Kevin Knudson: Welcome to My Favorite Theorem, a podcast about mathematics, favorite theorems, and other random stuff that we never know what it’ll be. I’m one of your hosts, Kevin Knudson. I’m a professor of mathematics at the University of Florida. This is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah, where I forgot to turn on the heat when I first woke up this morning. I've got separate systems. So it is very cold in the basement here where I am recording.

KK: Well, yeah, it's cold in Florida this morning. It was, you know, in the mid-60s. It's very pleasant. I'm still in short sleeves. Our listeners can't see this, but I'm in short sleeves. Evelyn’s in a sweater. And our guest is in a jacket in his attic.

EL: Yes.

KK: So today we are happy to welcome Nira Chamberlain all the way from the UK. Can you tell everyone about yourself a little bit?

Nira Chamberlain: Yes, hello. My name is doctor Nira Chamberlain. I’m a professional mathematical modeler. I'm also the president-designate of the Institute of Mathematics and its Applications.

KK: Fantastic. So tell us about the IMA a little bit. So we have one of those here, but it's a different thing. So what is it?

NC: Right. I mean, the Institute of Mathematics and its Applications is a professional body of mathematicians, of professional mathematicians, and it's a learned society. It's been around since 1964. And it is actually to make sure that UK has a strong mathematical culture and look after the interest of mathematicians by industry, government and academia.

KK: Oh, that's great. Maybe we should have one of them here. So the IMA here is something else. It’s a mathematics institute. But maybe the US should have one of these. We have the AMS, right, the American Mathematical Society.

EL: Or SIAM might be more similar because it does applications, applied math.

KK: Yeah, maybe.

EL: Yeah, we’ve kind of got some.

NC: So we asked you on for lots of reasons. One is, you know, you're just sort of an interesting guy. Two, because you’re an applied mathematician, and we like to have applied mathematicians on as much as we can, Three you actually won something called the Great Internet Math-Off this summer, of which Evelyn was a participant.

EL: Yes. So he has been ruled—he’s not just an interesting guy, he has been officially ruled—the most interesting mathematician in the world…among people who were in the competition. The person who ran it always put this very long disclaimer asterisk, but I think Nira definitely has some claim on the title here. So, yeah. Do you want to talk a little bit about the big, Great Internet Math-Off?

NC: Yes, we have let's say an organization, a group of mathematicians that do a blog, Aperiodical, and they decided to start this competition called the big internet math-off. And it’s a as a knockout tournament, 16 mathematicians, and they put up something interesting about mathematics. It was put up on the internet, it was there for 48 hours, the general public would vote for what they found was their most favorite or most interesting, and the winner would progress to the next round, and it was four rounds all together. And if you reach to the final end, and you win it, you get the title “World's Most Interesting Mathematician.” And when I was invited, I thought, “Oh, isn't this really for those mathematicians that are pure mathematicians and those public communicators and those into puzzles? I mean, I'm a mathematical modeler, I’m in applied mathematics, so what am I really going to talk about?” And then when I saw that when I was actually introduced as the applied mathematician and everybody else was, let's say the public communicator, and here's the applied mathematician. It was almost like: then here's the villain—boo!

I thought, “Okay, there you go.” I’m thinking, “All right, what we're going to do is I'm actually going to stick to being an applied mathematician.” So three out of the four topics I actually introduced were about applied mathematics, and yes, the fourth topic was actually about the history of mathematics. And I was fortunate enough to get through each of the rounds and win the overall competition. It was very interesting and very good.

EL: Yeah, and I do wish — I think you look very interesting right now—I wish our listeners could see that you've got headphones on that make you look a little bit like a pilot, and behind you are these V-shaped beams, I guess in the attic, where I can totally imagine you, like, piloting some ship here, so you're really looking the part this morning, or this afternoon for you.

NC: Thank you very much indeed. I mean that’s what I call my mathematics attack room, which is the attic, and I have 200 math books behind me. And I’ve got three whiteboards in front of me, quite a number of computer screens. And I’ve got all my mathematical resources all in one place.

KK: Okay, so I just took a screenshot. So maybe with your permission, we’ll put this up somewhere.

So this is a podcast about theorems. So, Nira what is your favorite theorem?

NC: Okay, my favorite theorem is actually to do with the Lorenz equation, the Lorenz attractor. Now it was done in the 1960s by a meteorologist called Edward Lorenz. And what he wanted to do was to take a a partial differential equation, see if he could make some simplifications to it, and he came up with three nonlinear ordinary differential equations to actually look at, let's say, the convection and the movement, to see where we can actually use that to do some meteorological predictions. And then he got this set of equations, went to work solving it numerically, and then he decided, “Actually, I’d better restart my computer game because I've done something wrong.” So he went back, he restarted the computer, but he actually changed the initial conditions by a little bit. And then when he came back, he actually saw that the trajectory of the solution was different from what he had started with. When he went back and started checking, he actually saw that the initial conditions only changed by a little bit, and what was this? It was probably one of the first examples of the “butterfly effect.” The butterfly effect is saying that if, let's say, a butterfly flaps its wings, then that will prevent a hurricane going into Florida — topical.

KK: Yeah, it’s been a rough month.

NC: Yeah, or, if, let's say, another butterfly flaps its wings, then maybe another hurricane may go into Salt Lake City, for example. And this is, like I said, an example of chaotic behavior once you choose certain parameters now. The reason why I like this theorem so much is I was actually introduced to this topic when I was in my final year of my mathematics degree. And it probably was one of the introductions to the field of mathematical modeling, recognizing that when you actually model reality, mathematics is powerful, but also has its limitations. And you’re just trying to find that boundary between what can be done and what can't be done. Mathematical modeling has a part to play in that.

KK: Right. What's so interesting about meteorological modeling is that I've noticed that forecasts are really good for about two days.

NC: Yeah.

KK: So with modern computing power, I mean, of course, as you pointed out, everything is so sensitive to initial conditions, that if you have good initial data, you can get a good forecast for a couple of days, but I never believe them beyond that. It's not because the models are bad. It's because the computation is so precise now that the errors can propagate, and you sort of get these problems. Do you have any sense of how we might extend those models out better, or is it just a lost cause, is it hopeless?

NC: It's probably a lost cause. I agree with you to a certain extent. But it's a case of when we're dealing with, let’s say, meteorological equations, if they have chaotic behavior, if you put down initial conditions, and it's changed, you know, it's going out and it's changing, it just shows that, yeah, we may have good predictions to begin with, but as we go on into the future, those rounding errors will come, those differences will come. And it's almost like, let's use an analogy, let's say you go to whatever computer algebra software you have, and you get π, and let’s say you square root it 10 times, and then you raise it to a power 10 times, and then if you square root it 100 times and then you raise it to a power 100 times, and if you keep on repeating that, then actually, when you come back to the figure, you're thinking, “Is this actually π?” No, it's not. And also different calculator and different computer algebra softwares, you’ll see that they will have actually their difference. It’s that point where in terms of when we're doing, predicting a weather system, because of the chaotic behavior of the actual nonlinear differential equations, coupled with those rounding errors, it is very difficult to do that long-term weather forecast. So nobody can really say to me, “By the way, in five years’ time, on the 17th of June, the weather will be this.” That’s very much a nonsense.

KK: Sure, sure. Well, I guess orbital mechanics are that way too, right? I mean, the planetary orbits. I mean, we understand them, but we also can't predict anything in some sense.

NC: Yeah.

KK: Right, right. Living in Florida, I pay a lot of attention to hurricane models. And it's actually really fascinating to go to these various sites. So windy.com is a good one of these. They show the wind field over the whole planet if you want. And they'll also, when there are hurricanes, they have the separate models. So the European model actually turns out to be better than the American one a lot, which is sort of interesting because hurricanes affect us a lot more than— I mean the remnants get to the UK and all of that. But so you’re right, it's sort of interesting, the different implementations—the same equations, essentially, right, that underlie everything get built into different models. And then different computing systems have the different rounding error. And the models, they’re sort of, they're usually pretty close, but they do diverge. It's really very fascinating.

NC: Yeah, I mean, over in the United Kingdom, we had an interesting case in 1987 where the French meteorology office says, “By the way, people in the north of France, they should be aware that there's going to be a hurricane approaching.” While the British meteorologic office was saying, “Oh, there's no way that there's going to be a hurricane. There's no hurricane. Our model says there’s going to be no hurricane.” So the French are saying there’s going to be a hurricane. The British say there’s not going to be a hurricane. And guess what? The French were right and a hurricane hit the United Kingdom.

And because of that what they did is that now the Met Office, which is the main weather place in Britain, what they've done is they put quite a number of boats out in the Atlantic to measure, to come up with a much more accurate measure of the weather system so that they can actually feed their models, and they also use more powerful models because he equation itself remains the same, it’s the information that actually goes into it which is which is the difference, yeah? So in terms of what you said in the American models, it's all dependent on who you get the measurements from because you may not get exactly the measurement from the same boat. You may get it from a different boat, from different boats in a different location, different people. This is where you come to that human factor. Some people will say, “Oh, round it to this significant figure,” while someone else will say, “Round it to that significant figure,” and guess what? All of that actually affects your final results.

EL: Yeah, that matters.

NC: Yes.

KK: So do you do this kind of modeling yourself, or are you in other applications?

NC: Oh, I'm very much in other applications. I mean, I'm still very much a a mathematical modeler. I mean, my research now is to do with—to minimize the probability of artificial intelligence takeover. That’s what my current research I'm doing at Loughborough University.

EL: Well that, you know, the robots will have you first in the line or something in the robot uprising.

NC: Well, we talk about robots, but this is quite interesting. When we're talking about, let's say, artificial intelligence takeover, everybody thinks about the Hollywood Terminator matrix, I Robot, you know, robots marching down the street. But there are different types of AI takeovers, and some of them are much more subtle than that. For instance, one scenario is, let's say for instance, you have a company, and they decide to really upgrade their artificial intelligence, their machine learning, to a certain sense it's more advanced than their competition is. And by doing so, they actually put all their competitors out of business. And so what you have is you have this one company almost running the world economy. Now the question is, would that company make decisions (based on its AI), would they make decisions that are conducive with social cohesion? And you can't put your hand on your heart and say, “Absolutely, yes, because a machine, it’s largely, like, 1-0, it doesn't really care about the consequences of social cohesion. So henceforth, we can actually to a model of that, saying could we ever get to a situation where one company actually dominates all different industrial sectors and ends up, let’s say, running the world economy? And if that's the case, what can what strategies can we actually implement to try and minimize that risk?

EL: It sounds not entirely hypothetical.

KK: No, no. Well, you know, of course the conspiracy theorists types in the US would have you believe that this already exists, right? The Deep State and the Illuminati run everything, right?

EL: But getting back to the Lorenz system and everything, you were saying that this is one of the earliest examples of mathematical modeling you saw. Was it one of the things that inspired you to go that direction when you got your PhD?

NC: Yes, so I was doing that as part of my final year mathematics degree, and I thought, well, this whole idea that, you know, here’s applied mathematics, using mathematics in the real world, saying that there are problems that some people say it's impossible, you can't use mathematics. And you're just trying to push the boundaries of mathematics and say, “This is how we actually model reality.” It was one of the things that actually did inspire me, so Edward Lorenz actually inspire me, just saying, wait a minute, applied mathematics is not necessarily about: here’s a problem, here’s an equation, put the numbers in the right places, and here's a solution. It's about gaining that insight into the real world, learning more about the world around you learning more about the universe around you through through mathematics. And that's what inspired me.

KK: And it's very imprecise, but that's sort of what makes it intriguing, right? I mean, you have to come up with simplifying assumptions to even build a model, and then how much information can we extract from that?

NC: That’s one of the key things about mathematical modeling. I mean, you're looking at the world. The world is complex, full of uncertainty, and it’s messy. And you are making some simplifying assumptions, but the key thing is: do you make simplifying assumptions to an extent that it actually corrupts and compromises your solution, or do you make simplifying assumptions that say, “Actually, this gives me insight into how the world actually works.”? And recognizing which factors do you include, which factors do you exclude, and bring a model that is what I call useful.

KK: Right. That’s the art, right?

NC: Yeah, that's the art. That's the art of mathematical modeling.

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

NC: I chose to pair it with the Jamaican dish called ackee yam and saltfish[16:48] . Now the reason why is with ackee yam and saltfish, if you cook it right, it is delicious, but if you cook it wrong, the Ackee turns out to be poisonous and that’s a bit like the Lorenz equation.

KK: What is ackee? I don't think I know what this is.

NC: Okay. Ackee is actually a vegetable, but if you actually were to look at it, it looks like scrambled egg, but it's actually a vegetable. It's like a yellow vegetable.

EL: Huh.

KK: Interesting.

NC: And yam, it’s like an overgrown, very hard potato. It’s looks like a very overgrown, hard potato.

KK: Sure, yeah.

NC: And saltfish is just a Jamican saying for cod. Even though you could really say ackee yam and cod, they don't call it cod, they call it saltfish.

KK: Okay. All right. So I've never heard ackee.

EL: Yeah, me neither.

KK: I mean, I knew that yams, so in the United States, most people will call sweet potatoes yams, they’ll use those two words interchangeably. But of course, yams are distinct, and I think they can be poisonous if you don't cook them right, right? Or some varieties can. But so ackee is something separate from the yam.

NC: Yeah.

EL: Also poisonous if you don't cook it right.

NC: Absolutely.

KK: So can you actually access this in England, or do you have to go to Jamaica to get this?

NC: Yes, we can access this this in England because in England we have a large West Indian diaspora community.

KK: Sure, right.

NC: And also we do get lots of variety of foods from different countries around the world. So we can, it's relatively easy to to access ackee yam. And also we’ve got quite a number of Caribbean restaurants, so definitely there they are going to cook it right.

KK: So it's interesting, we have a Caribbean restaurant here in town in Gainesville, which of course we're not as far away as you are, but they don't try to poison us. The food is delicious.

EL: That you know of.

KK: Well that's right. I love eating there. The food is really spectacular. But this is interesting.

EL: And is this a family recipe? Do you have roots in in the West Indies, or…

NC: Yes, my parents were from from Jamaica. I still have relatives in Jamaica, and my wife’s descent is Jamaican. Now and again we do have that Caribbean meal. I thought, “Well, what shall I say as a food? I thought, “Well, should I go for the British fish and chips?” I thought, “No, let's go for ackee yam and saltfish.”

KK: Sure, well and actually I think your jacket looks like a Jamaican-influenced thing, right? With the black, green, and yellow, right?

NC: Yes, absolutely. And that's because it's quite cold in the in the attic. This is the same style of jacket as the Jamaican bobsled team, so I decided to wear it, as it’s quite cold up here.

EL: Yeah, Cool Runnings, the movie about that, was an integral part of my childhood. My brother and sister and I watched that movie a lot.

So I’m curious about this ackee vegetable, like how sensitive are we talking for the dependence on initial conditions, the dependence on cooking this correctly to be safe? Is it pretty good, or do you have to be pretty careful?

NC: You have to be pretty good, you have to be pretty careful. As long as you follow the instructions you’re okay, but in this case, if you don't cook it long enough, you don't cook it at a high enough temperature, whatever you do, please do not eat it cold, do not eat it raw.

EL: Okay.

KK: Like actually it might kill you, or it just makes you really sick?

NC: It will make you really sick. I haven't heard— well let’s put it this way, I do not wish to carry out the experiments to see what would happen.

KK: Understood.

EL: Yes.

KK: Well this has been great fun. I've learned a lot.

EL: Yeah

KK: Thanks for joining us, Nira.

NC: Thank you very much indeed for inviting me.

Kevin Knudson: Welcome to My Favorite Theorem, a podcast about mathematics, theorems, and, I don't know, just about anything else under the sun, apparently. I'm Kevin Knudson. I'm one of your hosts. I'm a professor of mathematics at the University of Florida. This is your other host.

Evelyn Lamb: Hi, I'm Evelyn lamb. I'm a freelance math and science writer based in Salt Lake City. So how are things going?

KK: It's homecoming weekend. We're recording this on a Friday, and for people who might not be familiar with Southeastern Conference football, it is an enormous thing here. And so today is is a university holiday. Campus is closed. In fact, the local schools are closed. There's a big parade that starts in about 20 minutes. My son marched in it for four years. So I've seen it. I don't need to go again.

EL: Yeah.

KK: I had brunch at the president's house this morning, you know. It's a big festive time. I hope it doesn't get rained out, though. It's looking kind of gross outside. How are things for you?

EL: All right. Yeah, thankfully, no parades going on near me. Far too much of a misanthrope to enjoy that. Things are fight here. My alarm clock-- we're also recording in the the week in between the last Sunday of October and the first Sunday of November.

KK: Right.

EL: In 2007, the US switched when it went away from Daylight Saving back to Standard Time to the first Sunday of November. But my alarm clock, which automatically adjusts, was manufactured before 2007.

KK: I have one of those too.

EL: Yeah, so it's constantly teasing me this week. Like, "Oh, wouldn't it be nice if it were only 7am now?" So yeah.

KK: All right. Well, yeah, first world problems, right?

EL: Yes. Very, very much.

KK: All right. So today, we are thrilled to have Skip Garibaldi join us. Skip, why don't you introduce yourself?

Skip Garibaldi: My name is Skip Garibaldi. I'm the director at the Center for Communications Research in La Jolla.

KK: You're from San Diego, aren't you?

SG: Well, I got my PhD there.

KK: Ish?

SG: Yeah, ish.

KK: Okay.

SG: So I actually grew up in Northern California. But once I went to San Diego to get my degree, I decided that that was really the place to be.

KK: Well, who can blame you, really?

EL: Yeah, a lot to love there.

KK: It's hard to argue with San Diego. Yeah. So you've been all over. For a while you're at the Institute for Pure and Applied Math at UCLA.

SG: Yeah, that was my job before I came to the Center for Communications Research. I was associate director there. That was an amazing experience. So their job is to host conferences and workshops which bring together mathematicians in areas where there's application, or maybe mathematicians with different kinds of mathematicians where the two groups don't really talk to each other. And so the fact that they have this vision of how to do that in an effective way is pretty amazing. So that was a great experience for me.

KK: Yeah, and you even got in the news for a while. Didn't you and a reporter, like, uncover some crime syndicate? What am I remembering?

SG: That's right. Somehow, I became known for writing things about the lottery. And so a reporter who was doing an investigative piece on lottery crime in Florida contacted me, and I worked closely with him and some other mathematicians, and some people got arrested. The FBI got involved and it was a big adventure.

KK: So Florida man got arrested. Never heard of that. That's so weird.

SG: There's a story about someone in Gainesville in the newspaper article. You could take a look.

KK: It wasn't me. It wasn't me, I promise.

EL: Whoever said math wasn't an exciting field?

KK: That's right.

Alright, so, you must have a favorite theorem, Skip, what is it?

SG: I do. So you know, I listened to some of your other podcasts. And I have to confess, my favorite theorem is a little bit different from what your other guests picked.

EL: Good. We like the the great range of things that we get on here.

SG: So my favorite theorem for this podcast answers a question that I had when I was young. It's not something that is part of my research today. It's never helped me prove another theorem. But it answers some question I had from being junior high. And so the way it goes, I'm going to call it the unknowability of irrational numbers.

So let me explain. When you're a kid, and you're in school, you probably had a number line on the wall in your classroom. And so it's just a line going left to right on the wall. And it's got some markings on it for your integers, your 0,1,2,3, your -1,-2,-3, maybe it has some rational numbers, like 1/2 and 3/4 marked, but there's all these other points on that number line. And we know some of them, like the square root of two or e. Those are irrational, they're decimals that when you write them down as a number-- like π is 3.14, we know that you can't really write it down that way because the decimal keeps on going, it never repeats. So wherever you stop writing, you still haven't quite captured π.

So what I wondered about was like, "Can we name all those points on the number line?

EL: Yeah.

SG: Are π and e and the square root of two special? Or can we get all of them? And it comes up because your teacher assigns you these math problems. And it's like "x^2+3x+5=0. Tell me what x is." And then you name the answer. And it's something involving a square root and division and addition, and you use the quadratic formula, and you get the answer.

So that's the question. How many of those irrational can you actually name? And the answer is, well, it's hard.

EL: Yeah.

SG: Right?

KK: Like weirdly, like a lot of them, but not many.

SG: Exactly!

EL: Yeah.

SG: So if we just think about it, what would it mean to name one of those numbers? It would mean that, well, you'd have to write down some symbols into a coherent math problem, or a sentence or something, like π is the circumference of a circle over a diameter. And when you think about that, well, there's only finitely many choices for that first letter and finitely many choices for that second letter. So it doesn't matter how many teachers there are, and students, or planets with people on them, or alternate universes with extra students. There's only so many of those numbers you can name. And in fact, there's countably many.

EL: Right.

KK: Right. Yeah. So are we talking about just the class of algebraic numbers? Or are we even thinking a little more expansively?

SG: Absolutely more expansive than that. So for your audience members with more sophisticated tastes, you know, maybe you want to talk about periods where you can talk about the value of any integral over some kind of geometric object.

KK: Oh, right. Okay.

SG: You still have to describe the object, and you have to describe the function that you're integrating. And you have to take the integral. So it's still a finite list of symbols. And once you end up in that realm, numbers that we can describe explicitly with our language, or with an alien language, you're stuck with only a countable number of things you can name precisely.

EL: Yeah.

KK: Well, yeah, that makes sense, I suppose.

SG: Yeah. And so, Kevin, you asked about algebraic numbers. There are other classes of numbers you can think about, which, the ones I'm talking about include all of those. You can talk about something called closed form numbers, which means, like, you can take roots of polynomials and take exp and log.

KK: Right.

SG: That doesn't change the setup. That doesn't give you anything more than what I'm talking about.

EL: Yeah. And just to back up a sec, algebraic numbers, basically, it's like roots of polynomials, and then doing, like, multiplication and division with them. That kind of thing. So, like, closed form, then you're expanding that a little bit, but still in a sort of countable way.

SG: Yes. Like, what kinds of numbers could you express precisely if you had a calculator with sort of infinite precision, right? You're going to start with an integer. You can take it square root, maybe you can take its sine, you know. You can think about those kinds of numbers. That's another notion, and you still end up with a countable list of numbers.

KK: Right. So this sounds like a logic problem.

SG: Yes, it does feel that way.

KK: Yeah.

SG: So, Kevin and Evelyn, I can already imagine what you're thinking. But let me say it for the benefit of the people for whom the word "countable" is maybe a new thing thing. It means that you can imagine there's a way to order these in a list so that it makes sense to talk about the next one. And if you march down that list, you'll eventually reach all of them. That's what it means. But the interesting thing is, if you think about the numbers on the number line, we know going back to Cantor in the 1800s that those are not countable. You use the so-called diagonalization argument, if you happen to have seen that.

KK: Right.

EL: Yeah. Which is just a beautiful, beautiful thing. Just, I have to put a plug in for diagonalization.

KK: Oh, it's wonderful.

SG: I've been thinking about it a lot in preparation for this podcast. I agree.

KK: Sure.

SG: So what that means is that that's the statement, these irrational numbers, you can't name all of them, because there are uncountably many of them, but only countably many numbers you can name.

It sort of has a hideous consequence that I want to mention. And it's why this is my favorite theorem. Because it says, it's not just that you can't name all of them. It's just much worse than that. So the reason I love this theorem is not just that it answers a question from my childhood. But it tells you something kind of shocking about the universe. So when you--if you could somehow magically pick a specific point on the number line, which you can't, because you know, there's--

KK: Right.

SG: You have finite resolution when you pick points in the real world. But pretend you could, then the statement is the chance that the number you picked was a number you could name precisely is very low. Exactly. It's essentially zero.

KK: Yeah.

SG: So the technical way to say this is that the countable subset of real numbers has Lebesgue measure zero.

KK: Right.

SG: So I was feeling a little awkward about using this as my theorem for your podcast, because, you know, the proof is not much. If you know about countable and uncountable, I just told you the whole proof. And you might ask, "What else can I prove using this fact?" And the answer is, I don't know. But we've just learned something about irrational numbers that I think some of your listeners haven't known before. And I think it's a little shocking.

EL: Yeah, yeah. Well, it sounds like I was maybe more of a late bloomer on thinking about this than you, because I remember being in grad school, and just feeling really frustrated one day. I was like, you know, transcendental numbers, the non-algebraic numbers are, you know, 100% of the number line, Lebesgue measure one, and I know like, three of them, essentially. I know, like, e, π, and natural log two. And, you know, really, two of them are already kind of, in a relationship with each other. They're both related to e or the natural log idea. It's just like, okay, 2π. Oh, that's kind of a cheap transcendental number.

Like there's, there's really not that much difference. I mean, I guess then, in a sense, I only know, like, one irrational number, which is square root of 2, like, any other roots of things are non-transcendental, and then I know the rationals, but yeah, it's just like, there are all these numbers, and I know so few of them.

SG: Yeah.

KK: Right. And these other these other things, of course, when you start dealing with infinite series, and you know, you realize that, say, the Sierpinski carpet has area zero, right? But it's uncountable, and you're like, wait a minute, this can't be right. I mean, this is, I think why Cantor was so ridiculed in his time, because it does just seem ridiculous. So you were sitting around in middle school just thinking about this, and your teacher led you down this path? Or was it much later that you figured this out?

SG: Well, I figured out the answer much later. But I worried about it a lot as a child. I used to worry about a lot of things like, your classic question is--if you really want to talk about things I worried about as a child--back in seventh grade, I was really troubled about .99999 with all the nines and whether or not that was one.

EL: Oh yeah.

SG: And I have a terrible story about my eighth grade education regarding that. But in the end, I discovered that they are they are actually equal.

KK: Well, if you make some assumptions, right? I mean, there are number systems, where they're not equal.

SG: Ah, yeah, I'd be happy--I'm not prepared to get into a detailed discussion of the hyperreals.

KK: Neither am I. But what's nice about that idea is that, of course, a lot depends on our assumptions. We we set up rules, and then with the rules that we're used to, .999 repeating is equal to one. But you know, mathematicians like sandboxes, right? Okay, let's go into this sandbox and throw out this rule and see what happens. And then you get non Euclidean geometry, right, or whatever.

SG: Right.

KK: Really beautiful stuff.

SG: I have an analogy for this statement about real numbers that I don't know if your listeners will find compelling or not, but I do, so I'm going to say it unless you stop me.

KK: Okay.

EL: Go for it.

SG: Exactly. So one of the things I find totally amazing about geology is that, you know, we can see rocks that are on the surface of the earth and inspect them, and we can drill down in mines, and we can look at some rocks down there. But fundamentally, most of the geology of the earth, we can't see directly. We've never seen the mantle, we're never going to see the core. And that's most Earth. So nonetheless, there's a lot of great science you can do indirectly by analyzing as an aggregate, by studying the way, earthquake waves propagate and so on. But we're not able to look at things directly. And I think that has an analogy here with the number line, where the rocks can see on the surface are the integers and rationals. You drill down, and you can find some gems or something, and there's your irrational numbers you can name, and then all the ones you'll never be able to name, no matter how hard you try, how much time there is, how many alternate universes filled with people there are, you'll never be able to name, somehow that's like the core because you can't ever actually get directly at them.

EL: Yeah. I like this analogy a lot, because I was just reading about Inge Lehmann who is the Danish seismologist (who I think of as an applied mathematician) who was one of the people who found these different seismic waves that showed that the inner core had the liquid part--or I guess the core had the liquid part and then the solid inner core. She determined that it couldn't all be uniform, basically by doing inverse problems where, like, "Oh, these waves would not have come from this." So that's very relevant to something I just read. Christiane Rousseau actually wrote a really cool article about Inge Lehmann.

SG: Yes, that's a great article.

EL: So yeah, people should look that up.

KK: I'll have to find this.

EL: great analogy. Yeah.

KK: So, we know now that this, this is a long time there for you. So that's another question we've already answered. So, okay, what does one pair with this unknowability?

SG: Ah, so I I think I'm going to have to pair it with one of my favorite TV shows, which is Twin Peaks.

EL: Okay.

SG: So I watch the show, I really enjoy it. But there's a lot of stuff in there that just is impossible to understand.

And you can go read the stuff the people wrote about it on the side, and you can understand a little bit of it. But you know, most of it's clearly never meant to be understood. You're supposed to enjoy it as an aggregate.

KK: That's true. So you and I are the same age, roughly. We were in college when Twin Peaks was a thing. Did you did you watch it then?

SG: No, I just remember that personal ads in the school paper saying, "Anyone who has a video recording of Twin Peaks last week, please tell me. I'll bring doughnuts."

EL: You grew up in a dark time.

SG: Before DVRs, yeah.

KK: That's right. Well, yeah. Before Facebook or anything like that. You had to put an ad in the paper for stuff like this, yeah.

EL: Yeah, I'm really, really understanding the angst of your generation now.

KK: You know what, I kind of preferred it. I kind of like not being reached. Cell phones are kind of a nuisance that way. Although I don't miss paying for phone calls. Remember that, staying up till 11 to not have to pay long distance?

SG: Yeah.

KK: Alright, so Twin Peaks. So you like pie.

SG: Yeah, clearly. And coffee.

KK: And coffee.

SG: And Snoqualmie.

KK: Very good.

SG: I don't know if you--

KK: Sure. I only sort of vaguely remember-- what I remember most about that show is just being frustrated by it, right? Sometimes you'd watch it and a lot would happen. It's like, "Wow, this is bizarre and weird, and David Lynch is a genius." And then there'd be other shows where nothing would happen.

SG: Yes.

KK: I mean, nothing! And, you know, also see Book II of Game of Thrones, for example, where nothing happens, right? Yeah. And David Lynch, of course, was sort of at his peak at that time.

SG: Right.

KK: All right. So Twin Peaks. That's a good pairing because you're right, you'll never figure that out. I think a lot of it was meant to be unknowable.

SG: Yes. Yeah. Have you seen season three of Twin Peaks? The one that was out recently?

KK: No, I don't have cable anymore.

SG: About halfway through that season, there's an episode that is intensely hard to watch because so little happened on it. And if you look at the list the viewership ratings for each episode, there's a steep drop-off in the series at that episode. So this is like the most unknowable part of the number line if you if you follow the analogy.

KK: Okay. All right. That's interesting. So I assume that these these knowable numbers are probably fairly evenly distributed. I guess the rationals are pretty evenly distributed. So yeah.

So So our listeners might wonder if there's some sort of weird distribution to these things, like the ones that you can't name, do they live in certain parts? And the answer is no, they live everywhere.

SG: Yes. That's absolutely right.

EL: I wonder, though, if you can kind of--I'm thinking of continued fraction representations, where there is an explicit definition of number that's well-approximable versus badly-approximable numbers. I guess those are approximable by rationales, not by finite operations or closed form. So maybe that's a bad analogy.

KK: Mm hmm.

SG: Well, if you or your listeners are interested in, thinking about this question some more, then you can Google closed-form number. There's a Wikipedia entry to get people started. And there are a couple of references in there to some really well-written articles on the subject, one by my friend Tim Chow, that was an American Mathematical Monthly, and another one by Borwein and Crandall that was in the Notices of the AMS that's for free on the internet.

EL: Oh, great.

KK: Okay, great. We'll link to those.

EL: And actually, here's, this question, I'm not sure, so I'll just say, is this the same as computable or is closed form a different thing from computable numbers?

SG: Yeah, that's a good question. So there's not a widely-agreed upon definition of the term closed form number. So that's already a question. And then I'm not sure what your definition of computable is.

EL: Me neither.

SG: Okay.

EL: No, I've just heard of the term computable. But yeah, I guess the nice thing is no matter how you define it, your theorem will still be true.

SG: That's right. Exactly.

EL: There's still only countable.

KK: And now we've found something else unknowable: are these the same thing?

SG: Those are really hard questions in general. Yeah. That's the main question plumbed in those articles and referred to is: if you define them in these different ways, how different are they?

EL: Oh, cool.

SG: If you take a particular number, does it sit in which set? Those kinds of questions. Yeah, those are really hard usually, much like you said, what are the transcendental numbers that are--are certain numbers transcendental or not can be a hard question to answer.

EL: Yeah, yeah, even if you think, "Oh yeah, this certainly has to be transcendental, it takes a while to actually prove it."

SG: Yes.

KK: Or maybe you can't. I wonder if some of those statements are even actually undecidable, but again, we don't know. All right, we're going down weird rabbit holes here. Maybe David Lynch could just do a show.

SG: That would be great.

KK: Yeah, there would just be a lot of mathematicians, and nothing would happen

SG: And maybe owls.

KK: And maybe owls. Well, this has been great fun. Thanks for joining us before you head off to work, Skip. Our listeners don't know that it's you know, well, it's now nine in the morning where you are. So thanks for joining us, and I hope your traffic isn't so bad in La Jolla today.

SG: Every day's a great day here. Thank you so much for having me.

KK: Yeah. Thanks, Skip.

Evelyn Lamb: Hello and welcome to My Favorite Theorem, a math podcast where we ask mathematicians what their favorite theorem is. I’m one of your hosts, Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

EL: I’m all right. It’s fall here, or hopefully getting to be fall soon.

KK: Never heard of it.

EL: Yeah. Florida doesn’t have that so much. But yeah, things are going well here. We had a major plumbing emergency earlier this month that is now solved.

KK: My big news is that I’m now the chair of the math department here at the university.

EL: Oh yes, that’s right.

KK: So my volume of email has increased substantially, but it’s and exciting time. We’re hiring more people, and I’m really looking forward to this new phase of my career. So good times.

EL: Great.

KK: But let’s talk about math.

EL: Yes, let’s talk about math. We’re very happy today to have Michèle Audin. Yeah, welcome, Michèle. Can you tell us a little bit about yourself?

Michèle Audin: Hello. I’m Michèle Audin. I used to be a mathematician. I’m retired now. But I was working on symplectic geometry, mainly, and I was interested also in the history of mathematics. More precisely, in the history of mathematicians.

EL: Yeah, and I came across you through, I was reading about Kovalevskaya, and I just loved your book about Kovalevskaya. It took me a little while to figure out what it was. It’s not a traditional biography. But I just loved it, and I was like, “I really want to talk to this person.” Yeah, I loved it.

MA: I wanted to write a book where there would be history and mathematics and literature also. Because she was a mathematician, but she was also a novelist. She wrote novels and things like that. I thought her mathematics were very beautiful. love her mathematics very much. But she was a very complete kind of person, so I wanted to have a book like that.

KK: So now I need to read this.

EL: Yeah. The English title is Remembering Sofya Kovalevskaya. Is that right?

MA: Yeah.

KK: I’ll look this up.

EL: Yeah. So, what is your favorite theorem.

MA: My favorite theorem today is Stokes’ formula.

EL: Oh, great!

KK: Oh, Stokes’ theorem. Great.

EL: Can you tell our listeners a little bit about it?

MA: Okay, so, it’s a theorem, okay. Why I love this theorem: Usually when you are a mathematician, you are forced to face the question, what is it useful for? Usually I’ll try to explain that I’m doing very pure mathematics and maybe it will be useful someday, but I don’t know when and for what. And this theorem is quite the opposite in some sense. It just appeared at the beginning of the 19th century as a theorem on hydrodynamics and electrostatics, some things like that. It was very applied mathematics at the very beginning. The theorem became, after one century, became a very abstract thing, the basis of abstract mathematics, like algebraic topology and things like that. So this just inverts the movement of what we are thinking usually about applied and pure mathematics. So that’s the reason why I like this theorem. Also the fact that it has many different aspects. I mean, it’s a formula, but you have a lot of different ways to write it with integrals, so that’s nice. It’s like a character in a novel.

KK: Yeah, so the general version, of course, is that the integral of, what, d-omega over the manifold is the same as the integral of omega over the boundary. But that’s not how we teach it to students.

MA: Yeah, sure. That’s how it became at the very end of the story. But at the very beginning of the story, it was not like that. It was three integrals with a very complicated thing. It is equal to something with a different number of integrals. There are a lot of derivatives and integrals. It’s quite complicated. At the very end, it became something very abstract and very beautiful.

KK: So I don’t know that I know my history. When we teach this to calculus students anymore, we show them Green’s theorem, and there are two versions of Green’s theorem that we show them, even though they’re the same. Then we show them something we call Stokes’ theorem, which is about surface integrals and then the integral around the boundary. And then there’s Gauss’s divergence theorem, which relates a triple integral to a surface integral. The fact that Gauss’s name is attached to that is probably false, right? Did Gauss do it first?

MA: Gauss had this theorem about the flux—do you say flux?

KK: Yeah.

MA: The flux of the electric—there are charges inside the surface, and you have the flux of the electric field. This was a theorem of Gauss at the very beginning. That was the first occurrence of the Stokes’ formula. Then there was this Ostrogradsky formula, which is related to water flowing from somewhere. So he just replaced the electric charges by water.

KK: Sort of the same difference, right? Electricity, water, whatever.

MA: Yes, it’s how you come to abstraction.

KK: That’s right.

MA: Then there was the Green theorem, then there is Stokes’ formula that Stokes never proved. There was this very beautiful history. And then in the 20th century, it became the basis for De Rahm theory. That’s very interesting, and moreover there were very interesting people working on that in the various countries in Europe. At the time mathematics were made in Europe, I’m sorry about that.

KK: Well, that’s how it was.

MA: And so there are many interesting mathematicians, many characters, different characters. So it's like it's like a novel. The main character is the formula, and the others are the mathematicians.

EL: Yeah. And so who are some of your favorite mathematicians from that story? Anyone that stands out to you?

MA: Okay, there are two of them: Ostrogradsky and Green. Do you know who was Green?

EL: I don't know about him as a person really.

MA: Yeah, really? Do you know, Kevin? No.

KK: No, I don't.

MA: Okay. So nobody knows, by the way. So he was he was just the son of a baker in Nottingham. And this baker became very rich and decided to buy a mill and then to put his son to be the miller. The son was Green. Nobody knows where he learned anything. He spent one year in primary school in Nottingham, and that’s it. And he was a member of some kind of, you know, there are books…it’s not a library, but morally it’s a library. Okay. And that’s it. And then appears a book, which is called, let me remember how it is called. It’s called An essay on the application of mathematical analysis to the theories of electricity and magnetism. And this appears in 1828.

EL: And this is just out of nowhere?

MA: Out of nowhere. And then the professors in Cambridge say, “Okay, it’s impossible. We have to bring here that guy.” So they take the miller from his mill and they put him in the University of Cambridge. So he was about, I don’t know, 30 or 40. and of course, it was not very convenient for the son of a baker to be a student with the sons of the gentlemen of England.

KK: Sure.

MA: Okay. So he didn't us stay there. He left, and then he died and nobody knew about that. There was this book, and that’s it.

KK: So he was he was 13 or 14 years old when he wrote this? [Ed. note: Kevin and Evelyn had misheard Dr. Audin. Green was about 35 when he wrote it. The joys of international video call reception!]

MA: Yeah. And then and then he died, and nobody knew except that—

KK: Wow.

MA: Wow. And then appears a guy called Thomson, Lord Kelvin later. This was a very young guy, and he decided to go to Paris to speak with a French mathematicians like Cauchy, Liouville.

And then it was a very long trip, and he took with him a few books to read during the journey. And among these books was this Green book, and he was completely excited about that. And he arrived in Paris and decided to speak of this Green theorem and this work to everybody in Paris. There are letters and lots of documentation about that. And then this is how the Green formula appeared in the mathematics.

EL: Interesting! Yeah, I didn't know about that story at all. Thanks.

KK: It’s fascinating.

MA: Nobody knows. Yeah, that's very interesting.

KK: Isn’t what we know about Stokes’ theorem, wasn't it set as an exam problem at Cambridge?

MA: Yeah, exactly. So it began It began with a letter of Lord Kelvin to Stokes. They were very friendly together, on the same age and doing say mathematics and physics and they were very friendly together and and they were but they were not at the same at the same place of the world writing letters. And once Thomson, Kelvin, sent a letter to Stokes speaking of mathematics, and at the very end a postscript where he said: You know that this formula should be very interesting. And he writes something which is what we now know as the Stokes theorem.

And then the guy Stokes, he had to make a problem for an exam, and he gave this as an examination. You know, it was in Cambridge, they have to be very strong.

KK: Sure.

MA: And this is why it’s called the Stokes’ formula.

EL: Wow.

KK: Wow. Yeah ,I sort of knew that story. I didn't know exactly how it came to be. I knew somewhere in the back of my mind that it had been set as an exam problem.

MA: It’s written in a book of Maxwell.

KK: Okay.

EL: And so the second person you mentioned, I forget the name,

MA: Ostrogradsky. Well, I don’t know how to pronounce it in Russian, and even in English, but Ostrogradsky, something like that. So he was a student in mathematics in Ukraine at that time, which was Russia at that time, by the way. And he was passing his exams, and the among the examination topics there was religion. So he didn't go for that, so he was expelled from the university, and he decided to go to Paris. So it was in 1820, something like that. He went to Paris. He arrived there, and had no exams, and he knew nobody, and he made connections with a lot of people, especially with Cauchy, who was a was not a very nice guy, but he was very nice to Ostrogradsky.

And then he came back to Russia and he was the director of all the professors teaching mathematics in military schools in in Russia. So it was quite important. And he wrote books about differential calculus—what we call differential calculus in France but you call calculus in the U.S. He wrote a book like that, and for instance, because we were speaking of Kovalevskaya when she was a child, on the walls of her bedroom there were the sheets of the course of Ostrogradsky on the wall, and she read that when she was a little girl. She was very good in calculus.

This is another story, I’m sorry.

KK: No, this is the best part.

MA: And so, next question.

KK: Okay, so now I’ve got to know: What does one pair with Stokes’ theorem?

MA: Ah, a novel, of course.

EL: Of course!

KK: A novel. Which one?

MA: Okay, I wrote one, so I’m doing my own advertisement.

EL: Yeah, I was hoping we could talk about this. So yeah, tell us more about this novel.

MA: Okay, this is called Stokes’ Formula, a novel—La formule de Stokes, roman. I mean, the word “novel” is in the title. In this book I tell lots of stories about the mathematicians, but also about the about the formula itself, the theorem itself. How to say that? It’s not written like the historians of mathematics like, or want you to write. There are people speaking and dialogues and things like that. For instance, at the end of the book there is a first meeting of the Bourbaki mathematicians, the Boubaki group. They are in a restaurant, and they are having a small talk, like you have in a restaurant. There are six of them, and they order the food and they discuss mathematics. It looks like just small talk like that, but actually everything they say comes from the Bourbaki archives.

EL: Oh wow.

MA: Well, this is a way to write. And also this is a book. How to say that? I decided it would be very boring if the history of Stokes’ formula was told from a chronological point of view, so it doesn’t start at the beginning, and it does not end at the end of the story. All the chapters, the title is a date: first of January, second of January, and they are ordered according to the dates. So you have for instance, it starts with the first of January, and then you have first of February, and so on, until the end, which is in December, of course. But it’s not during the same year.

EL: Right.

MA: Well, the first of January is in 1862, and the fifth of January is in 1857, and so on. I was very, very fortunate, I was very happy, that the very end of the story is in December because the first Bourbaki meeting was in December, and I wanted to have the end there.

Okay, so there are different stories, and they are told on different dates, but not using the chronology. And also in the book I explain what the formula means. You are comparing things inside the volume, and what happens on the surface face of the volume. I tried to explain the mathematics.

Also, in every chapter there is a formula, a different formula. I think it’s very important to show that formulas can be beautiful. And some are more beautiful than others. And the reader can just skip the formula, but look at it and just points out that it's beautiful, even if I don't understand it completely.

There were different constraints I used to write the book, and one of them was to have a formula, exactly one formula in every chapter.

EL; Yeah, and one of the reasons we wanted to talk to you—not just that I read your book about Kovalevskaya and kind of fell in love with it—but also because since leaving math academia, you've been doing a lot more literature, including being part of the Oulipo group, right, in France?

MA: Yes. You want me to explain what it is?

EL: Yeah, I don't really know what it is, so it'd be great if you could tell us a little more about that.

KK: Okay. It's a group—for mathematicians, I should say it’s a set—of writers and a few mathematicians. It was founded in 1960 by Raymond Queneau and François Le Lionnais. The idea is to the idea is to find constraints to write some literary texts. For instance, the most famous may be the novel by George Perec, La Disparition. It was translated in English with the title A Void, which is a rather long novel which doesn’t use the letter e. In French, it is really very difficult.

EL: Yeah.

MA: In English also, but in French even more.

EL: Oh, wow.

MA: Because you cannot use the feminine, for instance.

EL: Oh, right. That is kind of a problem.

MA: Okay, so some of the constraints have a mathematical background. For instance, this is not the case for La Disparition, but this is a case for for some other constraints, like I don't know, using permutations or graph theory to construct a text.

KK: I actually know a little about this. I taught a class in mathematics and literature a few years ago, and I did talk about Oulipo. We did some of these—there are these generators on the internet where you can, one rule is where you pick a number, say five, and you look at every noun and replace it by the one that is five entries later than that in the dictionary, for example. And there are websites that will, you feed it text, and it's a bit imperfect because it doesn't classify things as nouns properly sometimes, it's an interesting exercise. Or there was another one where—sonnets. So you would you would create sonnets. Sonnets have 14 lines, but you would do it sort of as an Exquisite Corpse, where you would write all these different lines for sonnets, and then you could remove them one at a time to get a really large number, I forget now however many you do so, yeah, things like that, right?

MA: Yeah, this is cent mille milliards, which is 10 to the 14.

KK: That’s right, yeah. So 10 different sonnets. But yeah, it’s really really interesting.

MA: The first example you gave then, which is called in French “X plus sept,” X plus seven, you do you start from a substantive, a noun, you take the seventh in a dictionary following it.

KK: That’s right.

MA: It depends on the dictionary you use, of course.

KK: Sure.

EL: Right.

MA: So that's what they did at the beginning, but now they're all different.

KK: Sure.

EL: Yeah, it's a really neat creative exercise to try to do that kind of constraint writing.

MA: That forms a constraint, the calendar constraint I used to in this book, is based on books by Michelle Grangaud, who is a poet from the Oulipo also, and she wrote Calendars, which were books of poetry. That's where the idea comes from.

EL: Yeah, and I assume this, your novel has been translated into English?

MA: Not yet.

EL: Oh, okay.

MA: Somebody told me she would do it, and she started, and I have no news now. I don’t know if she were thinking of a published or not. If she can do something, I will be very grateful.

EL: Yeah, so it’s a good reason to brush up your French, then, to read this novel.

And where can people find—is there writing work of yours that people can find on a website or something that has it all together?

MA: Okay, there is a website of the Oulipo, first of all, oulipo.net or something like that. Very easy to find.

KK: We’ll find it.

MA: Also, I have a webpage myself, but what I write is usually on the Oulipo site. I have also a site, a history site. It’s about history but not about mathematics. It’s about the Paris Commune in 1871. It has nothing to do with mathematics, but this is one of the things I am working on.

EL: Okay. Yeah, we'll share that with with people so they can find out more of this stuff.

MA: Thank you.

KK: Alright, this has been great fun. I learned a lot today. This is this is the best part of doing this podcast, actually, that Evelyn and I really learn all kinds of cool stuff and talk to interesting people. So we really appreciate you to appreciate you taking the time to talk to us today, and thanks for persevering through the technical difficulties.

MA: Yes. So we are finished? Okay. Goodbye.

EL: Bye.

Evelyn Lamb: Hello, and welcome to my favorite theorem, a math podcast where we asked mathematicians to tell us about their favorite theorems. I'm one of your hosts, Evelyn Lamb. 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 you doing, Evelyn?

EL: I’m all right. I had a really weird dream last night where I couldn't read numbers. And I was like, trying to find the page numbers in this book. And I kept having to ask someone, "Oh, is this 370?" Because it looked like 311 to me. For some reason those are two of the numbers that like somehow, yeah, those numbers don't look the same. But yeah, it was so weird. I woke up, and I opened a book. And I was like, "Okay, good. I can read numbers. Life is ok." But yeah, it was a bit disorienting.

KK: That's weird. I’ve never had anything like that.

EL: So how about you?

KK: Well, I don't know. I was in California earlier this week, so I'm trying to readjust to Florida after what was really nice in California. It’s just gruesomely hot here and gross. But anyway, enough about that. Yeah.

EL: Yeah. So today, we're very happy to have Anil Venkatesh joining us. Hi, Anil, can you tell us a little bit about yourself?

Anil Venkatesh: Hi, Evelyn. Hi, Kevin. Yes, I am an applied mathematician. I teach at a school called Ferris State University in Michigan. And I am also a musician, I play soccer, and I’m the lead Content Developer for a commercial video game.

EL: Oh, wow. And I how I ran across your name is through the music connection. Because you sometimes give talks at the Joint Math Meetings and things like that. And I think I remember seeing one of your talks there. But I didn't know about the game developing. What game is that?

AV: It's called Star Sonata. And I'll plug it maybe at the end of the episode. But it actually relates because the theorem I'm going to talk about, well, I ran across it in my development work, actually.

EL: Oh, cool. So let's get right to it.

AV: Okay. Well, I'm going to talk about the Shapley value, which is due to Lloyd Shapley. The paper came out in 1953, and there's a theorem in that paper. It did not come to be known as the Shapley theorem, because that's a different theorem. But it's an amazing theorem, and I think the reason theorem didn't gain that much recognition is that the value that it kind of proved something about is what really took off.

So should I tell you a little bit about what the Shapley value is, and why it's cool?

KK: Yeah, let’s have it.

AV: Well, so actually, I picked up this book that came out in ’88, so quite a long time after the Shapley value was originally introduced. And this book is amazing. It's got like 15 chapters. And each chapter is a paper by some mathematician or economist talking about how they use the Shapley value. So it's just this thing that really caught on in a bunch of different disciplines. But it is an econ result, which is why I took a while to actually track it down once I came up with the math behind it.

EL: RIght.

AV: So putting this into context, in 1953 people were thinking a lot about diplomacy, they were thinking about the Cold War, or ensuing Cold War. And so here's a great application of the Shapley value. So you have the United Nations. It’s got in the Security Council, five permanent members who can veto resolutions and then 10 rotating members. So for a resolution to pass, I don't know if this is exactly how it works now, but at least when the paper was written, you needed nine out of 15 to vote in favor.

KK: That’s still correct.

AV: And of those nine, you needed all five of the permanent members. So you couldn’t have any of those vetoes. So you might ask, How powerful is it to have the veto? Can we quantify the negotiating strength of possessing a veto in this committee?

KK: Okay.

EL: Okay.

AV: Okay. And yes, you can with the Shapley value, and it comes down to, well, do you want to hazard a guess? Like, how many times better is it to have a veto?

KK: Like a million.

AV: It's a lot better. You know, I didn't really have a frame of reference for guessing. It's about 100.

EL: Yeah, I don't know… Oh, how much?

AV: A hundred.

KK: I was only off by four orders of magnitude! That’s pretty good.

AV: Yeah.

EL: Yeah, not bad.

AV: So the way the Shapley value carries this out is you imagine out of 100 percent, let's apportion pieces of that to each of the 15 members according to how much power they have in the committee.

KK: Okay.

AV: And so if it was 20% to each of the permanent members, there wouldn't be any left for the remaining 10 voting, right? In actuality, it's 19.6% to each of the five permanent members.

KK: Okay.

EL: Wow.

AV: And then that last sliver gets apportioned 10 ways to the rotating members. And that's how we come up with roughly 100 times more powerful with the veto.

EL: Okay.

AV: I will tell you how this value is computed, and I'll tell you about the theorem. But I'll give you one more example, which I thought was pretty neat and timely. So in the US, laws get made when the House of Representatives and the Senate both vote with the majority in favor of the bill, and then the President does not veto that bill.

KK: Yes.

AV: Or if the president vetoes, then we need a two-thirds majority in both houses to override that veto. So you could ask, well, if you think of it just as the House, the Senate and the President, how much of the negotiating power gets apportioned to each of those three bodies when it comes to finally creating a law? And if you apply the Shapley value, you get ratios of 5:5:2, which means the president alone has a one-sixth say in the creation of a law.

EL: Okay. Yeah, I was when you said that I was thinking, I mean, if you do the Security Council one, the people with vetoes had almost one fifth each, so I was thinking maybe like being one third of the things that could veto, it would be about a third for the president, but that seemed too high.

AV: Yes. So if the if the it was not possible to override the veto, then it would be a little bigger, right?

EL: Right, right. Okay.

AV: Yes. Now, if you actually break this down on the individual basis, so you might think, okay, well, the house gets 5 out of 12 of the power, but there are so many people in the house, so each individual person in the house doesn't have as much power, right?

KK: Yes

AV: When it breaks down that way, so going individual representative, individual senator and President, the ratio goes like 2 to 9 to 350.

EL: Okay.

AV: So the President actually has way more power than any one individual lawmaker.

KK: Well, that makes sense, right?

AV: Yes, it does. And so, yeah. The great thing about the Shapley value is that it's not telling you things you don't know exactly, but it's quantifying things. So we know precisely what the balance of power is. Of course, you've got to ask, “Okay, so this sounds like a like a fun trick. But how is it done anyway?”

EL: Yeah.

AV: The the principle behind the Shapley value is just, it’s beautiful in its simplicity. The theory is this—and actually when I tell you this, it's going to remind you of a lemma that's already been on this podcast.

EL: Okay.

AV: More than one actually, this just a very standard kind of technique. So imagine all the possible orderings of voters. So suppose they come in one at a time and cast their vote. Under how many of these arrangements is a particular person casting the pivotal vote? The more more frequently the more arrangements in which Person A casts the typical vote, the more power Person A is allotted.

EL: Okay.

AV: That's it. So we actually just take an average overall possible orderings of votes and basically count up however many of those orderings involve a particular person casting the pivotal vote, and that's how we that's how we derive this breakdown of power.

EL: So this is a lot like having everyone at a vertex and looking at symmetries of this object, which is kind of reminding me of Mohammed Omar's episode about Burnside’s lemma. I assume so that's the one that you were thinking about.

AV: Yes, that’s the one I was thinking about.

EL: But you said another one as well.

AV: The other one hasn’t actually been on this podcast yet. And I could have talked about this one instead. But the Cohen-Lenstra heuristics for the frequency of ideal class groups of imaginary quadratic extensions also involves an idea, now this one gets a little deeper. but essentially, if you dig into the Shapley value, you notice that the bigger the group is, the less power each person has in it. And yeah, so there are various other twists you can ask using the Shapley value. So in the Cohen-Lenstra heuristics, you essentially divide by the automorphisms of a group, you weight things inversely by the number of automorphisms they have. Anyway, that one also evoked because you take sort of an average across all the groups of the same size. So, I'm not claiming that there's some kind of categorical equivalence between the Cohen-Lenstra heuristics and the Shapley value, but this idea of averaging over an entire space comes up in a bunch of different branches of mathematics.

KK: Sure.

EL: Yeah. Very cool. So, we've got the Shapley value now, and what is the theorem?

AV: The theorem, and this is what makes it all really pop, the theorem is why people, why the Shapley value is so ubiquitous. There is no other logical apportionment of 100 percent than the Shapley value’s algorithm.

EL: Okay.

AV: There is no other sensible way to quantify the power of a person in the committee.

EL: Interesting.

KK: What’s the definition of sensible?

AV: I’ll give it to you, and and when you hear—this is how weak the the assumptions are that already gave you this theorem, and that's why it's amazing.

KK: Sure.

AV: Efficiency: you must apportion all hundred percent

KK: Okay.

AV: Of course. Symmetry: if you rename the people but you don't change their voting rules, the Shapley value is not affected by that kind of game.

KK: Sure.

AV: Null player: if a person has no voting power at all, they get zero percent.

KK: All right.

AV: Obviously. And finally, additivity. That one takes a little bit more thinking about, but it's nothing crazy. It's just saying, like, if there are two different votes happening, then your power in the total situation is the sum of your power in the one vote and your power on the other vote. If there's more than one game being played, basically, the Shapley value is additive over those games.

KK: That's the weirdest one, but yeah, okay.

AV: Yeah, I looked at it. I thought a little bit about what to say. And then honestly, if you dig into it, you realize it's just, like, not saying anything amazing. You have to think about this: the Sheppey value, it's a function, right? So we're working in the space of functions, and weird things can happen there. So this is just asserting you don't have any really wild and woolly functions. We're not considering that.

EL: Okay.

AV: So you just have these assumptions. And then there's only one. And the way they prove it is by construction. They basically write down a basis of functions, and they write down a formula using that basis, and there can only be one because it's from a basis, and then they prove that formula has the properties desired. It’s a really short paper, it's like a 10 page paper with four references. It's amazing.

EL: You said this is the 1953 paper by Shapley?

AV: Yes, by Shapley.

EL: Yeah, was there another author too, or just?

AV: No, Shapley collaborated with many people on related projects, but the original paper was just by him.

EL: Yeah. So I assume people have maybe looked at Shapley values of individual voters, like in the US or in an individual state or local election. We're recording this in election season, a little bit before the midterm elections.

KK: Yeah, can’t end soon enough here.

EL: Yeah, I guess. Oh, I guess actually, that wouldn't be that interesting, because it would just be, I mean, within a state or something. But I guess, the Shapley value of someone in one state versus another state might be a fairly interesting question.

AV: Oh, yes. But even the Shapley value for one person in a certain district or another district, this gets into gerrymandering, for sure.

KK: Right.

AV: I don't know to what extent people have thought about the Shapley value applied in this way. I imagine they have, although I haven't personally seen it mentioned, or anything that looks like it in the gerrymandering math groups that have been doing the work.

KK: No, I mean, I've been working with them a little bit, too. I mean, not really. And yeah, of course, it sort of gets to things like, you know, the Senate is sort of fundamentally undemocratic.

EL: Right.

KK: I mean, the individual senators kind of have a lot power. But you know, the voters in Wyoming have a lot more, you know, their vote counts more than than a voter and say, Florida.

EL: Right? Or the voter in Utah versus the voter in Florida.

AV: I'm thinking about within a specific state, if you look at the different districts. I mean, I read a little bit about this. And I see that they're, they're trying to resolve kind of the tension between the ability to cast a pivotal vote and the ability to be grouped with people who are like-minded. I don't know, it seems like, I wonder whether there's some extent to which they're reinventing the wheel, and we already have a way to quantify the ability to cast a pivotal vote. There's only one way to do it.

EL: Interesting.

AV: I don't know. Yeah, I'm not super informed on that. But it feels like it would apply.

KK: Yeah. So what drew you to this through this? I mean, okay. So fun fact: Anil and I actually had the same PhD advisor, albeit a couple of decades apart, and neither of us works in this area, really. So what drew you to this?

AV: Well, that's why I mentioned my game development background. So this game, Star Sonata, is one of those massively multiplayer online role-playing games. It actually was created back in 2004, when World of Warcraft had just started. And basically the genre of game had just been created. So that's why the game started the way it did. But it's kind of just an indie game that stuck around and had its loyal followers since then.

And I also played the game myself, but several years ago, I just kind of got involved in the development side. I think initially, they wanted—Well, I was kind of upset as a player, I felt they’d put some stuff in the game that didn't work that well. So I said, “Listen, why don't you just bring me on as a volunteer, and I'll do quality assurance for you.” But after some time, I started finding a niche for myself in the development team, because I have these quantitative skills that no one else on the team really had that background in. So a little later, I also noticed that I actually had pretty decent managing skills. So here I am, I'm now basically managing the developers of the game.

And one of my colleagues there asked me an interesting question. And he was kind of wrestling with it in a spreadsheet, and he didn't know how to do it. So the question is this, suppose you're going to let the player have like, six pieces of equipment, and each piece of equipment, let's say it increases their power in the game by a percent. Power could be like, you know, your ability to kill monsters or something.

EL: Yeah.

AV: So the thing is, each piece of equipment multiplicatively increases your power. So your overall power is given by some product, let's say (1+a)(1+b)(1+c), and so on. One letter for each piece of equipment. So you write down this product, you have to use the distributed property to work out the the final answer. And it looks like 1 plus the sum of those letters plus a bunch of cross-terms.

KK: So symmetric functions, right?

AV: Yes, exactly. So what his question was, “Okay, now that we're carrying all six of these pieces of equipment, how much of that total power is due to each piece of equipment?”

EL: Okay.

AV: How much did each item contribute to the overall power of the player? The reason we want to know this is if we create a new piece of equipment the player can obtain, and we put that in, and then suddenly we discover that everyone in the game is just using that, that's not good game design. It's boring, right? We want there to be some variety. So we need to know a way to quantify ahead of time whether that will happen, whether a new a new thing in the game is going to just become the only thing anyone cares about. And they'll eschew all alternatives. So he asked me, basically, how can I quantify whether this will happen? And I thought about it. And as you can tell, what this is asking about is the Shapley value in a special case where all the actors contribute multiplicatively to the to the total. And I didn't know that at the time because I'd never learned about the Shapley value. I didn't really learn much econ.

KK: Sure.

AV: So I just derived it, as it turns out, independently, in this in this special case. And it works out in a very beautiful formula involving essentially the harmonic means of all those letters. So reciprocals of sums of reciprocals. The idea there—and I mean, I can give a real simple example—Like, suppose you have two items. One of them increases your power by 20%, and one increases by 30%. So your overall power is 1.2 times 1.3. So what does that get to? 1.56 So of that of that 56% increase, 20% goes to the one item, 30 goes to the other, but 6% is left over. And how should that be aportioned?

EL: Right.

AV: Well, if you think about it, you might think, “Well, okay, the 30 percent should get the lion's share.” And maybe so, maybe so. But then there's a competing idea: because that 30% was pretty big, the 20 percent’s effect is amplified, right? So it's not, there's not an immediately obvious way to split it. But you can kind of do it in a principled fashion. So once I wrote this down, you know, I gave it to my colleague, he implemented it, it improved our ability to make the game fun. But then I also started wondering, like, look, this is, this is nice and all, but someone must have thought of this before, you know? So I don't actually remember now, how I came across it, whether I just found it or somebody sent it to me. But one way or another, I found the Shapley value on Wikipedia. I read about it, and I immediately recognized it as the generalization of what I'd done. So, yeah.

EL: Oh, yeah. Well, and this seems like the kind of thing that would come up in a lot of different settings, too. A friend of mine one time was talking about a problem where, you know, they had sold more units and also increased price, or something. And, you know, how do you allocate the value of the increased unit sales versus the increase price or something, which might might be a slightly different, the Shapley value might not apply completely there.

AV: No, it does.

EL: Okay.

AV: Yes, that’s called the Aumann-Shapley pricing rule.

EL: Okay, yeah.

AV: Yeah. So, questions of fair division and cost allocation are definitely applications of the Shapley value. So, yeah.

EL: Neat. Thanks.

KK: Very cool. The other fun part of this podcast is that we ask our guests to pair their theorem is something What have you chosen to pair this with?

AV: Well, like many your guests, I really struggled with this question.

KK: Good.

AV: And the first thing I thought of, which won't be my choice, was a pie because you have to, you know, fairly divide the pie. I told this to one of my friends, and I explained what the Shapley value was, and she was like, “No, that's, that's a terrible idea, because you want to divide the pie equally.” But the Shapley value is this prescription for dividing unequally but according some other principle. So it won't be a pie. So I actually decided this morning, it's going I'm going to pair it with a nice restaurant you go to with your friends, but then they don't let you split the bill.

KK: Ah.

EL: Okay. Yeah, so you have to figure out what numbers to write on the back of the receipt for them read your credit cards. Or for the added challenge, you could decide, like, given the available cash in each person's wallet? Can you do that?

AV: Oh, don't even get me started.

KK: This is the problem, right? Nobody has cash. So when you're trying to figure out how to how to split the bill…People think that mathematicians are really good at this kind of thing, and in my experience, when you go to a seminar dinner or whatever, nobody can figure out how to split the bill.

AV: If I'm out with a bunch of people, and we have to split a bill, let it not be mathematicians, that’s what I say. Let it be anyone else.

KK: Yeah, because some people want to be completely exact and each person ordered a certain thing and it cost as much and you pay that, then you divide the tip proportionally, all this stuff. Whereas I'm more, you know, especially the older I get, the less I care about five or $10 one or the other.

AV: Yeah, well, I find it's good if I go out with a bunch of people who are kind of scared of math, because then they just let me do it. You know, I become the benevolent dictator of the situation

KK: That’s happened to me too, yeah.

EL: So, I don't remember where what city Ferris State is in.

AV: Well, it's in a town of Big Rapids, which is a little the Grand Rapids, which is a little bit more well-known

EL: Slightly grander. So, yeah, you're the slightly lesser rapids?

AV: So, there are at least five rapids in Michigan, like five different places named something rapids.

KK: Sure.

EL: So do you have a Big Rapids restaurant in mind for this pairing?

AV: You know, they're all really nice about splitting the bills there. So I was thinking something maybe in New York City or Boston.

KK: College towns are pretty good about this. In fact, they'll let you hand them five cards, and they'll just deal with it.

AV: Yeah, totally.

KK: Yeah, yeah, very nice. So your rapids are big but Grand Rapids’ rapids are grander.

AV: They’re much grander. Don’t get me started about Elk Rapids. I don't know how to compare that to the other two.

KK: Elk Rapids?

EL: Yeah, Big, Elk, and Grand, not clear what order those go in. [I guess Iowa’s got the Cedar Rapids.]

AV: Yes. I don't remember the other two rapids, but I know identified them at some point.

EL: Well thank you so much for joining us

AV: Thanks for inviting me. It was great.

EL: Yeah, I learned something new today for sure.

KK: Math and a civics lesson, right?

EL: Yes. Everybody go vote. Although this episode will already be out. [Ed. note: Evelyn said this backwards. Voting occurred before the episode, not vice versa.] But get ready to vote in the next election!

KK: Yeah, well, it's never ending right? I mean, as soon as one elections over, they start talking about the next one. Thanks, Anil.

EL: All right, bye.

AV: Thank you.

[outro]

Show notes

Evelyn Lamb: Welcome to My Favorite Theorem. I'm your host Evelyn lamb. I'm a freelance math and science writer based in Salt Lake City. Today I am by myself because I'm on location. I am in Washington DC right now for the Science Writers conference. That's the conference for the National Association of Science Writers and I'm really happy to be joined by Yen Duong, who is a also a science writer with a math background. So yeah, can you tell us a little bit about yourself?

Yen Duong: Yeah, so I am in Charlotte, North Carolina, and I work part time for North Carolina Health News. And the rest of my time, I am a freelance math and science writer like you.

EL: Yeah.

YD: And I just finished the AAAS Mass Media Fellowship this summer, and before that I got my Ph.D. at UIC in geometric group theory.

EL: Yeah, and the AAAS fellowship is the one, the way I started doing science writing as well. A lot of people, when you come to conferences like these, you find out a lot of people who are more senior in the field have also gone through this. So it's really great. The application deadline, I believe is in January. So we'll try to air this at a time when people can look into that and apply for it. But yeah, it's a great program that brings grad students in science, you know, math and other sciences, into newsrooms to learn a little bit about how the news gets made and how to report on science for a broader audience. So it was a great experience for me. It sounds like it was a great experience for you.

YD: Yeah, it's fantastic. It's 10 weeks, I think this coming year, the stipend will be $6,000. So that's great. It is paid. And for me, at least it jump started the rest of my career as a math and science writer.

EL: Yeah, definitely. And it’s nice to hear that it's being paid a little more. I lived in New York City for less than that. And that was difficult. Okay, so do you want to tell us about your favorite theorem?

YD: I've been listening to this podcast for a while. And it's like, okay, I'll do a really fancy one to be really impressive. And people will think I'm fancy. But I decided not to do that. Because I'm not that fancy. And I think it's silly to be that pretentious. So I'm going with one of the first theorems I learned, like as an undergrad, which was Ramsey theory, that the Ramsey number of R(3,3) equals six.

EL: Okay, great. So, yeah, tell us what a Ramsey number is.

YD: Okay, so this is from graph theory. And the idea of saying, R(3,3)=6, I’ll just do the whole spiel.

EL: Yeah, yeah. And please use your hands a lot. It's really helpful for the podcast medium when you’re—Yeah, I know. Like Ramsey theory. I’m, like, moving my hands all around to show you what everything is.

YD: I will attempt to not pen and paper and start drawing things. Luckily, we don't have any available, right now. Yeah. So the idea is that, let's say that you are trying to put together a committee of three people. And you either want all three people to pairwise know each other and have worked together before, or you want all three people to be relative strangers. What you don't want is one person in the middle and everyone talks to them. And then the other two people don't talk to each other. That's a bad committee. Yeah. So the question is, how many people do you need to look at to guarantee that you can find such a committee?

EL: Right, so how big is your pool going to be of people you're choosing?

YD: Exactly. So like, if I look at three people? Well, that's not great, because it's me, you and someone in the next room. And there you go. We don’t have a good committee. And if I look at 100 people, like okay, I'm pretty sure I can find this with 100 people. So what Ramsey theory does is use graph theory to answer this question. And so like I said, the giveaway was that the number is 6, and something that I really love about this theorem is that you can teach it to literal—I think I taught it to 10 year olds the summer.

EL: Nice.

YD: And it's just a really nice basic introduction to, in my opinion, the fun parts of math. These kids who are like, “Ugh, I have to memorize equations, and I hate doing this.” And then I start drawing pictures and I explain the pigeonhole principle, and like, “Oh, I get it, like, I can do this.” I’m like, “Yes, you can! Everyone can do math!”

EL: Yay. Yeah. So the, the proof for that is, is kind of like you, you take a hexagon, right? Or the vertices of a hexagon and try to build—what do you do to denote whether you have friends or strangers?

YD: So graph theory is when you have vertices, which are dots, and edges, which are lines in between dots, and you use it to describe data and information systems. So in this case, we can make each person a dot, so we'll put six dots on a piece of paper. I do not have paper. I am using my hands. So we’ll have six dots on a piece of paper, and we’ll draw a blue line for friends, and we can draw a red line for strangers. So now our question becomes, how many dots do I need to make either a red triangle or blue triangle. So if you have six dots, let's look at one person, and that person will be me. And I look out at this crowd of five people. So for at least three of those people, I will have the same color line going to them. So they might all be strangers, so I'll have five red lines, or one might be a stranger and four friends—one red and four blue—but in that case, I have three, at least three, blue ones. So I can just assume that one of them is blue. So we'll just say, “Okay, I’ve got three blue lines going out.” So now I look at those three friends of mine. And I look at the relationships that they have with each other. This is really hard without pen and paper.

EL: Yeah, but luckily, our listeners have all gotten out of pens, two colors of pens, and they are driving this at home. So it's fine.

YD: Excellent. Good job, listeners! So now you've got your three dots. And you've got three blue lines coming out of them to one common dot. So you've got four dots on your piece of paper. So if in between any of those three dots, I draw a blue line, we’ve got our blue triangle, and we're done. We've got our committee.

EL: Yeah.

YD: Therefore, if I want to make this a proof, I'd better draw red lines. Yeah, I should draw a red line. Yeah. So now I've got three dots. And I've got red lines. But now I have red lines between all three of them. And there's my committee. So that's it. That's the entire proof. You can do it in a podcast in a few minutes. You can teach it to 10 year olds. You can teach it to 60 year olds. And I love it because it's like the gateway drug of mathematics proofs.

EL: Yeah, it’s really fun. And yeah, you can just sit down at home and do this. And—spoiler alert: to do this for four, to get a committee of four people, it's a little harder to sit down at home and do this, right? Do you—I should have looked up

YD: Oh, the Erdos quote, right? Is that what you're talking about?

EL: Well, well, I you can do four. Yeah, there's an Erdos quote about I think getting to six. Or five.

YD: So the Erdos quote is, paraphrased: if aliens come to the earth, and they tell us that they're going to destroy us unless we calculate R(5,5), then we should get all of the greatest minds in the world together and try to calculate it and solve it. But if the aliens say that we should try to compute R(6,6), then we should just try to destroy the aliens first.

EL: Yeah, so I think R(4,4) is like something like 18. Like, it's doable. I mean, by a computer, I think, not by a person, unless you really like drawing very large graphs. But yeah, it's kind of amazing. The Ramsey numbers just grow so fast. And we've been saying R(3,3) or R(4,4), having the same number twice in those. There are also Ramsey numbers, right, where it’s not symmetric.

YD: Like R(2,3) or R(2,4), Okay, so well two is maybe not the greatest number for this. But yeah, you can do things where you say, Oh, I'm going to have either a complete—so I'll either have a triangle of red, or I'll have four dots in blue, and they'll all be connected to each other with blue lines, a complete graph on four dots or however many dots.

EL: Yeah. So they don't have to be the same number. Although, you know, usually the same number is sort of a nicer one to look at. So how did you learn this theorem?

YD: Let's see. So I learned this through—I’ll just tag another great program—Budapest semesters in mathematics.

EL: Nice

YD: From a combinatorics professor. So BSM is when college students in the U.S. and Canada can go to Budapest for a semester and learn math from people there and they hang out with all these others. It’s a nice study abroad program for math. So that's when I first learned it. But since then, I think I've taught it to just like a hundred people, hundreds of people. I tell it to people in coffee shops, I break it out at cocktail parties, it's just like, my like, math is fun, I promise! little theorem. I think I've blogged about it.

EL: So watch out. If you're in a room with Yen, you will likely be told about this theorem.

YD: Yeah, that's my cocktail party theorem, that and Cantor’s diagonalization.

EL: Yeah, well, and cocktail parties are a place where people often like, describe this theorem. Like, if you're having a party, and and you want to make sure that any [ed. note: Evelyn stated this wrong; there shouldn’t have been an “any”] three people are mutual acquaintances, or mutual strangers, although the committee one actually makes a lot more sense. Because like, who thinks through a cocktail party that way? It's just a little contrived, like, “Oh, I must make sure the graph theory of my cocktail party is correct.” Like, I know a lot of mathematicians, and I go to a lot of their parties, but even I have never been to a party where someone did that. So on this podcast, we also like to ask you to pair your theorem with something. And why have you chosen for R(3,3)?

YD: I thought really hard about it, by the way.

EL: Yes. This is a hard part.

YD: Yeah. So I decided on broccoli with cheese sauce.

EL: Okay. Tell us why.

YD: Because it is the gateway vegetable, just like this theorem is the gateway theorem.

EL: Okay.

YD: Yeah. Like, my kids sometimes eat broccoli with cheese sauce. And it's sort of like trying to introduce them to the wonderful world of Brussels sprouts and carrots and delicious things. I feel like the cheese sauce is sort of this veneer of applicability that I threw on with the committee thing.

EL: Oh, very nice. Yeah.

YD: Even with the situation of the committee, like no one has ever tried to make a committee of three people who’ve all worked together or three people who didn’t. But, you know, it makes it more palatable than just plain broccoli.

EL: Yeah, okay. Well, and honestly, I could kind of see that, right. Because, like, it can be really that third wheel feeling when you’re hanging out with two people who know each other better than, you know either of them or something. Yeah. So actually, I feel, yeah, if you were making a committee for something, I could see why you might want to do this. I feel like a lot of people are not so thoughtful about making their committees that they would actually be like, “Will the social dynamics of this committee be conducive to…?”

YD: This is why my husband and I don't host cocktail parties, because my way of doing it is like, let's just invite everyone we know. And he's like, no, but what if someone feels left out? And then he gets stuck in the graph theory of our cocktail party and then it doesn't happen.

EL: And he's not even a mathematician, right?

YD: Yeah.

EL: Should have been, turns out.

YD: Yes, that's true. Stupid computers.

EL: Yeah. So when you make broccoli with cheese sauce, how do you make it. Are you a broccoli steamer? Do you roast it?

YD: We're definitely, if it's going to have cheese sauce on it, you’ve got to steam it. But generally, we're more roasters because I prefer it rested with garlic and olive oil.

EL: Okay.

YD: So delicious. Broccoli with cheese sauce is really a last resort. It's like, man, the kids have not eaten anything green in like a week

EL: They need a vitamin.

YD: Let’s give them some broccoli.

EL: So one of our favorite recipes is roasted broccoli with this raisin vinaigrette thing. You put vinegar and raisins, and maybe some garlic, A couple other things in a blender.

YD: Wait, so you blend the raisins?

EL: Yeah, you make a gloppy sauce out of the raisins and everything. And I don't think you plump them first or anything. I mean, usually I kind of get in a hurry, and I’ll put them all in, the ingredients, and then go do something else, and then come back. So maybe they plump a little from the vinegar. But yeah, it makes like a pasty kind of thing. It kind of looks like olive tapenade. And I have actually accidentally mistaken the leftover sauce in the fridge for olive tapenade and have been a bit disappointed. You know, if you're expecting olives, and you’re eating raisins instead, you’re just not as happy. But yeah, it's a really good recipe. If you want to expand your broccoli horizons, maybe not as kid friendly.

YD: Actually, my kids do love raisins. So maybe if I put raisins on top of broccoli, they would like it more.

EL: Yeah, I think there's some cumin in it too, something? And we're talking about recipes, because both of us like to cook a lot. And in fact Yen's blog is called Baking and Math. And it's not like baking with math. Like, there's baking, and there's math.

YD: Yeah, it’s a disjoint union. It doesn’t make that much sense, but I'm still a big fan of it. And it's actually how we met.

EL: Yes.

YD: Yeah. Because you found me on the internet.

EL: Yeah, I found you on the internet. And it was when I was writing for the AMS Blog on Math Blogs. And I was like, this is a cool blog. And yeah, then we became internet friends. And then I realized a couple of years later like, I feel like I know this person, but we've never actually met. We met at Cornell, at the Cornell topology festival, and I was like, “Wow, you're tall!” I just realized I always think people are either shorter than I think or taller that I think unless they're exactly my height because I think my

YD: You expect everyone to be your height?

EL: Yeah, my default, the blank slate version is like, “Oh, this person is the same height as I am.” So yeah, I was like oh, you're taller than I am. And I expected you to be exactly my height because I have no imagination

YD: I’m trying to think if I was surprised by, maybe, no, I don't think you had blue hair, maybe you did? No.

EL: No, I probably had blond hair at that point, yeah.

YD: I remember we did acro yoga when we first met. That's a good thing to do when you first meet someone.

EL: Yeah.

YD: It was very scary. It wasn't leap of faith, but so is meeting a stranger on the internet.

EL: Yeah. But luckily we’re both great people.

YD: Yeah. I also signed up for that conference because you tweeted that you were going to go, and I though, “Oh, I might as well sign up and then I can meet you.”

EL: I should have asked for a commission from the festival, although they probably paid for your travel, so it'd be like a reverse commission. So people can find your writing at your blog Baking and Math. They can find you on Twitter, you’re yenergy. And where here can they find your your science and health writing?

YD: So I post a lot of my clips on my website, my professional website, so that's yenduong.com, and then I also write for North Carolina Health News if you're interested in exactly what it sounds like, North Carolina health news.

EL: Yeah I'm sure a lot of people are I read them and I'm not in North Carolina, but I have a body, so I am interested in health news.

YD: Yeah.

EL: So thanks a lot for joining me.

YD: Thanks for having me. It was super fun. Fun fact for podcast listeners: Evelyn and I did not know where to look during this conversation. We couldn’t tell, should we look at each other or at the recording device?

EL: Yeah, so we did some of both. All right. Bye.

YD: Bye.

Show notes

Evelyn Lamb: Hello and welcome to My Favorite Theorem. I’m your host Evelyn Lamb, and 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 all right. I had a lovely walk today. And there are, there’s a family of quail that is living in our bushes outside and they were parading around today, and I think they're going to have babies soon. And that's very wonderful.

KK: Speaking of babies, today is my son's birthday.

EL: Who’s not a baby anymore.

KK: He’s 19. Yeah, so still not the fun birthday, right? That's that's another two years out.

EL: Yes, in this country.

KK: In this country, yes. But our guest, however, doesn't understand this, right?

EL: Yes. Today we are very happy to have Katie Steckles from Manchester, England, United Kingdom. So hi, Katie. Can you tell us a little about yourself?

Katie Steckles: Hi. Well yeah I'm a mathematician, I guess. So I did a PhD in maths and I finished about seven years ago. And now my job is to work in public engagement. So I do events and do talks about maths and do workshops and talk about maths on YouTube and on the TV and on the radio and basically anywhere.

KK: That sounds awesome.

EL: Yeah, you’re all over the place.

KK: Yeah, that sounds like great fun, like no grading papers, right?

KS: A minimal amount of, yeah, I don’t think I’ve had to grade anything, no.

EL: Yeah, and you have some great YouTube videos. We’ll probably talk more about some of them later. Yeah. And and I have stayed at your apartment a few years ago, or your flat, in Manchester. Quite lovely. And yeah, it's great to have you on here and to talk with you again. So what is your favorite theorem?

KS: Okay, my favorite theorem is what's called the fold and cut theorem, which is a really, really nice piece of maths which, like the best bits of maths, is named exactly what it is. So it's about folding bits of paper and cutting them. So I first encountered this a couple years ago when I was trying to cut out a square. And I realize that's not a very difficult task, but I had a square drawn on a piece of paper and I needed to cut out just the square, and I also needed the outside bit paper to still be intact as well. So I realized I wasn't going to be able to just cut in from the edge. So I realized that if I folded up the bit of paper I could cut the square out without kind of cutting in from the side, and then I realized that if I folded it enough I could do that in one cut, one straight line would cut out the whole square. And I thought, “That’s kind of cool. I like that, that’s a nice little bit of maths.” And I showed this to another friend who’s also a mathematician, and he was basically like, “Isn't there a theorem about this?” I thought, “Ooh, maybe there is,” and I looked and the fold and cut theorem basically says that for any figure with straight line edges, you can always fold a piece of paper with that figure drawn on it so that you can cut out the whole thing with one cut, even if it's got more than one bit to it or a hole in it or anything like that. It's always possible with one cut, in theory.

EL: Yeah. So you discovered a special case of this theorem before even knowing this was a thing to mathematically investigate.

KS: Yeah, well, I was I was cutting out a square for math reasons, because that's everything I do. But I was actually trying to make a flexagon at the time, which as I'm sure you've all been there, but it was just because I needed this square hole. And I thought it was such a satisfying thing to see that it was possible in one cut. And my maths brain just suddenly went, “How can I extend this? Can I generalize this to other shapes?

KK: Sure.

KS: And it was just a nice kind of extension of that.

EL: Yeah. So I have a question for you. Did you, was your approach to go for the, like diagonal folds, or the folds that are parallel to the sides?

KS: Yeah, this is the thing. There are actually kind of two ways to do a square. So you can do, like, a vertical and a horizontal fold, and then you get something that needs two cuts, and then you can make one diagonal fold and just end up with the thing that you can do in one cut, but you can actually do it in two folds if you do two diagonal folds, but it's along the cut. I don't know what the payoff is there. It depends on how much time you want to spend cutting, I don't know.

EL: Okay.

KK: I was thinking as you're doing this, I've never—I know about this theorem, but I've never actually done it in practice, but never really tried, but I was as soon as you said the square, I started thinking, “Okay, what would I do here?” You know, and I immediately thought to sort of fold along the diagonals. But so in general, though, so you have some, you know, 75-sided figure, is there an algorithm for this?

KS: It’s pretty horrible, depending on how horrible the thing is. Like simple things are nice, symmetrical things are really nice, because youjust fold the whole thing off and then use, you know, just do the half of it. And so there are algorithms. So the proof is done by Eric Demaine and Martin Demaine. And they've essentially got, I think, at least two different algorithms for generating the full pattern given a particular shape. So I think one of them is based around what they call the straight skeleton, which is if you can imagine, you can shrink the shape in a very sort of linear way, so you shrink all of the edges down but keep them parallel to where they originally were, you’ll eventually get to kind of a skeleton shape in the middle of the shape, and that's sort of the basis of constructing all the fold lines. And it is sort of seems quite intuitive because if you think about, for example, the square, all your folds are going to need to either be bisecting an angle or perpendicular to a straight edge. Because if it bisects the angle, it puts one side of the shape on top of the other one. And if you go perpendicular to the edge, it’s going to put the edge straight on top of the edge. And I always kind of think about in terms of putting lines on top of where the lines are, because that's essentially what you're doing if you've got a thin enough bit of paper and a thick enough line, you can actually physically see it happening. So it's beautiful. And then the other method they have involves disks in each corner of the shape, I think, and you expand the disks until they're as big as they can be and touch the other disks. And that then gives you a structure to generate a fold pattern. But they have got algorithms. I haven't yet managed to find a simple enough implementation that you can just upload the picture to a website and it will tell you the whole pattern, which is a shame because I've come across some really difficult shapes that I would really like to be able to fold but haven't quite been able to do it by hand. I'm just going, “Ah, I could just put some maths on this and throw it in the computer program!” But I actually asked Eric Demaine because I was in email contact with him about this. And then the thing that happened was, there’s a TV show in the UK called Blue Peter. Their logo it's like a giant boat that’s called the Blue Peter. It's a big ship with about 20 sails on it. And they said we could talk about this nice piece of maths, and you could even maybe try and cut out our logo with one cut. And I said to myself, “Goodness me!” Because it's all curves as well, so I’d have to approximate it all by straight lines and then work out how to cut this whole thing, and I emailed Eric Demaine and I sent him the picture and asked him, “Like, do you have a program that you can use to just you know, take a figure, even if I send the shape the edge or whatever?” And in his reply, he was like, “Wow, well, that looks, no.”

I just love the fact that they asked me to do something that not even the mathematician that proved that it's possible for any shape was prepared to admit would be easy. And so yeah, I'm not sure if there is kind of a, I mean, I would love it. I’m not enough of a coder to be able to implement that kind of thing myself. I would love it if there was a way to, you know, put in a shape or word or picture and come up with a fold pattern. Yeah, no, I don't know if anyone's done that yet.

KK: Well, this is how mathematicians are, right? We just proved that a solution exists, you know, and then we walk away.

EL: And so I seem to remember you've done a video about this theorem. And one of the things you did in it was make a whole alphabet, making all of those out of one-cut shapes.

KS: Yeah, well, this was, I guess this is my kind of Everest in terms of this theorem. This is one of the reasons why I love it so much, because I put so much time into this as a thing. So essentially in the paper that Demaine and Demaine have written about this, they've got a little intro bit where they talk about applications at this theorem and times when it's been used. So I think it's maybe Harry Houdini used to do a five-pointed star with one car as part of his actual magic show. It was really impressive. And people watch me do it. And they go, “Wow, how do you do that?” Such a lovely little demo. They also mentioned in there that they heard of someone who could cut out any letter of the alphabet, and I saw that and thought, “Wow, that would be a really nice thing to be able to do!” You know, that would impress people because it's kind of like if you can do any shape, then the proof of that should be whatever shape you tell me, I can do. And of course, a mathematician would know that 26 things is not infinity things, but it's still quite a lot of things. It's an impressive demo. So I thought I would try and work that out. And I literally had to sit down and kind of draw out the shapes and kind of work out where all the bits went and how to fold them. And some are easy, some are nice ones to start off with, like I and C and L. As long as you’ve got a square sort of version of it, they're pretty easy to imagine what you’d do. And then they get more difficult. So S is horrible, because there’s no reflection symmetry at all. It's just rotation symmetry and you can't make any use of that at all. R is quite difficult, but not if you know how to do P, and P is quite difficult, but not if you know how to do F. And so it all kind of kind of builds gradually. And I worked out all of these patterns and and in fact, it was one of the reasons I was in communication with Eric Demaine. Because he'd seen the video and he said, “As well as being mathematicians, we collect fonts, like we just love different fonts, type faces, and we wondered if you could send us your fold patterns for your letters so that we can make a font out of them.”

EL: Oh wow.

KS: And I thought that was really nice, so they've got a list on their website of different fonts, and they’ve now got a fold-and-cut font which I’m credited for as well.

KK: Oh nice.

KS: So yeah, the video I did with Brady was for his channel Numberphile, which is as I understand it a hugely popular maths channel. I've done about five or six videos on there, and I've genuinely been recognized in the street.

EL: Oh wow. That’s amazing.

KS: I walked into a shop and the guy was like, “Are you Katie Steckles?” I said, “Yes?” Like, the customer service has gone way up in this place. And he said, “No, I’ve just been watching your video on YouTube.” It’s like, Oh, okay. So that was nice. So he asked me to come and do a few videos, and that was one of the things I want us to talk about. I said, “What do you want me to do? I mean, do you want me to spell out Numberphile or your name or whatever? Brady, who’s Australian, said, “No do, the whole alphabet.” His exact words were, “If you're going to be a bear, be a grizzly.” A very Australian thing to say, he was basically saying let's do the whole alphabet, it will be great. I think at that point it was early enough I wasn't 100 percent sure I would get them all right, but his kind of thing that he has about his videos is that they always write maths down on brown paper, so he had this big pile of brown paper there, and he cut it all into pieces for me, one for each letter. And it was such a wonderful kind of way to nod to that tradition of using brown paper. But I just sat there folding them all, and he filmed the whole thing, and he put it in as a time lapse, and then I cut each one, one cut on each bit of paper, and open them all up, and they all worked, so it was good. But it was this very long day crouched over a little table cutting out all of these letters. But people genuinely come and ask me about it because of that video, so that's quite nice.

EL: Yeah, well I think after I watched that video, I tried to do—I didn’t. H was was my kryptonite. I was trying to fold that, and I just at some point gave up. Like I kept having these long spindles coming out of the middle bar that I couldn't seem to get rid of.

KS: I think somewhere I have a photograph of all of my early attempts at the S. It’s just ridiculous. Like it's just a Frankenstein's monster parade of villains, just horrific shapes that don't even look like an S, and like how did I get this?

But it kind of gave me a learning process, and I think it was maybe just a few weeks of solidly playing around with things. I think I had one night in a hotel room while I was away working so that no one else around. I just spent the whole evening folding bits of paper. I don't know what the maid who cleaned the room the next day thought. The bin was full of bits of cut up paper. I've got like a big stacks of scrap paper at home that's like old printouts and things I don't need that I use for practicing the alphabet because I go through a lot of paper when I’m practicing.

KK: This is a really fun theorem. So you know, another thing we like to do on this podcast is ask our guests to pair their theorem with something. So what have we chosen to pair the fold-and-cut theorem with?

KS: Wow. So I know that you sometimes often pair things with foodstuffs, so I'm going to suggest that I would pair this with my husband's chili and cheddar waffles.

EL: Okay.

KS: And I’ll tell you why, so my reasoning is that I kind of feel like this is a really nice example, as a theorem, about kind of the way that maths works and the way the theorems work. So my husband's chili is a recipe that he's been working on for years. He comes from a family where they do a lot of cooking, and it was natural for him when he moved out to just have his own kind of recipes. His chili recipe is so good that we've taken his chili to parties and people have asked for the recipe. And I'm just like, there isn't one. It's not written down anywhere. It's just in his head. He has this recipe. And he's obviously worked really hard on on it and achieved this brilliant thing. And kind of the ability to do the alphabet, the ability to kind of make things using this theorem for me is my equivalent of that. It's my special skill I can show off to people with. Because, you know, I've put in that time and I've solved the problem. And one of my favorite things about maths is that it gives you that problem solving kind of brain, in that you will just keep working at something, you keep practicing until you get there. And then the reason why I’ve paired it with cheddar waffles is (a) because that is a delicious combo.

EL: That sounds amazing.

KK: Yeah.

KS: Yeah. As soon as we got a waffle maker, that was our first go at it, was “What can we put with this chili that will make it even better?” And I just found the recipe for cheddar waffles on the internet, because we don't have that, you know, we don't do that many waffles. We don’t really know how to make them. And but the fact that you can go online and just find a recipe for something, is a really nice kind of aspect of modern life.

This is one of the things about maths I appreciate is that once you prove the theory that kind of goes into a toolbox, and other people can then you know, look at that theorem and use it in whenever they're doing, and you kind of building your maths out of bits of things that other people have proved, and bits of things that you're proving, and it's sort of a nice analogy for that, I guess. So those are those are the two things about it. Now that we've got the fold-and-cut theorem, nobody needs to prove it again, and anyone can use it.

EL: Yeah. And I guess if it were a perfect analogy, in some ways, maybe the chili recipe is sort of like these algorithms for making them, they’re really—well maybe that’s not good because the algorithms seem really complicated and difficult. Here, it's more that the recipe is hidden in your husband's brain.

KK: Well, a lot of algorithms feel that way.

KS: It really is quite complex. So you get some more things out of the cupboards that I've never seen before and they all go back in again afterwards. There’s a lot to it that people don’t realize.

KK: It’s a black box. my chili recipe is a black box, too. I can't tell you what's in it I mean it’s probably not as good as your husband’s, though.

KS: It’s got roasted vegetables in it. Yeah, it's that's that's one of the main secrets if anyone's trying to recreate it. But then just a whole lot of other spices that only he can tell me

EL: My husband doesn't like soups with tomatoes in them very much. I mean sometimes he does. But I don't do chili very much. So yeah, I don't have a good chili recipe we have a friend who's allergic to onions and that's a nice excercise in, can you cook or modify your recipe and still have it taste like what it’s supposed to be? because without us yeah a lot of things that don't work and she must have a nightmare with it. Because like a lot of packaged foods, they've got it.

KK: Sure.

KS: They’ve got onion powder or stuff.

EL: Every restaurant.

KS: We made chili without, and it kind of works. It kind of works without onions. It was great. I think there was a bit more aubergine that went in and some new spices, just to give it a bit more oniony flavor, but it still works.

EL: Oh, nice. Yeah, cooking without onions is tough. Does it extend to to garlic—does it generalize to other things in the allium family?

KS: Yeah, it’s all alliums, so she can’t really have garlic either she can get away with a little bit of garlic, but not any reasonable amount. Yeah, it must be completely horrible. Actually it kind of reminds me of Eugenia Cheng, her first book was about maths and baking. But one of the really nice points that she makes about the analogy between recipes and maths, which we have apparently stumbled into is that, you know, understanding something in a maths sense means that you can take bits of it out and replace it with other things. You've got a particular problem and you go, “Okay, well, do we need to make this assumption, do we need this particular constraint? What happens if we relax this and then put something else in?” And that's how you explore kind of where you go with things. And if you relax a constraint and then find the solution, that maybe tell us something about the solution to the constraint problem, and things like that. So, you know, tweaking a recipe helps you to understand the recipe a bit more. And as long as you know roughly what goes in there and you've got something that is, you know, recognizably a chili, then, you know, it doesn't matter what you've changed, I guess.

KK: Yeah, so we also give our guests a chance to plug anything you're working on. You want to plug videos, websites, anything?

KS: Oh, I’m always working on a million different things. I guess probably the nicest thing for people to have a look at would be the Aperiodical, which is a website where I blog with two of my colleagues so we write— it's kind of a maths blog but aimed at the people who are already interested in maths, so it's one of the few things I do that is not an outreach project. Which is essentially it’s aimed at people who already are interested and want to find out what's going on, so we sometimes right like opinion pieces about things or, like, “Here’s a nice bit of maths I found,” and then sometimes we just write news. And there’s a surprising amount of maths news, it turns out. It's not just “They’ve discovered a new Mersenne prime again.” There are various other maths news stories that come up as well, so we write those up, and bits of competitions and puzzles and things as well and it's at aperiodical.com. And we get submissions. So if anyone else wants to write an article and have it go out on a blog that’s seen by, you know, a couple of thousand people a day or whatever, they’re welcome to send us stuff, and we’ll have a look at it.

EL: Yeah, it's a lovely blog, and you also organize and host the math blog carnival that is, like, every month a round-up of math blog posts and stuff like that.

KS: We sort of inherited that from whoever was running it before, the Carnival of Mathematics. Every month someone who has a maths blog takes it in turn to write a post, which is essentially just a bunch of blog posts that went out this month. And we have the submissions form and all the kind of machinery behind it is now hosted at Aperiodical and has been for a few years, so if you have a maths blog elsewhere, and you want to get an opportunity to put a post on your site that will be seen by a bunch of people because there's a bunch of people who just read it every month, then get in touch because we're always looking for hosts for future months. And essentially we just forward to your email address all the submissions that people put in during the month, and you can then write it up in kind of the first week of the next month.

EL: Yeah. And I always see something cool on there that I had missed during the month. So it's a nice resource.

KS: So one of the other non-outreach, I guess, maths things that I'm involved with is a thing called Maths Jam. Or in the U.S. the equivalent would be Math Jam. We do have both websites, basically. So I coordinate all the Math Jams in the world. So it's essentially a pub night for people who want to go and do maths in a pub with people. It's aimed at adults because a lot of kids already get a chance to go to math club at school and do maths puzzles and things in their classrooms, but adults who have finished school, finished university, don't often get that chance. So we basically go to the pub once a month or to a bar or restaurant somewhere that will allow us to sit around and drink do maths. And there are now I think, getting on for a hundred Maths Jams in the world. So we've got about 30 or 40 in the UK. And then they’re popping up all over. We just picked up one in Brazil, we’ve got three in Italy now, three in Belgium, and there are a few in the U.S. But what I'm going to say is that I’m very sad that we don't have more because I feel like it would be really nice if we had a whole load of U.S. jams. I think we've got more in Canada that we have in the USA, which interesting given the population sizes, or relative sizes.

EL: Right.

KS: I think Washington DC has just gone on hiatus because not enough people are coming along. So the organizer said, “I'm getting fed up of sitting in the pub on my own. No one else is coming. I'm just going to put it on hold for now.” And so if you live somewhere in the U.S. and you want to go meet with the people and do maths in an evening, essentially to start when you just need a couple of people you that know you can drag along with you to sit around the case no one else turns up. And we send out a sheet with some ideas of puzzles and things to do. And you can play games, chat about maths, and do whatever. People can bring stuff along. And all you need to do to organize it is choose a bar and send the email once a month. And those are the only requirements. And go to the pub once a month, but I think that's probably not a big ask if that's the kind of thing you're into. So if anyone is interested, you can email to [email protected] and I can send you all the details of what's involved. You can have a look on the website, mathsjam.com, or math-jam.com, if you want to have a look at what there is already, what’s near you.

EL: Yeah, it'd be nice to have more in the U.S.

KS: Yeah, well, I get a lot out of it. Even though it's kind of sort of my job, but also I always meet people and chat through things and share ideas and people always go, “Oh, that reminds me of this other thing I saw,” and they show me something I've not seen before. And it's such a nice way to share things. But also just to know that everyone else in the room is totally sympathetic to maths and will be quite happy for you to chat on about some theorem or whatever and not think you’re weird. It’s quite nice.

EL: Well thanks a lot for joining us. I enjoyed talking about the fold-and-cut theorem. It makes me want to go back and pick up that alphabet again and try to conquer Mount H, that felled me the last time.

KS: I can send you send you a picture of my fold pattern for each, but I’m sure you would much rather work it out for yourself. It’s such a lovely puzzle. It's a really nice little challenge.

EL: Yeah, it’s fun.

Show notes

Kevin Knudson: Welcome to My Favorite Theorem. I’m Kevin Knudson, professor of mathematics at the university of Florida, and this is your other host.

EL: Hi. I’m Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah.

KK: How’s it going, Evelyn?

EL: How are you today?

KK: I’m okay. I’m a little sleepy. So before I came to Florida, I was at Mississippi State University, and I still have a lot of good friends and colleagues there, and that basketball game last night. I don’t know if you guys saw it, but that last minute shot, last second shot that Notre Dame hit to win was just a crusher. I’m feeling bad for my old friends. But other than that, everything’s great. Nice sunny day.

EL: Yeah, it’s gray here, so I’m a little, I always have trouble getting moving on gray mornings.

KK: But you’ve got that nice big cup of tea, so you’re in good shape.

EL: Yes.

KK: That’s right. So today we are pleased to welcome Mike Lawler. Mike, why don’t you introduce yourself and tell everyone about yourself?

1:24 ML: Hi. I’m Mike Lawler. I work in the reinsurance division for Berkshire Hathaway studying large reinsurance deals. And I also spend a lot of my spare time doing math activities for kids, actually mostly my own kids.

KK: Yeah.

EL: Yeah.

KK: Yours is one of my favorite sites on the internet, actually. I love watching how you explain really complicated stuff to your kids. How old are they now? They’re not terribly old.

ML: They’re in eighth grade and sixth grade.

KK: But you’ve been doing this for quite a while.

ML: We started. Boy, it could have been 2011, maybe before that.

KK: Wow, right.

ML: I think all three of us on the podcast today, and probably everybody listening, loves math.

KK: One hopes.

ML: And I think there’s a lot of really exciting math that kids are really interested in when they see. It’s fun finding things that are interesting to mathematicians and trying to figure out ways to share them with kids.

EL: Yeah. Well I like, you always make videos of the things, so listening to your kids talking through what they’re thinking is really fun. Recently I watched one of the old ones, and I was like, “Oh my goodness! They’re just little babies there.” They’re so much bigger now. I don’t have kids of my own, so I don’t have that firsthand look at kids growing up the same way. They’re sweet kids, though.

ML: I have to say, one of the first, it wasn’t actually the first one we did, but it’s called Family Math 1, where we do the famous “How many times can you fold a piece of paper?” And, you know, they’re probably 4 and 6 at the time, or maybe 5 and 7, and yeah, it’s always fun to go back and watch that one.

EL: Yeah.

KK: I see videos of my son, he’s now 18, he’s off in college. When I see videos of him, he’s a musician, so when he was 10, figuring out how to play this little toy accordion we got him, I kind of get a little weepy. You know.

ML: It’s funny, I was picking him up somewhere the other day, and I confused him with a 20-year-old, my older son, and I just thought to myself: how did this happen?

KK: So, all right. Enough talking about kids, I guess. So, Mike, we asked you on to talk about your favorite theorem. So what is it?

ML: Well, it’s not quite a theorem, but it’s something that’s been very influential to me. Not in sharing math with kids, but in my own work. It comes from a paper from 1995 at a professor named Zvi Bodie at BU. And he was studying finance, and continues to study finance. And he published a paper showing that the cost of insurance for long holdings in the stock market actually increases with time. Specifically, if you want to buy insurance to guarantee your investments at least earn the risk-free rate, that cost of insurance goes up over time. And it just shocked me when I was just learning about finance, actually when I was just in grad school. And this paper has had a profound influence on me over the last 20 years. So that’s what I want to talk about today.

KK: Okay. I know hardly any of those words. I have my retirement accounts and all that, but like most good quantitatively-minded people, I just ignore them.

ML: Well, let’s take a simple example. Let’s just take actually the most simple example. Say you wanted to invest $100 in the stock market, and you thought, because you’ve read or you’ve heard that the stock market gives you good returns, you thought, “Well, in 10 years from now, I think I’ll probably have at least $150 in that account.” And you said, “Well, what I want to do is go out and buy some insurance that guarantees me that at least I’ll have that amount of money in the account. That’s the problem. That’s the math problem that Bodie studied.

KK: Right. So how does one price that insurance policy, I guess? So right, on the insurance side, how do they price that correctly, and on the consumer side, how do you know you’re getting a worthwhile insurance policy, I guess.

ML: Yeah, well this is kind of the fun of applied mathematics. So there’s a lot of theory behind this, and I think like a lot of good theories, it’s not named after the people who originally discovered that. So I think that’s important part of any theory. But then when you understand the theory, and you actually go into the financial markets, you have to start to ask yourself, “What parts of the theory apply here, and which ones don’t?” So the theory itself goes back to the early 1900s with a French mathematician and his Ph.D. thesis. His last name is Bachelier, and I’m probably butchering that. But then people began to study random processes, and Norbert Wiener studied those. And eventually all of that math came into economics, I think in the late 60s, early 1970s, and something called the Black-Scholes formula came to exist. The Black-Scholes formula is what people use to price this kind of insurance, sometimes called options. So that’s been around the financial markets since at least the early 1970s, so let’s call it 50 years now. And if you’re a consumer, I think you’d better be careful.

EL: Well I find, I don’t know a lot about financial math, but I’ve tried to read a few books about the financial crash, actually one of which you suggested to me, I think, All the Devils Are Here. And I find, even with my math background, it’s very confusing what they’re pricing and how they’re calculating these, how they’re batching all of these things. It just really seems like a black box that you’re just kind of hoping what’s in the box isn’t going to eat you.

ML: That’s a pretty good description. Yeah, Bethany MacLean’s book All the Devils Are Here is absolutely phenomenal, and Roger Lowenstein’s book, called When Genius Failed, is also an absolutely phenomenal book. You are absolutely right. The math is very heavy, and a lot of times, especially when you talk about the financial crisis, the math formulas get misused a little bit, and maybe are applied into situations where they might not necessarily apply.

KK: Really? Wall Street does that?

ML: So you really have to be careful. I think if you pull the original Black-Scholes paper, I think there are 7 or 8 assumptions that go into it. As long as these 7 or 8 things are true, then we can apply this theory. In theory we can apply the theory.

KK: Right.

ML: So when you go into the financial markets, a lot of times if you have that checklist of 7 things with you, you’re going to find maybe not all 7 are true. In fact, a lot of times, maybe you’re going to find not a single one of those things is true. And that is I think a problem that a lot of mathematicians have when they come into the markets, and they just think the theory applies directly, if you will.

KK: Right, and we’ve all taught enough students to know they’re not very good at checking assumptions, right? So if you have to check off a list of 6 or 7 things, then after the first couple, you’re like, “Eh, I think it’s fine.”

ML: Right. Maybe that seventh one really matters.

KK: Right.

EL: Yeah.

ML: Or maybe you’re in a situation where the theory sort of applies 95% of the time, but now you’re in that 5% situation where it really doesn’t apply.

KK: So should I buy investment insurance? I mean, I’ve never directly done such a thing.

ML: Well…

KK: I don’t know if it’s an option for me since I just have 401Ks, essentially.

ML: Well, it’s probably not a great idea to give investment advice over a podcast.

KK: Right, yeah, yeah.

ML: But from a mathematical point of view, the really interesting thing about Bodie’s paper is Black-Scholes is indeed a very complicated mathematical idea, but the the thing that Bodie found was a really natural question to ask about pricing this kind of insurance, ensuring that your portfolio would grow at the risk-free rate. In that situation, and you can see it in Bodie’s paper, the math simplifies tremendously. And I think that is a common theme across mathematics. When someone finds exactly the right way to look at a problem, all of a sudden the problem simplifies. And I’m sure you can probably give me 3 or 4 examples in your own fields where that is then the case.

KK: Sure. Well, I’m not going to, but yeah.

EL: So something when you told us that this was the theorem or quasi-theorem you were going to talk about, it got me wondering how much the financial world—I’ve been trying to think about how to phrase this question—but how much your natural tendencies as mathematicians actually carry over into finance. How much are you able to think about your work in finance and insurance as math questions and how much you really have to shift how you’re thinking about things to this more realistic point of view.

ML: I think it’s a great question because, you know, the assumptions and a lot of times the mathematical simplifications that allow you to solve these differential equations that stand behind the Black-Scholes theorem and generally stochastic processes, you know, you’re, that doesn’t translate perfectly to the real world. And you have to start asking questions like, “If this estimate is wrong, does it miss high? Does it miss low?” “In the 5% of the times it doesn’t work, do I lose all my money?

EL: Right.

ML: And so those, I can tell you as an undergrad I was also a physics major, and I spent a lot of time in the physics lab, and there’s not one single person who was ever in lab with me who misses me. I was a mathematician in the labs. But doing some of these physics experiments really teaches you that applying the theory directly, even in a lab situation, is very difficult.

KK: Right.

EL: Right. And your Ph.D. was in pure math, right?

ML: Right, it was sort of mathematical physics. In the late 90s, people were really excited about the Yang-Mills equations.

KK: Mirror symmetry.

ML: Work that Seiberg and Witten were doing. So I was interested in that.

EL: So your background is different from what you’re doing now.

ML: Oh, totally. You know, I, it’s kind of a hard story for me to tell, but I really loved math from the time I was in fifth grade all the way up through about my third year of graduate school.

EL: Yeah, I think that could be a painful story.

ML: I don’t know why, I really don’t know why, I just kind of lost interest in math then. I finished my Ph.D., and I even took an appointment at the University of Minnesota, but I just lost interest, and it was an odd feeling because from about fifth grade until—what grade is your third year of graduate school?

KK: Nineteenth.

ML: Nineteenth grade. I really got out of bed every morning thinking about math, and I sort of drifted away from it. But my kids have brought me back into it, so I’m actually really happy about that.

KK: Well that’s great. So, what have you chosen to pair with your quasi-theorem, we’re calling it?

ML: Well, you know, so for me, this paper of Bodie’s goes back, and it sort of opened a new world for me, and for the last 20 years I’ve been studying more about it and learning more about it and all these different things, so I got to thinking about a journey. I have books on my table right now about this paper. So the journey I want to highlight is—and I think a lot of people can understand who are outside of math—is an athletic journey. I’m going to bring up a woman named Anna Nazarov, who represents the United States on the national ultimate frisbee team, which is a sport I’ve been around. And four years ago, she made it almost to being on the national team and got cut in the last minute and wrote this very powerful essay about her feelings about getting cut and then turned around and worked hard and improved and won three world and national championships in the last four years as a result of that work.

KK: Wow.

ML: Yeah, you know, it’s hard to compare world championships to just your plain old work. I think people in math understand that you kind of roll up your sleeves and over a long period of time you come to understand mathematics, or you come to understand in this case how certain mathematics applies, and so I want to pair this with that kind of athletic journey, which I think, to the general public, people understand a little bit better.

EL: Yeah, so I played ultimate very recreationally in grad school. There was a math department pickup ultimate game every week, and playing with other math grad students is my speed in ultimate. I really miss it. When you, I can tell, follow ultimate, and I often read the links you post about ultimate frisbee, I’m like, oh, I kind of miss doing that. But a few years ago, I did get to, I happened to be in Vancouver at the same time that they were doing the world ultimate championships there and got to see a couple games there, and it’s really fun, and it’s been fun to follow the much-higher-level-than-math-grad-student ultimate playing thing through the things you’ve posted.

ML: Yeah, it’s neat to follow an amateur sport, or not as well-known a sport because the players work so hard, and they spend so much of their own money to travel all over the world. You know, I think a lot of people do that with math. Despite the topic of today’s conversation, most people aren’t going into math because of the money.

KK: Well this has been great fun. Thanks for joining us, Mike. Is there anything, we always want to give our guest a chance to plug anything. We already kind of plugged your website.

EL: We’ll put links to your blog in the show notes there, and your Twitter. But yeah, if there’s anything else you want to plug here, this is the time for it.

ML: No, that’s fine. If you want to follow Mike’s Math Page, it’s a lot of fun sharing math with kids. And like I said, I sort of lost interest in math in grad school, but sharing math with kids now is what gets me out of bed in the mornings.

KK: Great.

EL: Yeah.

KK: All right. Well, thanks again, Mike.

ML: Thank you.

Show notes

Kevin Knudson: Welcome to My Favorite Theorem. I’m your cohost Kevin Knudson, professor of mathematics at the University of Florida. I am joined by cohost number 2.

Evelyn Lamb: I am Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City. So how are you?

KK: I’m okay. And by the way, I did not mean to indicate that you are number 2 in this.

EL: Only alphabetically.

KK: That’s right. Yeah. Things are great. How are things in Salt Lake?

EL: Pretty good. I had a fantastic weekend. Basically spent the whole thing reading and singing, so yeah, it was great.

KK: Good for you.

EL: Yeah.

KK: I didn’t do much. I mopped the floors.

EL: That’s good too. My floors are dirty.

KK: That’s okay. Dirty floors, clean…something. So today we are pleased to have Chawne Kimber on the show. Chawne, do you want to introduce yourself?

Chawne Kimber: Sure. Hi, I’m a professor at Lafayette College. I got my Ph.D. a long time ago at University of Florida.

KK: Go Gators!

CK: Yay, woo-hoo. I work in lattice-ordered groups.

KK: Lattice-ordered groups, very cool. I should probably know what those are, but maybe we’ll find out what they are today. So yeah, let’s get into it. What’s your favorite theorem, Chawne?

CK: Okay, so maybe you don’t like this, but it’s a suite of theorems.

KK: Even better.

EL: Go for it.

CK: So, right, a lattice-ordered group is a group, to begin with, in which any two elements have a sup and an inf, so that gives you your lattice order. They’re torsion-free, so they’re, once you get past countable ones, they’re enormous groups to work with. So my favorite theorems are the representation theorems that allow you to prove stuff because they get unwieldy due to their size.

EL: Oh cool. One of my favorite classes in grad school was a representation class. I mean, I had a lot of trouble with it. It was just representations of finite groups, and those were still really out there, but it was a lot of fun. Really algebraic thinking.

CK: Well actually these representations allow you to translate problems from algebra to topology, so it’s pretty cool. The classical theorem is by Hahn in 1909. He proved the special cases that any totally ordered Archimedean group can be embedded as a subgroup of the reals, and it kind of makes sense that you should be able to do that.

KK: Sure.

CK: And then he said that any ordered abelian group, so not necessarily lattice-ordered, can be embedded in what’s called a lexicographical product of the reals. So we could get into what that is, but those are called Hahn groups. They’re just huge products of the reals that are ordered in dictionary order that only live on well-ordered sets. So this conjecture, it’s actually a theorem, but then there’s a conjecture that that theorem is actually equivalent to the axiom of choice.

KK: Wow.

EL: Oh wow.

CK: Right?

EL: Can we maybe back up a little bit, is it possible to, for me, I really like concrete examples, so maybe can you talk a little bit about a concrete example of one of these archimedean groups? I don’t know how concrete the concrete examples are.

CK: No, they’re just really weird ways of hacking at the reals, basically, so they’re just subgroups of the reals. Think of your favorite ones, and there you go, the ones that are archimedean. And as soon as you add two dimensions of ordering, it’s even more complex, right? So the classical example that I work with would be rings of continuous functions on a topological space, and then you can build really cool examples because we all understand continuous functions, so C(X), real-valued continuous functions on a Tychonoff space, so T-3 1/2, whatever.

KK: Metric space.

CK: The axioms so you have enough continuous functions. So Gillman and Jerison in the 1950s capitalized on a theorem from the 1930s by Gelfand and Kolmogorov that said that the maximal ideals of C(X), if you take them in the hull-kernel topology, are isomorphic to the Stone-Čech compactification of the space that you’re working on. And so if you have a compact space to begin with, then your space is isomorphic to your maximal ideals. So then, just build your favorite—so C(X) is lattice-ordered, if you take the pointwise ordering, and then since the reals have a natural order on then, you pick up your sups and infs pretty easily. So there you’re starting to touch some interesting examples of these groups. Have I convinced you, Evelyn?

EL: Yeah, yeah.

CK: Okay, good. So they’re huge. You have to have some complexity in order to be able to prove anything interesting about them. So then there the Hahn embedding is pretty obvious. You just take the images of the functions. There’s too much structure in a ring like that, so maybe you want to look at just an ordered group to get back to the Hahn environment. So how can you mimic Hahn in view of Gelfand-Kolmogorov? So can we get continuous functions as the representation of an ordered group? Because the lex products that Hahn was working with are intractable in a strong way. And so then you have to start finding units because you have to be able to define something called a maximal sub-object, so you want it to be maximal with respect to missing out on some kind of unit. And so then we get into a whole series of different embedding theorems that are trying to get you closer to being able to deal with the conjecture I mentioned before, that Hahn’s embedding theorem is equivalent to the axiom of choice.

EL: Yeah, I’m really fascinated by this conjecture. It kind of seems like it comes out of nowhere. Maybe we can say what the axiom of choice is and then, is there a way you can kind of explain how these might be related?

CK: Yes and no.

KK: Let’s start with the axiom of choice.

CK: Yeah, so the axiom of choice is equivalent to Zorn’s lemma, which says that maximal objects exist. So that’s the way that I deal with it. It allows me to say that maximal ideals exist, and if they didn’t exist, these theorems wouldn’t exist. You use this everywhere in order to prove Hahn’s theorem, so that’s why it’s assumed to be possibly equivalent. This isn’t the part that I work on. I’m not a logician.

KK: So many things are equivalent to the axiom of choice. For example, the Tychonoff product theorem, which is that the product of compact spaces is compact. That’s actually equivalent to the axiom of choice, which seems a bit odd. I was actually reading last night, so Eugenia Cheng has this book Beyond Infinity, her most recent book, good bedtime reading. I learned something last night about the axiom of choice, which is that you need the axiom of choice to prove that if you have two infinities, two countable infinities, you want to think [they’re the same], it’s countable somehow. If they come with an order, then fine, but if you have two, like imagine pairs of socks, like an infinite collection of pairs of socks, is that countable? Are the socks countable? It’s an interesting question, these weird slippery things with the axiom of choice and logic. They make my head hurt a little bit.

CK: Mine too.

EL: So yeah, you’re saying that looking at the axiom of choice from the Zorn’s lemma point of view, that’s where these maximal objects are coming in in the Hahn conjecture, right?

CK: Absolutely.

KK: That makes sense.

CK: That’s kind of why I drew the parallel with this theorem about C(X), these maximal ideals being equivalent to the space you’re on. Pretty cool.

KK: Right. Because even to get maximal ideals in an arbitrary ring, you really need Zorn’s lemma.

CK: Right. And there’s a whole enterprise of people working to see how far you can peel that back. I did take a small foray into trying to understand gradations of the axiom of choice, and that hurts your head, definitely.

KK: Right, countable axiom of choice, all these different flavors.

CK: Williams prime ideal theorem, right.

KK: Yeah, okay.

EL: So what drew you to these theorems, or what makes you really excited about them?

CK: Well, you know, as a super newbie mathematician back in the day, I was super excited to see that these disparate fields of algebra and topology that everyone had told me were totally different could be connected in a dictionary way. So a characteristic of a ring can be connected is equivalent to a characteristic on a topological space. So all kinds of problems can be stated in these two different realms. They seem like different questions, but they turn out to be equivalent. So if you just know the way to cross the bridge, then you can answer either question depending on which realm gives you the easier approach to the theorem.

KK: I like that interplay too. I’m a topologist, but I’m a very algebraic one for exactly that reason. I think there are so many interesting ideas out there where you really need the other discipline to solve it, or looking through that lens makes it a lot clearer somehow.

EL: And was this in graduate school that you saw these, or as a new professor?

CK: Definitely grad school. I was working on my master’s.

KK: So I wonder, what does one pair with this suite of theorems?

CK: It’s a very hard question, actually.

KK: That’s typical. Most people find this the more difficult part of the show.

CK: Yeah. I think that if you were to ask my Ph.D. advisor Jorge Martinez what he would pair, he is very much a wine lover and an opera lover. So it would be both. You’d probably see him taking a flask into Lincoln center while thinking about theorems. So he loved to go to Tuscany, so I assume that’s where you get chianti. I don’t know, I could be lying.

KK: You do, yeah.

CK: Yeah, so let’s go with a good chianti, although that might make me sound like Hannibal Lecter.

KK: No fava beans.

CK: So we’ve got a chianti, and maybe a good opera because it’s got to be both with him. It’s hard for me to say. So he comes up to New York to do an opera orgy, just watching two operas per day until he falls down. I sometimes join him for that, and the last one I went to was Così fan tutte, and so let’s go with that because that’s the one I remember.

EL: If I remember correctly—it’s been a while since I saw or listened to that opera—there are pairs of couples who end up in different configurations, and it’s one of these “I’ll trick you into falling in love with the other couple’s person” that almost seems like the pairs being topology and algebra, and switching back and forth. I don’t know, maybe I’m putting ideas in your mind here.

CK: Or sort of the graph of the different couplings, the ordered graph could be the underlying object here. You never know.

EL: An homage to your advisor here with this pairing.

CK: Yeah, let’s do that.

EL: Well I must admit I was kind of hoping that you might pair one of your own quilt creations here. So I actually ran into you through a quilting blog you have called completely cauchy. Do you mind talking to us a little bit about how you started quilting and what you do there because it’s so cool.

CK: Yeah. Of course I chose that name because Cauchy is my favorite mathematician, and as a nerd there would be no other quilt blog named after a dead mathematician. So I am a little mortified that when you google “Cauchy complete,” as many students do, mine is actually the first entry that comes up on google.

KK: Excellent.

CK: I don’t know what that means, but okay. So yeah, when I applied for tenure, which is kind of a hazing process no matter where you are, no matter how good of a faculty member you are, I really wanted to have control, and you don’t have control at that point. And so I started sewing for fun, late at night, at 1 am, after everything kind of felt done for the day. I never imagined that I’d be doing what I’m doing today, which is using quilting to confront issues of social justice in the United States, and they’ve been picked up by museums and other venues. It’s this whole side hustle out there that I kept quiet for a long, long time. And then once I got promoted to full professor I came out of the closet.

KK: Were you concerned that having a side hustle, so to speak, would compromise your career? Because it shouldn’t.

CK: Yeah, I think something so gender-specific as quilting, something you associate with grandmas. At the end of the day, the guys I work with, I must say half of my quilts have four-letter words on them, you know, the more interesting four-letter words, so as soon as my guys saw them, they were totally on board with this enterprise, so I didn’t really need to be into the closet, but I didn’t want anybody to ever say, “Oh, she should have proved one more theorem instead of making that quilt.”

EL: Yeah.

KK: It’s unfortunate that we feel that way, right? I think that’s true of all mathematicians, but I imagine it’s worse for women, this idea that you have to work twice as hard to prove you’re half as good or something like that?

CK: Do we need to mention I’m also a black woman? So that’s actually how I was raised, you need to do three times as much to be seen as half as good, and that’s the way that I’ve lived my life, and it’s not sustainable in any way.

KK: No, absolutely not.

EL: But yeah, they are really cool quilts, so everyone should look at completely cauchy, and that’s spelled cauchy. There’s a mathematician named Cauchy. I actually have another mathematician friend with a cat named Cauchy, or who had a cat named Cauchy. I think the cat has passed away. Yeah, and I actually sew as well. I’ve somehow never had the patience for quilting. It just feels somehow like too little. I like the more immediate gratification of making a whole panel of a skirt or something. You do really intricate little piecing there, which I admire very much, and I’m glad people like you do it so I don’t have to.

KK: Sure, but Evelyn, you don’t have to make it little.

CK: You don’t.

KK: I’m sure you’ve seen these Gee’s Bend quilts, right, they’re really nice big pieces, and that can have a very dramatic effect too. But yeah, the intricate work is really remarkable. My wife has done a little quilting, and she always gets tired of it because of the fine stuff, but then she’s a book artist. She sets lead typing in her printing press by hand, and that’s fine, but piecing together little pieces of cloth somehow doesn’t work.

CK: It seems more futile, you take a big piece of fabric and cut it into small pieces so that you can sew it back together. That is kind of dumb when you think about it.

KK: Well, but I don’t know, you’ve got this whole Banach-Tarski thing, maybe.

EL: Bring it back around to the axiom of choice again.

CK: You guys are good at this.

KK: It’s not our first podcast. Well this has been great fun. Anything else you want to promote?

CK: No, I’m good.

KK: Thanks for joining us, Chawne. This has really been interesting, and we appreciate you being on.

CK: Great. Thank you.

EL: Thanks.

Show notes

Kevin Knudson: Welcome to My Favorite Theorem. I’m one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida, and here is your jet-lagged other host.

Evelyn Lamb: Hi, I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City. I’m doing pretty well right now, but in a few hours when it’s about 5 pm here, I think I will be suffering a bit. I just got back from Europe yesterday.

KK: I’m sure you will, but that’s the whole trick, right? Just keep staying up. In a couple of weeks, I’m off to that part of the world. I’ll be jet-lagged for one of the ones we have coming up right after that. That should be fun. I’ll feel your pain soon enough, I’m sure.

EL: Yeah.

KK: So today we are pleased to welcome James Tanton. James, why don’t you introduce yourself and tell everyone about yourself?

James Tanton: Hello. First of all, thank you for having me. This is such a delight. So who am I? I’m James Tanton. I’m the mathematician-at-large for the Mathematical Association of America, which is a title I’m very proud of. It’s a title I never want to give up because who doesn’t want to be a mathematician-at-large, wreaking havoc wherever one steps? But my life is basically doing outreach in the world and promoting joyous thinking and doing of mathematics. I guess my background is somewhat strange. You can probably tell I have an accent. I grew up in Australia and came to the US 30 years ago for my Ph.D., which was grand, and I liked it so much here I am 30 years later. My career has been kind of strange. I was in the university world for close to 10 years, and then I decided I was really interested in the state of mathematics education at all levels, and I decided to become a high school teacher. So I did that for 10 years. Now my life is actually working with teachers and college professors all across the globe, usually talking about let’s make the mathematics our kids experience, whatever level they’re at, really, truly joyous and uplifting.

EL: Yeah, I’ve wondered what “mathematician-at-large” entails. I’ve seen that as your title. It sounds like a pretty fun gig.

JT: So I was the MAA mathematician-in-residence for a good long while. They were very kind to offer me that position. But then I’m married to a very famous geophysicist, and my life is really to follow her career. She was off to a position at ASU in Phoenix, and then off we moved to Phoenix four years ago. So I said to the folks at the MAA, “Well, thanks very much. I guess I’m not your mathematician-in-residence anymore,” and they said, “Why don’t you be our mathematician at large?” That’s how that title came up, and of course I so beautifully, graciously said yes because that’s spectacular.

KK: Yeah, that sounds like a Michael Pearson idea, that he would just go, “No, no, we really want to keep you.”

JT: It’s so flattering. I’m so honored. It’s great because, you know, actually, it’s the work I was going to do in any case. I feel compelled to bring joyous mathematics to the world.

KK: Right. Okay, so this podcast is about theorems. So why don’t you tell us what your favorite theorem is?

JT: Okay, well first of all, I don’t actually have a theorem, even though I think it should be elevated to the status of one. I want to talk about Sperner’s lemma. So a lemma means, like, an auxiliary result, a result people use to get to other big ideas, but you know what? I think it’s charming in and of itself. So Sperner’s lemma. This was invented slash discovered back in the 1920s by a German mathematician by the name of Emanuel[] Sperner, who was playing with some combinatorial thinking in Euclidean geometry and came up with this little result.

Let me describe to you in one way first, not really the way he did it, because then I can actually explain a proof as well of the result. Imagine you have a big rubber ball, just a nice clean rubber surface, and you’ve got a marker. I’m going to suggest you just put dots all over the surface of the rubber ball, lots of dots all over the place. Once you’ve done that to your satisfaction, start connecting pairs of dots with little line segments. They’ll be little circular arcs, and make triangles, so three dots together make a triangle. Do that all over the surface of the sphere. Grand. So now you’ve got a triangulated sphere, a surface of a sphere completely covered with triangles. Each triangle for sure has only three dots on it, one in each corner, so no dots in the middle of the edges, please. All right? That’s step one.

Step two, just for kicks, go around and just label some of those dots with the letter A, randomly, and do some other dots with the letter B, randomly, just some other dots with the letter C—why not?—until each dot has a label of some kind, A, B, or C. And then admire what you’ve done. I claim if you look at the various triangles you have, you have some labeled BBB, and some labeled BCA, and some labeled BBA, and whatever, but if you find one triangle that is fully labeled ABC, I bet if you kept looking, in fact I know you are guaranteed, to find another triangle that’s labeled ABC. Sperner’s lemma says on the surface of a sphere that if there’s one fully labeled triangle, there’s guaranteed to be at least another.

EL: Interesting! I don’t think I knew that, or at least I don’t know that formulation of Sperner’s lemma.

JT: And the reason I said it that way, I can now actually describe to you why that is true because doing it on the surface of a sphere is a bit easier than doing it on a plane. Would you like to hear my little proof?

KK: Let’s hear it.

EL: Sure!

JT: Of course the answer to that has to be yes, I know. So imagine that these are really chambers. Each triangle is a room in a floor design on the surface of a sphere. So you’re in a room, an ABC room around you. You’ve got three walls around you: an AB wall, a BC wall, and an AC wall. Great. I’m going to imagine that some of these walls are actually doors. I’m going to say that any wall that has an AB label on it is actually a door you can walk through. So you’re in an ABC room, and you currently have one door you can walk through. So walk through it! That will take you to another triangle room. This triangle room has at least one AB edge on it, because you just walked through it, and that third vertex will have to be an A, B, or C. If it’s a C, you’re kind of stuck because there are no other AB doors to walk through, in which case you just found another ABC room. Woo-hoo, done!

EL: Right.

JT: If it’s either A or B, then it gives you a second AB door to walk through, so walk through it. In fact, just keep walking through every AB door you come to. Either you’ll get stuck, and in fact the only place you can possibly get stuck is if there’s exactly one AB door, in which case it was an ABC triangle, and you found an ABC triangle. Or it has another door to walk through, and you keep going. Since there’s only a finite number of triangles, you can’t keep going on indefinitely. You must eventually get stuck. You must get stuck in an ABC room. So if you start in one ABC room, you’ll be sure to be led to another.

EL: Oh, okay, and you can’t go back into the room you started in.

KK: That was my question, yeah.

JT: Could you possibly return to a room you’ve previously visited? Yes, there’s a subtlety there. Let’s argue our way through that. So think about the first room that you could possibly—if you do revisit a room, think of the first room you re-enter. That means you must have gone through an AB door to get in. In fact, if you’ve gone through that room before, you must have already previously used that AB door to go into and out of it. That is, you’ve used that AB door twice already. That is, the room you just came from was a previously revisited room. You argue, oh, if I think this is the first room I’ve visited twice, then the room you just came from, you’re wrong. It was actually that room that you first visited twice. Oh, no, actually it was the one before that that you first visited twice. There can be no first room that you first visited twice. And the only way out of that paradox is there can be no room that you visit twice.

EL: Okay.

JT: That’s the mind-bendy part right there.

EL: I feel like I need a balloon right now and a bunch of markers.

JT: You know, it’s actually fun to do it, it really is. But balloons are awkward. In fact, the usual way that Sperner’s lemma is presented, I’ll even not do it in the usual way. Sperner did it on a triangle. I’ll do it on any polygon. This time, this we can actually do with markers, and it’s really fun to actually do it. So draw a great big polygon on a page and then triangulate it. Fill its interior with dots and then fill in edges so you’ve got all these triangles filling up the polygon. And then randomly label the dots A, B, or C in a random, haphazard way. Make sure that you have an odd number of AB doors on the outside edge of that polygon. If you do that, no matter what you do, you cannot escape creating somewhere on the interior a fully labeled ABC triangle. The reason is, you just do this thing. Walk from the outside of the polygon through an AB door, an outside AB door, go along on a journey. If you get stuck, bingo! You’re on an ABC triangle. Or you might be led out another AB door back to the big space again. But if you have an odd number of AB doors on the outside, you’re guaranteed to have at least one of those doors not leading outside, meaning you’ve been stuck on the inside. It’s guaranteed to lead to an ABC triangle in the middle of the polygon.

EL: Okay, and this does require that you use all three—is it a requirement that you use all three letters, or does the odd number of things…I don’t know if my question makes sense yet.

JT: There’s no rules on what you except on the outside, please give me an odd number of AB doors.

EL: Okay.

JT: And there’s nothing special about the letters A and B. You could do an odd number of BC doors or an odd number of AC doors.

EL: Right.

JT: What you do on the interior is up to you. Label them all A, I dare you, and you’ll still find an ABC triangle.

EL: Okay.

JT: Isn’t that crazy?

KK: Okay, so why did Sperner care?

JT: Why did Sperner care? Well he was just playing around with this geometry, but then people realized, as one of your previous guests mentioned, the fabulous Francis Su, that this leads to some topological results, for example, the Brouwer fixed-point theorem, which people care about, and you should listen to his podcast because he explains the Brouwer fixed-point theorem beautifully.

EL: Yes, and he did actually mention to us in emails and stuff that he is actually quite fond of Sperner’s lemma also, so I’m sure he’ll be happy to listen to this episode.

JT: In some cases, Sperner’s lemma is kind of special because people knew Brouwer’s fixed-point theorem before Sperner, but they had very abstract nonconstructive proofs of the theorem. Fixed points, when you crumple pieces of paper and throw them on top of themselves, fixed points exist, but you can know that and not know to find them. Sperner’s lemma, if you think about it, is giving you a way to possibly find those ABC triangles. Just start on the outside and follow paths in. So it gives you a kind of hope of finding where those fixed points might actually lie, so it’s a very sort of constructive type of thinking on this topological result that is proved abstractly.

One thing that Francis Su did not mention is the hairy ball theorem, which I think is a lovely little application of Sperner’s lemma, which goes back to the spheres. Spheres—in my mind, this is how I was first thinking about Sperner’s lemma. So I don’t know if you know the hairy ball theorem. If you take a tennis ball with the little fur, the little hairs at, ideally, every point of the sphere, but that’s not really possible. But we can imagine in our mind’s eye a hairy ball. If you try to comb those hairs flat, tangent to the surface all the way around—well, maybe there would be a little angle, something like that. But as long as you don’t do anything crazy, you know, it’s a nice, smooth, continuous vector field on the surface of the sphere, just these hairs, close hairs go towards the same direction, very smoothly, nothing abrupt going on, then you are forced to have a cowlick, that is, one hair that sticks straight up. That is, you are forced to have a tangent vector that is actually the zero vector. You can actually prove that with Sperner’s lemma.

EL: Wow.

JT: Yeah, and the way you do that is: choose one point, like the North Pole, and imagine a little magnet there, and you can imagine all the magnetic fields make these two circular, the magnetic field of a dipole, sorry, I have to think back to my physics days. So you’ve got these natural lines associated with that magnet all over the sphere, so I suggest just triangulate the sphere. Just draw lots of little triangles all over it. And then at each vertex of the triangle, you’ve got this vector field, and you’ve got these hairs all over the vector field. At any point on the triangle, look at the direction the hair is pointing compared to the direction of the magnetic field. And you can label that either A, B, or C by doing the following. Basically you’ve got 360 degrees of possible differences of directions between those things. So if it’s in one of the first 0-120 degrees of counterclockwise motion, label it A. If it’s between the 120 and 240 mark, label it B. If it’s between the 240 to 360 mark, label it C. There is a way to label that triangulation based on the direction of the hairs on the surface of the sphere. Bingo! So we’ve just now proved that in any triangulation, you can argue that you arrange things at the pole as an ABC triangle, there’s this little thing you can arrange, then there has to be some other ABC triangle somewhere on the sphere. That is, there’s a little small region where you’ve got three hairs trying to point in three different directions. And do finer and finer triangulations. You actually argue the only way out of that predicament is there’s got to be one hair that’s pointing three directions at the same time, that is, the zero vector.

KK: That’s very cool.

EL: Yeah.

JT: I just love these. These things feel so tangible. I just want to play with them with my hands and make it happen. And you can to some degree. Try to comb a fuzzy ball. You have a hard time. Or look at a guinea pig. They’re basically fuzzy balls, and they always have a cowlick. Always.

KK: Are there higher-dimensional generalizations of this? This feels very much two-dimensional, but I feel there’s an Euler characteristic lurking there somewhere.

JT: Absolutely you can do this in higher dimensions. This works in any dimension. For example, to make this three-dimensional, stack all these tetrahedra together. Take a polyhedron, triangulate it. If there’s an odd number of ABC faces on the outside, then there’s guaranteed to be some ABCD tetrahedron in the middle. And higher dimensions. And people of course play with all sorts of variations. For example, I’ll go back to two dimensions for a moment, back to triangles. If three different people create their own labeling scheme, so you’ve got lots of ABC triangles around the place, then there’s guaranteed to be one triangle in the middle, so if you chose one person’s label for the vertex, the second person’s for the second vertex, the third person’s label for the third vertex, according to their labels, which are all different, that’s an ABC triangle in this sort of mixed labeling scheme. So they call these permutation results of Sperner’s lemma and so forth. Just mind-bendy, and in higher dimensions.

EL: So was this a love at first sight kind of theorem for you? What were your early experiences with it?

JT: So when did I first encounter it? I guess when I studied the Brouwer fixed-point theorem, and when I saw this lemma in and of itself—and I saw it in the light of proving Brouwer’s fixed-point theorem—it just appealed to me. It felt hands-on, which I kind of love. It felt immediately accessible. I could do it and experience it and play with it. And it seemed quirky. I liked the quirky. For some reason it just appealed to me, so yes, it appealed to all my sensibilities. And I also have this thing I’ve discovered about me and my life, which is that I like this notion that I’m nothing in the universe, that the universe has these dictates. For example, if there’s one ABC triangle, there’s got to be another one. I mean, that’s a fact. It’s a universal fact that despite my humanness I can do nothing about it. ABC triangles just exist. And things like the “rope around the earth” puzzle: if you take a rope and wrap it around the Equator, add 10 feet around the rope and re-wrap, you’ve got 19 inches of space. What I love about that puzzle, if you do it on Mars 10 feet from its Equator, it’s 19 inches of space. Do it for Jupiter: it’s 19 inches of space. Do it for a planet the size of a pea: it’s 19 inches of space. You cannot escape 19 inches. That sort of thing appeals to me. What can I say?

KK: So you are a physicist?

JT: Don’t tell anyone. My first degree was actually in theoretical physics.

KK: So the other fun thing we do on this podcast is we ask our guest to pair their theorem, or lemma in this case, with something. So what have you chosen to pair Sperner’s lemma with?

JT: You know, I’m going with a good old Aussie pavlova.

EL: Excellent.

JT: And I’ve probably offended all the people from New Zealand because they claim it’s their pavlova. But Australians say it’s theirs, and I’ll go with that since I’m an Aussie. And why that, you might ask?

EL: Well first can we say what a pavlova is in case, so I only learned what this was a couple years ago, so I’m just assuming—I was one of the lucky people who learned about this in making one, which was delicious, so yeah.

JT: First of all, it’s the most delectable dessert devised my mankind, or invented, or discovered. I’m not sure if desserts are invented or discovered. That’s a good question there. So it’s a great big mount of meringue, you just build this huge blob of meringue, and you bake it for two hours and let it sit in the oven overnight so it becomes this hard, hard outer shell with a soft, gooey meringue center, and you just slather it with whipped cream and your favorite fresh fruits. And my favorite fruits for a pavlova are actually mango and blueberries together. That’s my dessert. But you know, well maybe to think of it, there are actually two reasons. I happen to know it was invented in the 1920s, the same time Sperner came up with this lemma, which is kind of nice. But any time I bake one—I bake these things, I really enjoy baking desserts—it kind of reminds me of a triangulated sphere because you’ve got this mound of meringue, and you bring it out of the oven, and it’s got this crust that’s all cracked up, and it kind of looks like a triangulation of a polyhedron of some kind. So it has that parallel I really like. So pavlovas bring as much joy to my life as these quirky Sperner’s lemma type results, so that’s my pairing.

EL: So they’re not, so I went to this Australia-themed potluck party a couple years ago, and I decided to bring this because I was looking for Australian foods, so I got this. I was pretty intimidated when I saw the pictures, but it’s actually, at least I found a recipe and it looked good, and it worked the first time, more or less. I think you can handle it.

JT: It is a showstopper, but it’s so easy to make. Don’t tell anyone, it’s ridiculously easy, and it looks spectacular.

KK: Yeah, meringues look like something, but really, you just have to be patient to whip the whites into something, and then that’s it. It works.

JT: Then you’re done. It kind of works. You can’t overcook it. You can undercook it, but then it’s just a goopy delicious mess.

KK: Right. So we also like to give our guests a chance to plug various things. I’m sure you’re excited to talk about the Global Math Project.

JT: Of course I’m going to talk about the Global Math Project. Oh my goodness. You know, when I mention I’m kind of a man on a mission to bring joyous, uplifting mathematics to the world, I’m kind of trying to live up to those words, which is kind of scary. But let me just say something marvelous, really marvelous and humbling, happened last October. We brought a particular piece of mathematics to the world, a team of seven of us, the Global Math Project team, not knowing what was going to happen. It was all volunteer, grassroots, next to no funding, we’re terrible at raising funding, it turns out. But it really was believing that teachers, given the opportunity to have a real joyous, genuine, human conversation about mathematics with their students, that’s actually classroom-relevant mathematics. Classroom mathematics is a portal to the same mystery, delight, intrigue, and wonder, they will. Teachers are our best advocates across the globe for espousing beautiful, joyous, uplifting mathematics. So we presented a piece of mathematics called Exploding Dots, and we invited teachers all around the globe to do that, to have just some experience on this topic with their students during Global Math Week last October, and they did. We had teachers from 170 different countries and territories, all of their own accord, reach out to about 1.77 million students just in this one week. Phenomenal. And this is school-relevant mathematics. So we’re doing it again! Why not?

EL: Oh, great!

JT: So this year, 10/10, we chose that date because it’s a universal date. No matter how you read it, it’s the tenth of October. We’re going to go up to 10 million students with the same story of Exploding Dots. So I invite you, please look up Global Math Project, go Google Exploding Dots. See what we’re bringing to the world. And on its own accord, in the last number of months, we’ve now reached 4.6 million students across the globe, so 10 million students sounds outlandish, but you know what? We might actually do this. And it’s just letting the mathematics, the true, joyous mathematics, simply shine for itself, just get out of its way. And you know what? It happens. Math can speak for itself. Welcome to Global Math Project.

EL: Yeah. We’ll include that in the show notes, for sure.

KK: In fact, this is June that we’re taping this, recording this. Taping? I’m dating myself. We’re recording this in June. So just this weekend Jim Propp had a very nice essay on his Mathematical Enchantments blog about this, about Exploding Dots. I’d seen some things about it, so I knew a little about it. It’s really very lovely, and as you say relevant.

JT: I’m glad you mentioned Jim Propp. I was about to give a shout out to him as well because he wrote a beautiful piece, and it’s this Mathematical Enchantments blog piece for June 2018. Worth having a look at. Absolutely. What I love about this, it really shows, I mean, Exploding Dots is the story of place value, as simple as that. But it really connects to how you write numbers, what you’re experiencing in the early grades. It explains, if you think of it in one particular way, all the grade school algorithms one learns, goes through all of high school polynomial algebra, which is just a repeat of grade five, but no one tends to tell people that. Why stop at finite things? Go to infinite things, go to infinite series and so forth, and start getting quirky. Not just playing with 10-1 machines with base 10 and 2-1 machines with base 2, start playing with 3-2 machines and discover base 1 1/2, start playing with 2-negative 1 machines and discover base -2, and you get to unsolved research questions. So here’s this one simple little story, just playing with dots in boxes, literally—like me playing with dots on a sphere; I seem to be obsessed with dots in my life—takes you on a journey from K through 5 through 8 through 12 to 16, and on, all in one astounding fell swoop. This is mathematics for you. Think deeply about elementary ideas, and well, it’s a portal to a universe of wonder.

KK: That’s why we’re all here, right?

JT: Indeed. So let’s help the world see that together, from the young’uns all the way up.

KK: All right. Well, this has been great fun. I knew Sperner’s lemma, sort of in the abstract, but I never really thought of it too closely. So I’m glad that I can now prove it, so thank you for that.

EL: Yeah, I’m going to sit down and make sure you’re not pulling my leg about that. I think the odd number of AB’s is key here.

JT: Absolutely. The odd number of AB outside edges is key.

EL: Right.

JT: Because you could walk through the door and out a door, so pairs of them could cancel each other out. So play with them.

EL: I bet if I start drawing, I’ve been restraining myself from going over here to the side.

JT: Well you know, sketches work so well on a podcast.

KK: That’s right.

JT: Absolutely, play. That’s all mathematics should be, an invitation to play. Go for it.

EL: Yeah, thanks a lot for being here.

JT: My pleasure. Thank you so much.

[outro]

Show notes

Evelyn Lamb: Hello and welcome to My Favorite Theorem, a podcast where we ask mathematicians to tell us about their favorite theorems. I’m your host Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah. This is your other host.

Kevin Knudson: I’m Kevin Knudson, a professor of mathematics at the University of Florida. How’s it going?

EL: Great. I’m excited about a new project I’m working on that is appropriate to plug at the beginning of this, so I will. So I’ve been working on another podcast that will be coming out in the fall, may already be out by the time this episode is out. It’s with the folks at Lathisms, that’s L-A-T-H-I-S-M-S, which is a project to increase visibility and recognition of Hispanic and Latinx mathematicians. And our guest today is going to be a guest on that podcast too, so I’m very excited to introduce our guest, who is Erika Camacho. Hi, Erika. Can you tell us a little bit about yourself?

Erika Camacho. Sure. So I’m an associate professor at Arizona State University. My concentration is, well I’m a professor of applied mathematics, and my concentration is mathematical physiology, mainly focusing in the retina and modeling the retina and the deterioration of photoreceptors. And I’m in the west campus of Arizona State University, which is mainly focusing, it’s both a research and student focused institution, so it’s kind of like a hybrid between what you would call more of a research place and also a liberal arts education.

EL: Cool.

KK: Very nice. Which city is that in?

EC: We’re in Glendale, the west valley of Arizona, Phoenix greater area.

EL: I was in Arizona not too long ago, and the time zone is always interesting there because it’s exactly south of Utah, but I was there after Utah and most of the country went to daylight saving time, and most of Arizona doesn’t observe that, so it was kind of fun. I also went through part of the Navajo Nation there that does observe daylight saving time, so I changed time zones multiple times just driving straight north, which was kind of a fun thing.

EC: It is very confusing. Let’s say you have an event that you’re going to, and you’re driving to one where it’s say in some of the Navajo Nation, and you don’t realize that you might miss some of your event because of the time change. You’re just driving and you’re crossing the border where it changes to a different time zone. It takes a while to get adjusted to. I missed a flight one time for the same reason. I was not aware that Arizona didn’t observe daylight saving. Now I’m aware.

KK: I actually have a theory that someone could run a presidential campaign, and their sole platform is that they would get rid of daylight saving time, and they would win in a landslide.

EL: I mean, people have won on less.

KK: Clearly.

EL: So Erika, we invited you here not to chat about time zones or presidents but to chat about theorems. So what is your favorite theorem?

EC: Before I say my favorite theorem, like I said, I am an applied mathematician. So I focus on modeling. And in modeling, there’s a lot of complexities, a lot of different layers and levels where you’re trying to model things. So many of the systems you’re trying to develop as you create this model tend to be nonlinear models. Many times I’m looking at how different processes change over time. So many of the processes I work with are continuous. So I work with differential equations, and they tend to be nonlinear. Sometimes that’s where the complexity comes in, trying to analyze nonlinear systems, and the most accurate way, the way that we’re going to get the most insight into some of the behavior we’re looking for in terms of physiological systems that relate to the retina and retinal degeneration, one of the things that we’re really looking at is what happens in the long run? How is it that photoreceptors degenerate over time, and can we do something to stop the progression of blindness or the progression of certain diseases that would cause the photoreceptors to degenerate? So we’re really asking what are the long-term solutions of the system, and how did they evolve over time? So we’re looking for steady states. We’re looking for what is their stability and what are the changes in the processes or the mechanisms that govern those systems, which usually are defined by the parameters that end up actually leading to a change in the stability of the equilibria? And they could take the system to another equilibrium that is stable now. So in physiological terms, to another pathological state, or another state that we could hopefully do a few strategies to prevent blindness. So that’s the setting of where I come from, and when you asked me this question, what is my favorite theorem, it was hard because as applied mathematicians we utilize different theory. And all the theory is useful, and depending on what the question is, then the mathematics that are utilized are very different.

So I thought, “What is the theorem that is utilized the most in the case where we’re looking at nonlinear systems and we’re trying to analyze them? And one of the most powerful theorems out there, which is one that has almost become addicting, that you use it all the time, is the Hartman-Grobman theorem. I say addicting because it’s a very powerful theorem. It allows us to take a nonlinear system and in certain cases be able to analyze it and be able to get an accurate depiction of what’s happening around the equilibrium point, what is the qualitative behavior of the system, what are the solutions of the system, and what is their stability. Because you’re looking at, in most cases, a continuous system, you can map it and be able to kind of piece it together.

EL: So it’s been a long time since I took any differential equations. I’m a little embarrassed, or did any differential equations.

KK: Me too.

EL: So can you tell us a little more about the setting of this theorem?

EC: So the Hartman theorem, like I said, is a theorem that allows us to study dynamical systems in continuous time. It’s very powerful because it gives us an accurate portrayal of the flow, solutions of the nonlinear system in a neighborhood around a fixed point, the equilibrium, the steady state. So I’m going to be using fixed point, equilibrium, and steady state interchangeably. In some cases, and in the cases where it does help, is in the cases where the equilibrium that we’re looking at, the eigenvalues of the linearized system, or the nonlinear system that we’re looking at, actually has nonzero real part. In other words, we’re looking at hyperbolic equilibrium points. That’s when we could actually apply this system.

KK: Okay.

EC: This theorem, otherwise we cannot. That is, for certain cases? The standard technique is you look at your nonlinear system, you linearize it through a process, and you’re able to then shift your equilibrium point to the origin, and now you’re considering the linearized system, and that system, the Jacobian you obtain through the linearization has eigenvalues that have nonzero real part. Then you’re able to apply the Hartman theorem, which tells you that there is this homomorphism from the nonlinear system, the flow and the solutions of the nonlinear system, locally, to the actual linear system. And now everything that you get that you would normally be able to see analyzed in a linear system locally, you’re able to do it for the nonlinear system. So that’s where the powerful thing comes in. Like I said, the gist of it is that the solutions of the nonlinear system can actually be approximated by a linear system, but only in a neighborhood of the equilibrium point. And this is only in the case where we have hyperbolic equilibrium fixed points. But that is very powerful because that allows us to really get a handle on what’s going on locally in a neighborhood of the steady state. For us, we’re looking at, say, how certain diseases progress in the long run, where are we heading? Where is the patient heading, in terms of blindness? And it really allows us to be able to move in that direction in terms of understanding what is going on. And like I said, it’s powerful not just because it’s telling us about the stability, but it’s actually telling us the qualitative structure of the solution and the behavior, right, of your solutions, locally are the same in the linear case and the nonlinear case because of this topological equivalence.

KK: That’s pretty remarkable. But I guess the neighborhood might be pretty small, right?

EC: Right. The neighborhood is small.

KK: Sure.

EC: In nonlinear systems, you have plenty of different equilibrium points around those neighborhoods, right, but again remember that your solutions in the phase space are changing continuously, so you are able to kind of piece together what is going on, more or less, but for sure you know what’s going on in the long term behavior, you know what’s going on around that neighborhood, and for given initial conditions, which is really key in math applications because sometimes we’re asking what happens for different initial conditions. What are the steady states? What do the solutions look like in the long run? What do things look like, what is going on for different initial conditions?

KK: So if you’re modeling the retina, how many equations are we talking? How big are these systems?

EC: Well, that’s the thing. In the very most simplified case, where you’re able to divide the photoreceptors into the rods and cones, then you have two populations.

KK: Okay.

EC: And in one of the cases we’re looking at the flow of nutrients, so we are also considering the retinal pigment epithelium cells, which is another population, so you have three equations in that case. So that’s a more simplistic situation, but it’s a situation where we have been able to really get a sense of what’s going on in terms of degeneration in these two classes of photoreceptors that undergo a mutation. So one of the diseases I work on is retinitis pigmentosa, and the reason why that is a very complicated case that we haven’t been able to really get a handle on and be able to come up with better therapies and better ways of stopping degeneration of the photoreceptors—in fact there is no cure for stopping photoreceptors from degenerating—is because the mutation happens in the rods. The rods are the ones that are ill. Yet the cones die, which are perfectly healthy. And trying to understand how is it that the rods actually are communicating with the cones that ends up also killing them is an important part, and with a very simplistic model for an undiseased case, we were able to actually, before biologically this link was discovered, that in fact the photoreceptors produced this protein that is called the rod-derived cone viability factor, that helps the cone survive, and we were able to show that mathematically just by analyzing the equilibria and being able to look at different things in the long run, and the invariant spaces, and being able to show what we know just by basic biology of what happened to the rods and the cones and then realized that the communication had to be a one-way interaction from the rods to the cones. So that’s one of the models that we have. And then once we had that handled, we were able to introduce the disease and look at a four-dimensional system.

Now we’re looking internally at the metabolic process inside the cones because there’s a metabolic process. So the rods produce this protein. How is that protein taken by the rods, and what does it do once it’s inside the rods? For that we really need to look inside the metabolic process and the kinetics of the cones and also the rods. There, if you’re just considering the cones, you’re looking at 11-12 differential equations.

KK: Wow.

EL: Wow.

EC: With many parameters. So at that point we’re going to a much higher dimension. And that’s where we currently are. But that has given us a lot of insight, not just in how the rods help the cones but how is it that other processes are being influenced, getting affected? And again, where the Hartman-Grobman theorem applies is to autonomous systems, where time is not explicit in the equations.

EL: Okay.

KK: This is fascinating.

EL: Math gets this kind of rap for being really hard, but then you think, like, math is so much simpler than this biological system. Your rods being sick make your cones die!

EC: But I think the mathematics is essential. There’s a big cost in taking certain experiments to the lab, just to be able to understand what is going on. There’s a cost, there’s a time dependence, and math bypasses that. So once you have a mathematical model that is able to predict things. That’s why you start with things that are already known. Many times the first set of models that I create are models that show what we already know. They’re not giving any new insight. It’s just to show that the foundation is ready and we can build on it, now we can introduce some new things and be able to ask questions about things we don’t know. Because once we are able to do that, really it’s able to guide us to places, or at least indicate what kinds of lab studies and experiments should be run and what kinds of things should be focused on. And that’s one of the things we do. For example, one of the collaborators I work with is the Vision Institute of Paris, so the institute and the director there, and the director of genetics as well. And we have this collaboration where I think working together has really helped guide their experiments and their understanding of where they should be looking, just as they helped me really understand what are the types of systems we need to consider and what are the things that we can neglect, that we don’t have to really focus on? And I think that’s the thing, mathematics is really powerful to have in biological system, I think.

EL: Yeah.

EC: And my favorite theorem can be used to gain insight into photoreceptor degeneration in very complicated systems. Another thing that’s interesting about the Hartman-Grobman theorem is that one of the things that is really powerful is that you don’t have to find a solution, a solution to the nonlinear system, to get an understanding of what’s going on and get an insight into the qualitative behavior of those solutions. And I think that is really powerful. Do you have any questions?

EL: So, I mean, a lot. But something I always think is interesting about applied mathematicians is that often they end up working in really different application areas. So did you start out looking at retinas and that kind of biological system, or did you start out somewhere else in applied math and gradually move your way over there?

EC: So when I started in applied math, what I really liked was dynamical systems. Yes, the first project I worked on as a graduate student was actually looking at the cornea and how different light intensities affect the developing cornea. And for that I really had to learn about the physiology of the eye and the physiology of the retina. But then I did that for graduate school, and initially once I went out of graduate school, in my postdoc I was working on how different fanatic groups get formed.

EL: Oh wow, really different application.

EC: Which was in Los Alamos. I was looking at what are the sources of power that allow groups that can become terrorists, for example, to really become strong. What are the competing forces? So it was more a sociological application, but again using dynamical systems to try to understand it. Later on I moved on to a more general area of math biology, looking at other different systems and diseases, but then I went back through an undergraduate project in an REU. Usually the way I work with undergraduates is I make them be the ones that ask the question, that select the application. And I tell them, you have to go learn all about it because you have to come and teach me. And then from there I’m going to help you formulate the questions that can be put into a mathematical equation and that can be modeled somehow. And they were very interested, they wanted to do something with a PDE. And they thought, well, something with the retina. And they thought that retinitis pigmentosa would be perfect for modeling with a PDE and being able to analyze it that way. And then as I learned more about the disease, the interesting thing is that when the cones begin to die, like the rods are the ones that are sick, but when the cones begin to die, there is no spatial dependence anymore. They don’t die in a way that you can see this spatial dependence. It’s really more random, and it’s more dependent on the fact that there is this lack of protein that is not being synthesized by the rods anymore. Many times what happens is there is this first wave of death of photoreceptors where the rods die, and when most of then die, when 90 percent of them are gone, then the cones begin to die. And then there are all these other things about, yes, you can think about wakes and the velocity of them, but there is not this spatial dependence. Initially it is, but that’s when only the rods are dying. But when we are really interested in asking the questions about why the cones die, there’s not that case anymore.

KK: It’s sort of a uniformly distributed death pattern, as it were? What I love about this is, you know, here’s a problem that basically a second-year calculus student can understand in some sense. You have two populations. We teach them this all the time. You have two populations, and they’re interacting in some way. What’s the long-term behavior? But there are still so many sophisticated questions you can ask and complicated systems there. Yeah, I can see why your undergrads were interested in this, because they understood it immediately, at least that it could be applied. And then they brought this to you, and now you’re hopefully going to cure RP, right?

EC: Right, well another thing is that you can understand, and you can use math that is not very high-level to start to get your hands dirty. And for example, now that we’re looking at this multi-level layer where you’re looking at the molecular level and also at the cellular level, then you’re really asking about multi-scale questions and how can we better analyze the system when we have multiple scales, right? And then there are sometimes questions about delay. So the more focused and the more detailed the model becomes, the more difficult the mathematics becomes.

KK: Sure.

EC: And then there are also questions, for example, without the mathematics, there’s a lot of interesting mathematics going on, I’m sorry, I mean without the biology, that you could analyze with mathematics. We did a project like that with a collaborator where the parameter space was not really relevant biologically, but the mathematics was very interesting. We had all this different behavior. We had not just equilibrium points, but we had periodic solutions, torus, we had all of this, and what is going on? And a lot of this happened in a very small region, and it just became more of a mathematical kind of analysis rather than just a biological one.

EL: Yeah, very cool. So another part of this program is that we like to ask our guest to pair their theorem with something, you know, food, beverage, music art, anything like that. So have you chosen a pairing for the Hartman-Grobman theorem?

EC: I thought about it a lot, because like I said it’s such a powerful theorem, and I go back to the idea of it’s addicting. I think anyone who’s worked in dynamical systems in the nonlinear case in a continuous timeframe definitely utilizes this theorem. It comes to a point where we are doing it automatically. So I thought, what is something that I consider very addicting, yet it looks very simple, right? It’s elegant but simple. But once you have it, it’s addicting. And I could not think of anything else but the Tennessee whiskey cake. Have you ever had it?

KK: No.

EL: No, but it sounds dangerous.

EC: It is delicious. It’s funny, I don’t like whiskey, and I had it when I went to San Antonio to give a talk one time. I was like, well, okay, everyone wanted it, so I decided to go with it. I usually pick chocolate because that is my favorite.

EL: Yeah, that’s my go-to.

EC: I love chocolate. So I said, well, let me try it. It was the most delicious thing. Now I want to be able to bake it, make it. I had a piece, and I want more.

KK: So describe this cake a little bit. Obviously I get that it has bourbon in it.

EC: The way it’s served is it’s served warm, and it has vanilla ice cream. It has nuts, and it has this kind of butterscotch or sometimes chocolate sauce over it. And it’s very moist. It has those different layers. I also think, right, in terms of complexity, it has these different layers. In order to get a sense of the power of it, you have to kind of go through all the layers and have all of them in the same bite. And I feel like that with the Hartman theorem, right, that the power of it is really to apply it to something that has nonlinearity, that is really complex, and something that you know you might not be able to get a handle on the solutions analytically, but you still want to be able to say what is going on, what is the behavior, where are we heading? To somehow be able to infer what the solutions are through a different means, to be able to go around, and it gives you that kind of ability.

KK: And this is where whiskey helps.

EC: Well the whiskey’s the addicting part, right?

EL: So have you made this cake at all, or do you usually order it when you’re out?

EC: Usually I order it when I’m out. But I want to make it. So my mom’s birthday is coming up on August 3rd, and I’m going to try to make it. I was telling my husband, “We’re going to have to make it throughout the next few days because I’m pretty sure we’re going to go through a few trials.”

KK: Absolutely.

EC: I can never get it right.

EL: Even the mistakes will be rewarding, just like math.

KK: And again the whiskey helps.

EC: But that’s an interesting question. I thought, what can I pair it with? And the only thing I could think of was something that’s addicting or something that has multiple layers but that all of them have to be taken at once, that you’re allowed to look at all of them at once.

EL: This fits perfectly.

KK: Sounds great. Well I’ve learned a lot today. I’ve never thought about modeling the eye through populations of rods and cones, but now that you say it, I guess sure, of course. And now I have to look up Tennessee whiskey cake.

EL: Yeah, it’s really good. You should try it.

KK: I’m going to go do that.

EL: It’s almost lunch here, so you’re definitely making me hungry.

KK: Well thanks a lot for joining us.

EL: Thanks a lot for being here.

EC: Well thank you so much for having me here. I really enjoyed it.

Show notes