the ability to sort of question the vantage that you're looking at something from because there might be a better way of doing something. And “better” here is now a subjective term, right? Like, mathematically, the theorem says, I don't care. But practically, right, there is a better and a worse way of doing something. And so you're going to end up having choices, as you go about doing math research, that are going to be better or worse in a practical sense,

KK: So it's clear listening to you that you work in an interdisciplinary environment, right? Because, you know, the mathematician loves the duality, but you know, practically, the engineer wants to find the most efficient way to do it. Yeah, very cool.

EL: So I wanted to come back to something you said a little while ago, and I think it just percolated through my brain. So you said that, like, if your your capacities on these are all integers, then the process terminates. And the fact that you needed that “if” actually just made me really nervous. So is it the case that if you don't have integers, and possibly even rational numbers or something like this? So like, maybe if you had like a bunch of irrational or transcendental values on these, that it might not terminate? And then you’d be sad.

LM: Yep, yep, yep. Yeah, then you'd be sad.

EL: Or maybe you would be sad. I don't know.

LM: Well, yeah. Okay. So I actually, I'm not sure of the answer if it's rational. I know that if you allow irrational values on the edges, you can definitely have examples that don't terminate. And basically what ends up happening — okay, so the way it is pushing the flow through your residual graph, there's a setup that allows you to essentially undo pushing through edges. So these residual graphs get this sort of backwards edge that's added to them. There’s the flow that you've already pushed through that you could undo and the flow that's available, which is the capacity minus the flow that you've done to this point. And so finding a path in this graph could push new stuff through, or essentially undo pushing things through and move it to some other path in your graph. So what ends up happening if you — again, if you create incredibly contrived examples — is that finding these paths in this graph can keep updating and then undoing the update. And you end up with this vicious cycle where you kind of like spin stuff around in circles, and you just never get there. And it's got all sorts of weird behavior, like not only it, like, doesn't decrease, it doesn't convert, like, it's, it's all kinds of nasty.

KK: Right? I would imagine with rational, though, you can do the standard trick of just scaling everything to be an integer. Like, that's probably okay.

LM: Oh, yeah. Because especially if you've got a finite starting graph, yeah. Yeah. Right.

EL: Oh, “if you've got a finite…” you're putting all these “ifs” that I wouldn’t even think you’d need!

LM: Let's have everybody be finite before I say something false.

EL: That’s fine with me.

LM: I am going to bet somebody thought about that.

EL: Oh, I'm sure. But, you know, since you started saying this with water pipes, I just immediately thought of, like, okay, a city water system. And I just know that that is finite. So yeah, in my mind, this is always going to be finite.

KK Right. And you'd sort of create one source by having it all come in your case, Evelyn from the Wasatch, right?

EL: Nice job.

KK: And then it all goes to the water treatment plant. That's it. That's your whole, you’ve got this crazy network.

EL: Or that eventually to the the Great — the Lessening — Salt Lake as it dries out.

KK: Yeah. All right. So also on this podcast is, we ask our guests to pair their theorem with something. And you mentioned what it was going to be ahead of time and I'm really intrigued. So let's look. What pairs well with max flow, min cut?

LM: Okay, so we're gonna pair this with the cross-strung harp, which I promise is going to make sense. So give me a moment. So just for some backstory, so my undergraduate, I was actually a harp performance major. That was my previous life. And I sort of backdoored my way into the math department.

EL: So I have to interrupt say that my sister is a harpist, was a harp performance major, is now working on a PhD in music theory, but teaches harp and has a harp studio and yeah, that’s such a cool coincidence.

LM: You probably might already know some of the stuff I'm about to say. So that's super exciting. Because, yeah, most people you talk to have no idea what I'm talking about. This is great. Yeah. So I spent the first part of my professional life, I was going to be a professional harpist. I got about halfway through and was like, “No, that's not for me.” I got into, started taking math classes. Ended up in a math PhD program, unbeknownst to me. And here we all are, right? Yeah, so the thing with harps is if you look at a regular harp, what you're actually looking at are, like, the white keys on the piano. So no chromatics, no sharps and flats. It's basically in one key, right? So there was an issue in, sometime around the 1800s, where they were trying to figure out how to make a chromatic scale possible on a harp. So prior to that, there was sort of a trick where you could put things like levers on each of the strings and they would shorten each string by a little bit. And so you could basically get, if you tuned your string in natural, you could get the sharp. If you tuned your string flat, you could get the natural, but that was it. So you could do the key of B — the key of F and the key of D, and then things got hard. And so in the 1800s, they were trying to figure out how to deal with this problem, and so there were two solutions that showed up at about the same time. So there was the cross-strung harp, which is not what you've seen before, and the double-strung harp which is what you've probably seen before. So if you go to an orchestra concert, somebody most likely has the cross-strong harp, or sorry, a double action harp, which is, you have two sets of essentially these twisty peg things that go on top of the string. And so now you engage a pedal, and the pedal, if you push it down once it engages one of these twisty bits, and if you engage it a second time, you get two of them. And so now you have three notes on every string. And so you tune it in such a way that now you've got flat, natural and sharp on every string.

EL: And there was much rejoicing.

LM: And there was much rejoicing, right. So this was invented by a watchmaker who was trying to solve this problem, because the mechanics of this are absolutely nuts. Like, I don't even know how the inside of my harp works. At the same time, so there was another option that was starting to show up, which was the the cross-strung harp, which was literally — you're going to have to Google this later — but it looks like two harps with two sets of strings that meet at an X in the middle sort of interlaced.

EL: Whoa!

LM: Yeah. I've seen these things. My brain goes fuzzy. Like, you can't focus on any of the strings, right? But the idea is that now you can move your fingers up or down to get sharps and flats. And so now you have access to all the strings. And it involves different techniques and things like that. And so there were these two companies. So it was Erard at the Erard company. So Erard was the watchmaker that invented the double action harp. And the Pleyel company had invented the cross-strung harp, and they were fighting about who was basically going to win the business.

KK: Harp domination.

LM: Yeah. There's some there's some catty stuff in the harp world, let me tell you. And so each of the companies, so there are actually two very famous pieces for harp that were commissioned by each company to prove that their harp was better. And so the Pleyel company, commissioned the Debussy dances, and the Erard company commissioned the Ravel Introduction and Allegro. And these two pieces are basically made to be — I'm not going to say easy, but like, manageable on your harp and not good on the other one, right? Right, so fast-forward 100 years, and the double-action clearly won out, right? So that's what everybody has now. But we still play both pieces. So if you go to college for harp performance, you're going to play both. And so the Ravel isn't, you know, again, it's not easy, but it is manageable, because you are playing it on the instrument it was written for. And then you get to the dances. And it is a combinatorial nightmare, because basically, what you're trying to do is figure out these fingerings, and how to play these different chromatic notes at the same time, when you don't have access to the other set of strings. You are now limited to trying to play the note on another string that you might have needed a half a second ago to do whatever.

So I promised that this was going to come back to something that made sense. So the whole point of this is that I was thinking of the cross-strung harp because it's one of these two solutions for the same problem. And not necessarily that one of them is better than the other, but in certain cases, there is definitely a priority. One of them really, truly did better.

EL: Yeah. That's so cool. And I have to ask one more question about the cross-strung harp. I don't want to totally derail this, but I'm going to slightly. So you know, when I watch my sister play like her right hand, you know, as on one side of the strings that are left hand is on the other. So with the cross-strung harp, then, your right hand and left hand — if you were playing, like on the diatonic, would have one would have to be up and one would have to be down, right? Because the strings go…

LM: Yeah. I will admit I've never played one.

KK: You’d switch them, right?

EL: Yeah, like if you move to flats, you move like this.

LM: You kind of want to be able to do both at once. So your technique would change entirely. I don't actually know how you would deal with this. If you want to Google even more things, there’s also a thing called a Welsh triple, and the triple — so what this is is now, this will make your brain hurt if you try to look at this thing — it’s now three rows of perpendicular vertical strings. So instead of being crossed through the middle, they’re, like, up, straight up. And so now you've got white notes on the outside, that are spaced wider than a normal harp, with sharps and flats down the middle. And so now you have to reach through the strings to try to grab.

EL: That seems bad.

LM: Oh, and I’ve played one of those too. And it's like your eyes just can't focus. Like I can't decide which layer of strings to be looking at. It was it was rough,

KK: But I'm sure that's mostly a function of having grown up playing the one that you play.

LM: That’s true.

KK: If you grew up playing the cross-strung or the West triple, that would just feel absolutely, totally natural.

EL: Yeah, but I guess you'd have to do something for visual markers, like so the harp has, is it the F and C strings are a different color to help you? Because on the piano, you can look at where the black keys are, and that tells you what notes you're playing. On the harp if they're all the same color, it's just a nightmare. And I guess on the Welsh triple, maybe you'd have to…

LM: I don’t remember what the coloring was on the Welsh triple. Yeah, on a modern harp, C’s are red and F’s are black. And then if you get older strung harps, they used to be switched, and that also makes my brain hurt.

EL: Oh no!

LM: It’s very much what you're used to.

EL: Yeah. Well, that's so cool. And what a funny coincidence that you’re harpist, just like my wonderful sister.

LM: Hi to your sister.

EL: Hi, Rachel!

KK: Do you still play much?

LM: Oh, not much anymore, unfortunately. I've moved my harp around to multiple cities and houses. Yeah, so I basically, I kind of put myself through grad school playing weddings. So that was nice. And then after, you know, kids and everything else, it's just kind of turned into less so.

KK: Well, you’ll come back.

LM: It might come back.

KK: Yeah. As someone who's now an empty-nester, I can tell you it does happen. So the other thing is, you know, my kid’s about to go off to grad school for music composition. So apparently just like Evelyn’s sister, he’s going to go be a theoretical musician, I guess. He's a percussionist, though, you know, and I can't understand how he plays drumset right. That’s, like, two arms and two legs, doing four different things. But harp sounds pretty bad too.

LM: Yeah, I thought I had to move around a bunch of stuff. They are so much worse. They got, yeah, that was that was always the trick. So when I was in college, the harpists saw got our own practice rooms, so we didn't have to move the harps around. Everybody else had to, like, fight for it. But the the percussionists also got their own practice rooms where they got, like, nobody wants to move around, you know, the xylophone.

KK: Yeah, although I've certainly moved Gus’s drum kit plenty of times.

LM: Oh, yeah. No, if you've gotten to the point that you've bought vehicles based on your instrument…

EL: Yep. I was going to say that's Rachel’s, I think Rachel has taken a harp to a car place to, like, test in the lot to see how easy it is to load.

LM: I have done that. I've purchased vehicles entirely because it could fit my harp in the back and I've confused — oh my goodness, I love showing up to car dealerships like this. It's my new favorite thing.

EL: Yeah. So before we end this episode, I just had — I didn't get a chance to say this earlier. But Max and Min can both be names, and I just feel like max flow and min cut just sound really snappy and sound like either a superhero duo or maybe, like, superhero antagonists or something. I’ve just got to put this idea out there for like, you know, a graphic novel about Max Flo and Min Cut.

LM: I want to read that book. Absolutely.

KK: Or Max Flow sounds like a news reporter from the ‘20s. You know, like His Girl Friday, right? You know, Cary Grant and Rosalind Russell or Max Flow and Min Cut in there. Yeah. So we also give our guests a chance to plug anything, or where can we find you on the line or …

LM: Oh, I definitely should have thought of that in advance. I don't know. Come see MSU. I like her. I like our new department. That's probably the first thing I should plug. I'm on Twitter more than I really should be. So you can always find me there. [Her handle: @elizabethmunch] That's — Yeah. I love Math Twitter. It's a happy place.

KK: Mostly.

LM: Yeah, I think that's mostly it. The other thing I guess I should plug is, so I'm very involved in the women in computational topology network. So if anybody is interested in anything in that general direction, so this is a group for not just women. So we're aiming for a broader community of gender minorities and a place for people to do math in a supportive environment because that, in my opinion, is the reason I am still here, and so I want to create that for other people.

EL: Cool, yeah.

KK: I can verify Liz is well-known for her no assholes rule.

LM: I yes — oh good, we can swear on this podcast.

KK: Sure. Why not?

LM: You just did. I do have a tendency to curse like a sailor, and so my husband before I started this was like, “Remember, Liz, you need to not swear on that.” I was like, “Okay.” I have a very strict “does not collaborate with assholes” rule. And it is gotten me very far in life and made for much better research.

KK: Life’s too short not to have that.

LM: Exactly. Yeah, exactly.

EL: So now are we going to go back and rerecord, like, the swear version of max flow min cut? Just kidding.

LM: Now that I'm allowed to curse for the whole thing?

KK: It’s pretty light swearing.

LM: You’ve got your R rating, don’t worry.

KK: No, no, no, R rating requires f bombs.

LM: See? We’re doing okay.

KK: So PG— so movie ratings for PG-13, you’re allowed one f-bomb. If there's more than one it gets an automatic R.

LM: Good to know. So we’ll hold off on the f-bombs, just in case this turns into a movie.

KK: That’s why you don't hear it as much in PG-13 movies because they they deliberately leave them out to get a PG-13 rating so the teenagers can come. But you can use any other swear you want. Anything. But just one f-bomb.

EL: All right. Well, with that important knowledge…

LM: We’ve gone on so many tangents at this point. I probably got math in here somewhere, right? Somebody did math.

EL: Yes. Our math, hair, harp, and movie ratings podcast can now conclude.

KK: That’s right.

EL: Great to have you, Liz.

LM: Thank you. Thank you both for having me. I really appreciate it.

KK: Take care.

[outro]

On this episode, we were happy to talk with Liz Munch, an applied mathematician at Michigan State University, about the max flow, min cut theorem. Here are some links you might enjoy after you listen to the episode.
Munch's website and Twitter account
The Women in Computational Topology Network
The Max-flow Min-cut theorem at Brilliant.org
The Ford-Fulkerson algorithm on Wikipedia
The cross-strung harp on Wikipedia
Harp.com's history of the harp

Evelyn Lamb: Hello and welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb. I am a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. How are you, Evelyn?

EL: I’m all right. It's been raining a little here, which is very good, because we are in a perhaps somewhat historic drought and every bit of moisture we can get is fantastic. Probably not very close to your experience in Florida right now.

KK: It’s been pretty dry. But yeah, it's not really an issue for us. I mean, it’s actually really lovely right now, and I'm looking forward to kayaking some this week.

EL: Oh, fun.

KK: And my son graduates college in two weeks. And yeah, all kinds of fun stuff on the horizon for us. So anyway, let's talk math, though.

EL: That is exciting. Yes. We are very happy today to have Érika Roldan joining us. So yes, Érika, would you like to introduce yourself?

Érika Roldán: Yeah, thank you. Thank you very much for the invitation. I'm super happy to be here. And, well, I'm a postdoc right now. I finished my PhD thesis in 2018. And then I started jumping here and there, from Mexico to the states and now Europe. I’m at Munich, Technical University of Munich, and École polytechnique fédérale de Lausanne in Switzerland is my co-host. And yeah, I have this fellowship, the Marie Curie fellowship, for 11 more months, and then jumping again. Yes.

EL: Yeah. Well, that's very exciting. And what is your field of research?

ÉR: Well, it's stochastic topology and topological and geometric data analysis. I think most of the time, most of the brain time, goes to that. But also, there is something that is related because it gives extremal examples — you will never see them typically when you're using kind of random processes — but these extremal examples allow is to contrast with random ones, so I also do extremal topological combinatorics a bit.

EL: Okay, and I also am familiar with some work that you've done in recreational mathematics, which I guess might have to do with this extremal combinatorics too. And so if I can self-promote and Érika-promote a little bit, a couple of the puzzles that Érika has worked on appeared in the mathematics-themed calendar that I put together a couple years ago, which is still available for purchase through the bookstore of the American Mathematical Society and which is not specific to a year, so you can still enjoy this calendar whatever year it is when you are listening to this episode. So anyway, yeah, you did a couple of fun puzzles. I don't know if you want to talk about any of those. I am actually blanking right now on there's one with like, polyomino things, right?

ÉR: Yes. So there are two. One of them actually has a very special place in my heart because it was my first paper, and I wrote it for the Gathering for Gardner, this meeting that is every two years in Atlanta. It is a wonderful meeting. It is the first mathematical community, research community, that I got in contact with. And yeah, I did a complete analysis and characterization of a type of puzzles with colored cubes, and you have to stack them, and you have to do a tower and have some interesting coloring properties. And the most famous one is called Instant Insanity, just to give you the name. The name has a little bit of a clue of how interesting it is to try to solve them by trial and error. And yeah, I guess I did some computations and everything to characterize all possible different kinds of puzzles like this. And, yes, that's one of the of the entries of the calendar, this this wonderful calendar. Thank you for sending it to me. I enjoy it very much. And yes, and the other one was about maximally many holes with polynomials.

EL: Yes, that’s right.

ÉR: And for sure, we're going to go back to polynomials. Because it's related with the story that I want to talk about today about my favorite theorem.

EL: Yes. And you've provided the perfect segue now. What is your favorite theorem?

ÉR: Yes. Well, first of all, I want to say that I was thinking, and I changed my mind different times during the past two weeks. And I decided that I wanted to talk today about my favorite theorem, choosing it in particular for my mother to be able to follow the podcast. So today's the 10th of May. And, yes, because of the corona crisis, I think a lot of people have not been able to travel, and I haven't seen my mother for more than, almost a year and a half. And this is a way of sending her all my love and appreciation. So my favorite theorem, okay, so one of the things that my mom used to do, or does still very, very well, is shuffling cards. So every time that we gather together with my grandma, and in Mexico, this is very common that you gather with your family very often. It doesn't matter what, you always want to be in a huge crowd, just eating and playing. And yeah, she's a very good shuffler of cards. And my favorite theorem is a theorem by Perci Diaconis, the theorem for today is by Persi Diaconis, where he proves, and actually there is like a set of different papers that have different ways of modeling this shuffling, the usual shuffling that we know, and they prove how many shuffles you need to do to be a sampling from a fairly uniform distribution from all possible ways of having 52 cards of a deck in a certain order.

KK: This is a very famous result, and it's a surprising number, you're gonna tell us the number, right?

ÉR: Actually, it depends. It depends on the distribution that we use. And then this depends on the algorithm. And but yes, it's a very famous theorem. And it's very well-appreciated by by mathemagicians. Persi Diaconis is one of the most well-known mathemagicians.

KK: I have to say, I'm not surprised this is your favorite theorems. I've been at various TDA [topological data analysis] conferences with you. And topologists like to play fast and loose with distributions, and you are always quizzing us about which distribution we're trying to use. It's always “Wait a minute, wait a minute, what's the distribution?”

EL: So,

KK: Go ahead, Evelyn.

EL: Yeah. Um, so for someone who has not thought about card shuffling a whole lot, what do you mean by different distributions?

ÉR: So let me withdraw the word “distributions” for the first part.

EL: Okay.

ÉR: So you want to play poker or any other game, and you want to shuffle the cards in such a way that you feel comfortable betting $100. So, I have the deck of cards. And let's say that I decide to shuffle like this: I take the uppermost card, I take it, and then I put it in one of the possible positions that is not on top, or could be on top actually. And I choose where to put it uniformly randomly. That means I select one of the possible 51 spots, 52 spots, and I put it there. And let's say that I do this five times. Question for both of you: Will you bet $100 with me in this poker game?

KK: Uh, no.

EL: Probably not.

KK: It doesn’t feel very well-shuffled.

ÉR: Exactly. And I think no one will do it. Like, it doesn't matter if you know or not probability. We have a sense of when things are shuffled, even if we haven't heard the word “probability.” And so, let me just tell you that actually, for doing this, if I do this 10 minutes, you will start getting more and more comfortable, right?

EL: Yeah.

ÉR: At some point, you can say, “Okay, let's play.” That's enough after an hour, right? And the solution, or roughly the number from starting from which you feel comfortable and theoretically can be proven that is the kind of right stopping time, is around 200 shuffles.

EL: So this is not a very efficient way to shuffle, in case that wasn't already clear to everyone. One card at a time. Not so good.

ÉR: Yeah, well, in a Mexican family it will be okay, because everyone will be talking everything. No one will get, just, bored. But perhaps if you are there just for playing, it's not the right way to go. Now, in general, I want to say here, the right rate. In general, if you have n cards, with this shuffling you have to wait nlogn time. So n times logn is the rate that with this kind of shuffle will give you a uniform distribution as a convergence, and when you start to feeling that yeah, it is uniformly sampled. Now let's compare these to the actual way that my mom shuffles the cards. And this is commonly referred to as a riffle shuffle. So you take the deck of cards, and then you kind of try to cut it in into two packs, half perhaps. And then what you do is you hold each one of the packets with each one of your hands. And then you make — I’m trying to generate a mental image that everyone knows — and then you start with your hands, like, trying to put one after the other one, alternating from the packets from one and the other one. And there are different things to do here to convince you that this is for sure a different way of shuffling than the first one. That is one thing, and the other thing is, so if I do it one time, perhaps you will not be very happy. But whenever we're playing with friends usually is what? Five times, six times.

EL: Yeah.

KK: It depends on how many times I drop them. You know, when you do it, you do it the down and then you do that little flip where you reverse the cards to get that little “shhhh” sound. It took me for—I was an adult before I could finally ever do that. But then a few cards always go flying, right?

ÉR: And this is one of the things that I love seeing my mom doing because I think she has some secrets that she doesn't tell us because she does it so well. Like amazingly well.

KK: My mom was good at that too, actually. That's interesting. Okay.

ÉR: We have these two different ways of shuffling. And we we kind of feel that, or we're comfortable with saying a riffle shuffle shuffles faster than the other way. And so now I want to use to use this shuffling perspective to go and shuffle other mathematical structures. Because what happens is that I ended up using this way of shuffling objects, or mathematical entities, in my research. And that was a very pleasant outcome of my PhD thesis. Yeah.

EL: So you're saying that like with these polyominoes with holes and stuff, there's some way to think about shuffling polyominoes, or do you shuffle entire configurations of them?

ÉR: Yes, exactly. Yes. It's super fun. So okay, let me start with Tetris.

EL: Okay.

ÉR: I think if we start with Tetris, everyone who's listening to us will go directly to the picture, the image of what is a polyomino.

EL: Oh, that's right. Yeah, we didn't actually say what a polyomino is. So in Tetris, you know that each picture is made up of four blocks. And so that's a tetromino is. So it's like a polyomino, I guess, is something that's made up of any number of these blocks.

ÉR: Yes, yes. Exactly. So. I think my, my favorite definition of a polyomino was the first mathematical definition given by Solomon Golomb when he was a PhD student. He gave a talk, and he said, well, a polyomino is a rook-wise, connected subset of squares of the infinite checkerboard.

EL: Okay.

KK: Rook-wise.

ÉR: So you can imagine you select a finite number of squares, and then you place a rook in any one of the squares, and you have to be able, with only rows and column moves, to go and visit any other square of the structure. So rook-wise connected.

EL: Yup.

ÉR: Yeah. And with Tetris, what happens is that we — let's now come back to to the mindset of sampling. And when we're playing Tetris, there is some entity that is giving us one polyomino after another one. Question: I don't know if people have thought about this. But it's a fun thing to do. I did one morning, and I was like, “I don't know.” Yes. So how is this done? How does the game decide how to give you the next polyomino?

KK: It’s not just random?

EL: Yeah, I guess I assumed it was — there was a however many Tetris pieces there are. I actually can't think off the top of my head, but six or seven, maybe different pieces?

ÉR: Seven.

EL: Okay, then just so a 1/7 chance of each one, is that not the case?

ÉR: Well, you soon will be able to play the game that you're describing, because I'm programming a version with uniform distribution, just to make people really mad and upset.

EL: Okay.

KK: I can see, actually, now that I think about it, if it were just random, uniform, that would be bad. You’d lose pretty quickly, right? I mean, now that I think about it, often you'll get that long, the straight four in a row right when you need it, right? And and if it were uniform, you wouldn't necessarily expect that to happen.

ÉR: It could, you could have bad luck. Yeah. You could have had as long as you want bad luck.

KK: Right? Sure. That's right.

ÉR: Right. So the algorithm is very interesting. What they do is, they take the seven, the possible seven polyominoes with four squares that they give, and then they shuffle these seven in a way that you have a one over seven factorial, each one of the possibilities of the order has one over seven factorial, that means it’s a uniformly distributed sample. Okay, so what what can happen? Well, the worst that you can wait for that “I” shape is that if you have at the beginning of a set of seven, and then in the next set of seven, you have it at the end. That’s the worst that can happen in terms of these wonderful pieces that allow you to kill four rows.

KK: Interesting. Okay, so they're just choosing a random shuffle. Oh, wow. Well, that is better.

ÉR: It is for playing.

EL: I guess with, like, computer games like this. We, as you're saying earlier, Kevin, we don't actually want it completely random because we don't want to get, you know, five squares in a row or something like this, which can definitely happen when you play it randomly. Especially when you're me and my brother when we were, like seven and nine years old, always going in and playing for two hours to try to erase the other ones’ high scores.

KK: I still remember, so when I was in graduate school, I had a Nintendo Gameboy. My mother gave me one of these, and I played Tetris. And I can still hear that song. The music that went along with it, and I would go to sleep at night picturing these damn pieces just falling. I can hear this song over and over and over my head.

ÉR: That is a thing I think that happened when we were doing a PhD because I was always with my Nintendo playing Tetris and procrastinating, beautifully procrastinating. And it turns out.

EL: I have a question. How do you — like, did you find how Tetris is programmed? Or did you just do some — Like, how did you figure out that this is how Tetris works?

KK: Yeah.

ÉR: Well, so I started reading about it. But first of all, I started playing, and I started noticing this pattern, right? This pattern, because you never see — I think, I think what what tells you immediately that it cannot be a random distribution, uniform random period, is that you will never see three times the the same shape.

EL: Okay, you could see two in a row, if it happened, right at the end and then the beginning, but you can't see three in a row.

ÉR: You can’t see three in a row! So this is one of the immediate signs. So whenever you don't see three in a row, and you have a decently small set — and we're going to go back on the size of the set, because it plays a role. So it plays a role, the distribution that you have in the set, and it plays a role also the size of the set. And so in this case, because it's such a small set, not seeing three in a row is like, uh-uh, something fishy is happening here. This cannot be the uniform distribution, period. Yes.

EL: Okay.

ÉR: Yeah, and this is with four squares. And now imagine that we three get a very nice contract by a company that tells us, I want you to build a very nice Tetris, but it has to be for any number of tiles. And then we start discussing, okay, so let's see, one of the things that we need to do is to extend, or just to apply perhaps this algorithm, but it has to work for any n, any number of squares, let's say 57 squares, right? So, first step, go and find all polyforms with 57 squares.

KK: That’s a lot.

EL: Yeah.

ÉR: That’s a lot. Yes. And I kind of, yeah, I kind of, in purpose, choose 57. Because 57 is the first amount of squares that we as humans, don't know how many polyominoes there are with 57 squares.

EL: Okay.

KK: All right.

EL: It’s just too too many for for us to have sorted them out somehow.

ÉR: It’s too many for the patience that people and computers and computer time have. Like, 56, I think was two years.

EL: Wow.

KK: So there's some algorithm but it is not super-efficient.

EL: It hasn't stopped yet.

ÉR: It’s a little bit worse. Because even even if we wanted to say, Okay, let's cut at 56 because we know how many there are, or up to 56 and we deliver the game like this, right? So the problem is that this algorithm can count them. But this algorithm cannot really store the pieces or actually look at them, which is something that is possible to do. This is one of the magics of mathematics and computer science, that we can say something about something that we cannot see. And then we know up to 28 tiles, Tomás Oliveira e Silva in in 2015, was able to look at them and, for example, say the symmetries that they have because it's unknown, really, how many of them are going to have which kind of symmetries and so on and so forth, all the geometric and topological properties that you can have with polyominoes. And yes, so I guess we have to talk to the manager and say to the manager, we have a problem here. I don't think we can create Tetris for any amount of tiles.

EL: Yeah. Plus, I mean, it would be — I bet a lot of these tiles, as they get more pieces in them, have holes in the middle that you can just never plug. Because you can you can have rook-wise moves that will sort of surround this hole. Which I am thinking of because of your puzzle from the calendar. So I guess, do you happen to know like, at what size Tetris stops being fun?

KK: Eight.

ÉR: There is the answer. Well, with eight, perhaps not yet. Because almost all of them will not have holes.

KK: Right. But you’re still going to get that one with a hole every once in a while. And then you're done. Well, not yet. But it's, it's gonna cause problems.

ÉR: Super difficult.

EL: Yeah.

ÉR: Yes. And that is a very, very good observation, because actually, one of the things that is a mathematical truth is that most polyominoes have holds.

KK: Sure.

EL: Okay.

ÉR: Meaning with probability one, as n goes to infinity, if you take a polynomial at random, it will have most likely a linear amount of holes with respect to the number of tiles.

KK: Okay.

ÉR: And this is one of the topics that I started studying in my PhD, is how does the number of holes grow with respect to the number of tiles? And because we're asking a question upon a set that we don't know how many there are, and we don't know how to actually look at all of them. Right?

EL: Yeah.

ÉR: So the only way that we can say something is, for example, to have actual numbers, is sample. Sample and tell me. Sample them. Yeah, but we have another problem. How do you sample from a set that you don’t know?

EL: Yeah, that doesn't seem easy.

ÉR: Yeah. And that is huge. That is humongous. Because it’s — the number of polyforms is growing exponentially fast with respect to the number of tiles.

KK: Right? That's what I was going to say. I mean, even, you know, if you're up to, like, 10, the number is probably so large that you can't construct a reasonable game out of it, right? Yeah. So I mean, so Tetris is — so I have this game with blocks called Katamino, where they’re pentominoes, right? And that's fun, and challenging enough. Yeah. Okay. So now I understand why you're interested in stochastic topology, this feels very, if you're looking for holes in these things, you've set up some rules. And now the question is when do the Betti numbers change and things like that.

ÉR: Yes, yes. And now that you mention polyominoes, I also like like putting here and there one puzzle, so that people, perhaps even build it in their homes or something. So when Matthew Kahle was in Mexico visiting for my candidacy exam, we exchanged puzzles, and one of the puzzles that we exchanged is you take an eight by eight, a chess board, a usual eight by eight chess board. And then you have and then you break it on the head — no. And then the story in Diedonne’s book, it goes like this, that there were two royal persons playing chessboard, one loses, and then breaks the chessboard on the head of the other one, and then you get pieces. And these pieces are the 12 pieces that you were referring to, Kevin, the pentominoes, plus the two by two square. And then one of my favorite puzzles, and now so because Matt did this, built this puzzle with bamboo, with a very, very thin bamboo material, and he gave it to me as an exchange of the puzzles that we had. We actually had a night of puzzles and games back there.

KK: That’s fun.

ÉR: A lot of polynomials in my mathematical life.

KK: Very cool.

EL: Yeah. So, another thing we like to do on this podcast is asks our mathematicians to pair their theorem or example or their mathematics with something else. And so, what would you like to pair with this theorem about shuffles? And it may be even with the applications to all of these polyomino puzzles.

ÉR: I will say that is drawing and 3D modeling.

EL: Okay.

ÉR: One of the things that I like is to run these run some of these algorithms for generating poly-cubes in 3D, and different algorithms. And then to get these 3D crazy structures, and I like doing the modeling. So I guess, is the is a way of I, I think, it’s random art?

KK: Sure. Sure.

EL: And what programs do you use to actually draw those?

ÉR: Oh, well, here, and this is one of the things that I have been really into it ,is there is a program called Maya. And it’s for doing 3D modeling and rendering. But it can be combined with Python.

KK: Okay.

EL: Okay. So do some other math with it, and then put it into Maya?

ÉR: Yes. And so I found someone to give me some classes on Maya. And then this person was always like, “But how do you manage to do 1000 cubes and decide where to put them?” And no, like, I just click and play, and I have an algorithm that decides to go all the way to 1000. So I guess it was a very interesting combination, knowing the kind of fast-track things that we do as mathematicians, and then mixing that with art. I like digital art. Yes.

EL: Nice.

KK: Very cool. Okay, so so we also like to give our guests a chance to plug anything, or where can we find you on the internet, or anything else you want to tell us let people know.

EL: Yeah, if people want to play anything that you made, or where they can find this frustrating Tetris game that you'll be coding once you make it.

ÉR: Yes. That is going to be available very soon. But also, because I’ve been developing these apps with a computer science collaborator, and one of the apps that I have, and that I will provide a link for people to play, is for finding polyforms formed by squares and triangles. So you can also play this in the infinite the triangle tessellation of the plane, or you can play this with any tessellation. You could play it even in hyperbolic spaces, you could play it in higher-dimensional spaces, but the app that I’m going to share stays with the square and the triangular grid. So, you will be able to find polyforms with maximally many holes and it will tell you if you are actually reaching the maximum number of holes that you can for that amount of tiles, so it has that property. And I guess another another random way that I have for for producing these random polygons is like a cell growth model. And what you do is you start with a cell, and then you add uniformly random one of the cells that are neighboring, so you can imagine this is a cell colony growing. And there is a huge area of research called first passage percolation that has been analyzing these kind of models. But what I do is I study the topology of this other way of randomly generating polyforms. And I have an app that I will also provide here so that people can think about it and perhaps come up with some conjectures that later, they can see in one of my papers with Benjamin Schweinhart and Fedor Manin.

EL: Okay, great.

KK: That’s fun.

EL: So, you’ll get homework after this episode, but it's not a quiz. So it's still okay.

KK: All right. Well, this has been a lot of fun, Érika. I always like to hear about these sorts of questions, even though I'm no good at them. I always like to talk to people who are good at counting and these kinds of things. So it's been it's been great fun talking to you. Thanks for joining us.

ÉR: Yeah. Thank you so much for the invitation. And yes, looking forward for seeing you in the USA soon.

EL: Yes. Bye.

ÉR: Bye.

[outro]

On this episode of My Favorite Theorem, we had the pleasure of talking with Érika Roldán, a Marie Curie fellow at Technical University Munich and École Polytechnique Fédérale de Lausanne, about shuffling cards and Tetris pieces.

To read about the mathematics of riffle shuffles, this article by Persi Diaconis is a good place to start. To get a copy of the American Mathematical Society page-a-day calendar, click here. (If you already have a calendar, check out Dr. Roldan's puzzles on August 28 and October 12.)

Dr. Roldán shared some other links to and explanations of some of the apps and videos she mentioned in the episode:

The COVID crisis has allowed me to start developing digital material for my research, teaching, and outreach on mathematics and its applications. It has also allowed me to collaborate as a developer of digital material (in Germany) with artists whose projects promote gender equality, and diversity & inclusiveness awareness. Here, I share some of the links to explore this digital playground (new digital material created will be posted soonish at my website: http://www.erikaroldan.net/):
1) https://000612693.deployed.codepen.website
Follow the link above to find Extremal Animals, that is, polyforms with maximally many holes. A polyform is built by gluing together squares or triangles (in the case of this app) by their edges. And a hole in a polyform (that mathematicians call the first Betti number in this 2D case) is a finite connected component of the complement of the polyform. To get some intuition, just build a polyform with squares with 7 tiles and one hole, or a polyiamond with 9 triangles and one hole. Could you create one hole with less tiles in any of these two cases?
Have a look at these papers for the maths behind this (Extremal Topological Combinatorics) puzzle of finding polyforms with maximally many holes:
https://arxiv.org/pdf/1807.10231.pdf
https://www.combinatorics.org/ojs/index.php/eljc/article/view/v27i2p56/pdf
https://arxiv.org/abs/1906.08447v1
2) https://000612976.deployed.codepen.website
Here a link to an app that has a model to generate a random polyform by a cell growth process called The Eden Model. Pay attention to how the holes are created and destroyed as time (the number of tiles) evolves. Do you have any conjectures about how the number of holes is changing concerning time? Have a look at this link to see if your conjectures are stated and/or proved in this paper:
https://arxiv.org/abs/2005.12349
3) My first proto-game with Unity was developed for the Film “Broken Brecht” directed and produced by Caroline Kapp and Manon Haase, for the Brechtfestival Augsburg, Germany (Mar 2021). This is a project that will be extended during 2021! Here some links to the festival, the proto-game, and an extract from the film that happens within the proto-game.
Brecht Festival
https://brechtfestival.de/brokenbrecht/
Extract from Broken Brecht
https://vimeo.com/542287814
Link to the 3 min Archive Video Game
https://simmer.io/@ErikaRoldanRoa/~56f30f68-048c-c027-7aa0-aeaca82508fc
4) Some 3D models created with Python & Maya to explore (random) cubical complexes.
https://sketchfab.com/erikaroldan