Did you do anything fun on your summer holidays? The mathematicians in this podcast spent some of their summer helping to create the perfect smoothie, getting the most sugar out of sugar cane, and attacking other important real-world problems.

They did all this attending the Graduate Modelling Camp, which is organised every year by the Newton Gateway to Mathematics in Cambridge. The camp gives early career mathematicians a chance to experience life as a mathematical modeller by challenging them to solve problems posed by industry.

We talk to Chris Breward, who has been helping to run the modelling camp for many years, to PhD students Emily Cook, Julian Glover, and Michael Smah, who attended this year's camp, and to Ashleigh Hutchinson who took part as a mentor. They all enjoyed the camp immensely — find out why in this podcast!

This content was produced as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI) and the Newton Gateway to Mathematics. The INI is an international research centre in Cambridge which attracts leading mathematicians from all over the world. The Newton Gateway is the impact initiative of the INI, which engages with users of mathematics. You can find all the content from the collaboration here.

There's been some huge news in the world of cosmology: for the first time scientists have detected a low frequency hum of gravitational waves. The new results were published by the North American Nanohertz Observatory for Gravitational Waves, NANOGrav for short. The NANOGrav team were not alone — they coordinated with collaborations in Europe, India, Australia, and China, which released similar findings at the same time.

In this podcast we find out what these new results mean, and why they're so exciting, with Michalis Agathos, Amelia Drew, and Ulrich Sperhake of the Stephen Hawking Centre for Theoretical Cosmology at the University of Cambridge. Join us on this fascinating, and slightly mind-bending, cosmic ride!

To find out more about the topics discussed in this podcast see:

• Maths in a minute: Gravitational waves

You might also want to listen to our recent podcast A new map of dark matter.

The illustration above is an artist's rendering of black hole binaries emitting gravitational waves, produced by Olena Shmahalo for NANOGrav and reproduced here under CC BY 4.0.

People don't usually think about maths and literature as related subjects, but it turns out that there are plenty of connections between the two. In this podcast we talk to mathematician Sarah Hart about her brilliant book Once upon a time: The wondrous connections between mathematics and literature.

Sarah tells us about the links between poetry and mathematical proof, the maths of Moby dick and the The luminaries, and why mathematical patterns and references can enhance your enjoyment of a book even when you're not aware of them.

To find out more about the problem of squaring the circle, which is mentioned in this podcast, see Mathematical mysteries: Transcendental meditation. To find out more about conic sections, see here.

"I think I'll stop here." This is how, on 23rd June 1993, Andrew Wiles ended his series of lectures at the Isaac Newton Institute (INI), our neighbour here at the Centre for Mathematical Sciences. The applause, so witnesses report, was thunderous. Wiles had just announced a proof that had eluded mathematicians for over 350 years: the proof of Fermat's Last Theorem.

Wiles' announcement, 30 years ago today, was a thrilling moment in mathematical history. But Fermat's Last Theorem is not just the story of one person. Jack Thorne, who works on new mathematics that builds on Wiles' proof, told us that it is actually a story of people talking to each other over a period of centuries.

To celebrate 30 years since that exciting moment, we were lucky enough to speak with Andrew Wiles and Jack Thorne, and also to Tom Körner, who was there the day Wiles announced the proof.

This is a special joint episode with the INI's Living Proof podcast, made in collaboration with our friend Dan Aspel, from the INI.

You can find out more about Fermat's Last Theorem in the article that accompanies this podcast, and in this collection of further reading.

This podcast was produced as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI) – you can find all the content from the collaboration here.

The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.

Chocolate and mayonnaise are two of our all time favourite foods, so we were very happy to get the chance to talk to Valerie Pinfield, Professor of Engineering at Loughborough University, who has used maths to work on both chocolate and mayonnaise.

We talked to Valerie at the Isaac Newton Institute for Mathematical Sciences in Cambridge where Valerie is currently co-organising a research programme on the mathematical theory and applications of multiple wave scattering. As we will find out, this has a huge range of applications, from understanding mayonnaise to making invisibility cloaks.

While Valerie is a professor at the University of Loughborough now, her career path has also involved work in industry and time out for kids, so we also asked her for some advice for women mathematicians and scientists wondering how to build their career.

This podcast was produced as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI) – you can find all the content from the collaboration here.

The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.

You might have heard in the news recently that musician Ed Sheeran was being sued for similarities between his song Thinking out loud from 2014, and Marvin Gaye's song Get it on from 1973. But, given the way we write music to fit into specific genres, is it possible to write unique music with the limited quantity of notes and chords available? In this podcast Oli Freke, our favourite music correspondent, answers this question and plays us real examples of the connections between maths and music.

You can find out more about the maths in music in Oli's articles – How many melodies are there? and Sine language. And you can find more of Oli's music, and his book "Synthesizer Evolution", here!

The music in this podcast comes from, of course, Oli Freke! The track is called Funk Off.

Did you know that we don't know what 85% of the stuff in our Universe is made of? This mysterious 85% is known as dark matter. We can't see it because it doesn't emit or reflect light, but we know it's there because it exerts a gravitational pull on stars and galaxies, and also bends the path of light.

In this podcast we talk to Blake Sherwin about a brand new map of dark matter that has been produced by a consortium of scientists using data from the Atacama Cosmology Telescope which sits high up in the Chilean Andes. Sherwin is part of that consortium, and he is also Professor of Cosmology and Astrophysics the University of Cambridge and a member of the Stephen Hawking Centre for Theoretical Cosmology at Cambridge.

Sherwin tells us how you go about producing a map of something you can't see, in how far the new map solves the so-called crisis of cosmology, and whether we will ever find out what dark matter is made of.

To find out more about dark matter, read Maths in a minute: Dark matter and What is dark matter? And you can find all the details about the research by Blake, his PhD student Frank Qu, and their colleagues in their series of papers here, here and here.

The music in this podcast is by eusa and the track is called Plankton. you can find their music on Soundcloud.

Was the mathematical modelling projecting the course of the pandemic too pessimistic, or were the projections justified? Matt Keeling tells our colleagues Ed Hill and Laura Guzmán-Rincón from SBIDER about some of the COVID models that fed into public policy.----more----

Matt Keeling

----more----We're very pleased to host this episode of SBIDER Presents, one of the podcasts produced by the Zeeman Institute for Systems Biology & Infectious Disease Epidemiology Research (SBIDER). You can find out more about the work Matt Keeling is discussing in this podcast in our article Shining a light on COVID modelling. And you can hear more from Ed and Laura in our previous podcasts On the mathematical frontline: Ed Hill and Climate change and ready meals: Challenges for epidemiologists.

This podcast is part of our collaboration with JUNIPER, the Joint UNIversity Pandemic and Epidemic Response modelling consortium. JUNIPER comprises academics from the universities of Cambridge, Warwick, Bristol, Exeter, Oxford, Manchester, and Lancaster, who are using a range of mathematical and statistical techniques to address pressing questions about the control of COVID-19. You can see more content produced with JUNIPER here.

In a tank in an underground laboratory in Cambridge a little green alga is executing a powerful breaststroke. It belongs to a group of algae called volvocales and it doesn't have a brain. So how can it coordinate its tiny little "arms" to perform motions worthy of an Olympic swimmer?

Algae going through their paces.  (Movies: Kirsty Wan and Raymond E. Goldstein, for more information see this paper)

In 2019 we visited Ray Goldstein, Schlumberger Professor of Complex Physical Systems at Cambridge, and he explained how algae manage to synchronise their so-called flagella, what this means for human physiology, and how it sheds light on the evolution of multi-cellular organisms from single-celled ones.

You can also read our article on Goldstein's work with volvocales, or watch an interview with Goldstein in this video.

Sound effects in this podcast are from Robinhood76 and 16HPanskaKanclirova_Victoria on freesound.org.

This podcast was partially funded by the European Mathematical Society.

What is infinity? What is infinity plus 100? What is infinity plus infinity?

Today's podcast was inspired by questions sent in by our friend Ash. To answer Ash's questions we take a trip to our favourite hotel, and we revisit our 2012 interview with our late boss, John D. Barrow, when we asked him – does infinity exist? Listen to the podcast to find out how infinity can corrupt the youth, why subtracting infinities can give you the right answer, and the weirdness that might be lurking out there in the cosmos...

Image created by FAVIO.

You can read more about infinity on plus.maths.org. And if you have a question about life, the universe and everything you'd like us to answer – email us as [email protected] or contact us on twitter.