You are blue, and are surrounded by other blue people: swirling together in a dot, identical and indistinguishable. From somewhere above you hear the ticking of a clock, and suddenly find yourself and some of your fellows pulled upwards, sucked through a tube arcing high above...

Intrigued? That is a description of one of Jess Enright's adventures in her mathematical models. This is an exciting new approach that researchers are using to invite people into the worlds of their models, both to communicate their research to the people outside of academia, but also for the researchers themselves to reflect on what aspects of reality these models actually do, and don't, describe.

These adventures in model land build on the work of Erica Thompson in her book, Escape from Model Land: how mathematical models can lead us astray and what we can do about it. Any mathematical description of a process in the world around us is a mathematical model: whether it's describing the processes in our climate, the spread of a disease through a population or the movement of water across a landscape. They are incredibly useful and key to research in modern mathematics and science. But these mathematical models are, by necessity, simplifications of the real world.

Erica's book inspired geoscientist Chris Skinner to use principles of role-playing games to explore and communicate mathematical models. And this approach was a perfect fit with Jess' experience building board games to communicate her research - we event get to play some in this podcast at the huge UK Games Expo in Birmingham earlier this year!

Jess Enright, along with Emma Gort, in action at the UK Games Expo in Birmingham earlier in 2025. (Photo: Rachel Thomas)

In this podcast we talk to Jess (a reader in the school of Computing Science at the University of Glasgow and member of the JUNIPER partnership of disease modellers from across the UK), Chris (an independent geoscientist and researcher and a visiting fellow at York St John University)and Erica (Associate Professor of Modelling for Decision Making at University College, London) about their explorations of these ideas.

You can find out more information about the ideas discussed in the podcast here:

• Maths in a Minute: Mathematical model – a brief and an accessible introduction to mathematical models and where they are used.
• Escape from Model Land: how mathematical models can lead us astray and what we can do about it – Erica's book
• Adventures in Model Land– the framework developed by Chris, Jess and their colleagues for exploring mathematical models using table-top role-play games
• You can find some of the adventures in model land and other games that Jess took to the UK Games Expo in Birmingham

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.

There's been a lot of talk recently about whether artificial intelligence is becoming just as good as maths as humans are. But quietly in the background there's been another development regarding the use of computers in maths. It involves proof assistants: computer programmes that can check whether a mathematical proof is correct; whether it can be derived from a set of basic axioms of mathematics using only the rules of logic.

In this episode of Living proof we meet Kevin Buzzard, an expert on proof assistants at University College London. Kevin explains what proof assistants are, how using them is like playing a computer game, and why they turn maths into a highly collaborative pursuit. He also tells us about his effort to get a proof assistant to check one of the most famous results in all of mathematics — Fermat's Last Theorem — and how proof assistants and AI may team up to provide a powerful tool.

We met Kevin in the summer when he was taking part in a research programme called Big Proof at the Isaac Newton Institute for Mathematical Sciences (INI) in Cambridge. This programme, which attracted some of the best minds in modern mathematics, followed on from a pioneering workshop on the same topic which took place at the INI in 2017.

To find out more about the topics mentioned in this podcast, see the following articles:

• Proof assistants — This two part article, written by our brilliant summer intern Ben Watkins, is based on the interview with Kevin Buzzard and explores what proof assistants are.
• Maths in a Minute: Coding with Lean — Here's a simple walk-through of how to use a proof assitant called Lean.
• Pure maths in crisis? — In this article from 2019 Kevin Buzzard explains why he thinks that the standard of proof in research maths might not be as high as mathematicians would like to believe.
• How to (im)prove mathematics — This article explores how the simple notion of counting ends in a revolutionary new way of doing maths using proof assistants. This article is based on a talk by Terence Tao at a 2024 workshop at the INI which celebrated the mathematics of Tim Gowers as well as his 60th birthday.
• A very old problem turns 30! — This article explores Fermat's famous last theorem as well as the mathematics its proof has given rise to. It comes with a podcast featuring Andrew Wiles, who proved the result, and people who are now working on its legacy.
• You can find more background reading in our collection on proof assistants.

This content forms 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.

In this episode of the Living Proof podcast we're delighted to meet Moustapha Fall. Moustapha is the Center President of the African Institute for Mathematical Sciences (AIMS) in Senegal and winner of a prestigious Ramanujan Prize for Young Mathematicians from Developing Countries. He also plays an important role on the international stage as Member-at-Large of the Executive Committee of the International Mathematical Union.

Moustapha talks to us about about the amazing institution that is AIMS, about his outreach activities and advice for budding mathematicians, and about the challenges that face mathematicians in sub-Saharan Africa.

You can find the IMU-ICIAM report on fraudulent publishing, which Moustapha mentions in the podcast, here. The same team of authors has also drawn up recommendations on how to fight fraudulent publishing.