What are groups and what are they good for?

We love a game of billiards — or at least the mathematical version of it. It's a dynamical system that's just about basic enough to study but still poses lots of open questions. In this episode of Maths on the Move we talk to Giovanni Forni about chaos, periodicity and the many things we still hope to learn about billiards. We met Giovanni at the European Congress of Mathematics (ECM) in summer this year, which we attended with kind support of the London Mathematical Society. See here for more episodes of our Euromaths series which reports on the ECM. To find out more about mathematical billiards on Plus see   Chaos on the billiards table Playing billiards on doughnuts Playing billiards on strange tables   Here are a couple of academic papers by Forni and his collaborators:   Weakly Mixing Billiards, J. Chaika, G. Forni Weak Mixing in rational billiards, F. Arana-Herrera, J. Chaika, G. Forni.   This content was produced with kind support from the London Mathematical Society.  

As the days in the UK get shorter and darker we continue remembering the brilliant time we had in Seville last summer at the European Congress of Mathematics (ECM). In this episode of Maths on the move we talk to one of the mathematicians we met at the ECM, Jessica Fintzen, who won a prestigious EMS Prize at the Congress. Jessica tells us how to capture infinitely many snowflakes at the same time, the maths of symmetry and her work on representation theory, and why she likes doing handstands. To find out a little more about Jessica's mathematics, as well as her gymnastics, see this video. You might also like to look the following content relevant to topics discussed in the podcast: Groups: the basics Maths in a minute: Representing groups This content was produced with kind support from the London Mathematical Society.

The world is full of networks. We're part of them, our infrastructure is full of them, and there are even networks within our bodies (e.g. made from neurons). This summer the mathematician Richard Montgomery won a prestigious EMS Prize at the European Congress of Mathematics (ECM) for his work on the pure maths of networks, also known as graph theory. In this episode of Maths on the move Richard tells us about an amazing result he helped to prove to great acclaim, known as Ringel's conjecture, and why it's interesting to take graphs to the extreme. You might also want to read this article about Richard's work. To find out more about the event at the Isaac Newton Institute in honour of Tim Gowers, which is mentioned in the podcast, see here. This content was produced with kind support from the London Mathematical Society.

David Spiegelhalter, one of our favourite statisticians in the whole world, has a new book out. It's called The art of uncertainty: How to navigate chance, ignorance, risk and luck and published by Pelican Books. In this episode of Maths on the Move we talk to David about the book, touching on a huge range of topics — from double yolked eggs and the bay of pigs, to why it's useful to disagree and why uncertainty is personal. Enjoy! To find out more about some of the topics mentioned in this episode see, When being wrong is right — on the "tell me why I'm wrong" approach Struggling with chance — on the philosophy of probability Freedom and physics — on randomness and free will  

We recently found out why pieces of toast tend to land butter side down. It' because the physical factors at play, including the typical height of breakfast tables and the strength of the Earth's gravity, are just right to allow a piece of toast to perform one flip on its way to the floor: from butter side up to butter side down. The strength of the Earth's gravity is measured by the gravitational constant g, one of the constants of nature. These constants are special not just when it comes to toast. If their values were just a tiny bit different, life as we know it couldn't exist. This begs the question of why — why are the constants fine-tuned for our existence? Some people have taken this fine-tuning as evidence of the existence of a god who wanted us to be here, but there's also another explanation: perhaps our Universe is just one of many, all with different values for the constants of nature? If such a multiverse exists, then the existence of our Universe within it is no longer surprising. It's just one of many. All this reminded us of an interview we did in 2016 with astrophysicist Fred Adams at the FQXi international conference in Banff, Canada. In this episode of Maths on the move we bring you this interview. Adam tells us all about the multiverse and how knowledge about our own Universe can help us to calculate how many of those other universes could be similar to our own. We hope you enjoy it, but if it's too mind-boggling, have a piece of toast.   Fred Adams

A Gömböc is a strange thing. It looks like an egg with sharp edges, and when you put it down it starts wriggling and rolling around as if it were alive. Until not so long ago no-one knew whether Gömböcs even existed. Gabor Domokos, one of their discoverers, reckons that in some sense they barely exists at all. So what are Gömböcs and what makes them special? In this episode of Maths on the move we revisit an interview with Domokos from all the way back in 2009. We were reminded of this interview when we thought about what makes a good mathematical story and the story of the Gömböc has it all: beautiful mathematics, an exciting discovery, a beach holiday, romance (sort of) and even turtles. We hope you enjoy it! You can read the article that accompanies this this episode here. Gábor Domokos

Over the summer we've been incredibly lucky to have been working with Justin Chen, a maths student at the University of Cambridge who is about to start his Masters. Justin has done some great work on how to explain the concept of a mathematical group, and group theory as a whole, to non-mathematicians. In this episode of Maths on the move he tells us how groups are collection of actions, akin to walking around on a field, and why group theory is often called the study of symmetry. He also marvels at the power of abstraction mathematics affords us, tells us about what it was like diving into the world of maths communication, and what his plans are for the future. You can find out more about groups in the following two collections Justin has produced: Groups: The basics Groups: A whistle-stop tour You might also want to read Justin's article Explaining AI with the help of philosophy mentioned at the beginning of the podcast. It is based on an interview with Hana Chockler, a professor at King's College London, conducted at a recent event organised by the Newton Gateway to Mathematics and the Alan Turing Institute. This article was produced as part of our collaborations with the Isaac Newton Institute for Mathematical Sciences (INI) and the Newton Gateway to Mathematics. The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. 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.  

This summer we were very pleased to attend the European Congress of Mathematics (ECM), which took place in Seville, Spain, in July. We went to lots of fascinating talks and generally enjoyed the mathematical hustle and bustle. We also interviewed a range of interesting mathematicians about topics as diverse as mathematical billiards and topological data analysis, and we now bring you these interviews as part of our podcast. First up is the eminent Avi Wigderson, who has won many prestigious mathematical prizes, including an Abel Prize in 2021. Avi gave a great talk at the ECM about the role of errors in mathematical proofs. Traditionally, mathematical proofs need to be absolutely waterproof and errors are anathema. But as Avi told us, if you allow a certain level of error to creep in, you can do amazing things. For example, you can construct zero knowledge proofs, which allow you to prove something without giving any information away about what you're proving. And you can construct proofs that even if they're very long, can be checked for correctness by just reading a few pages. Find out more in this episode of Maths on the move. The photo of Avi Wigderson above is courtesy Cliff Moore/Institute for Advanced Study, Princeton, NJ USA/AbelPrize. This content was produced with kind support from the London Mathematical Society.  

Is there life elsewhere in the Universe? And how did life emerge here on Earth? These two questions are often considered separately, but answers to one shed important light on answers to the other. In their new book Is Earth exceptional: The quest for cosmic life, renowned astrophysicist Mario Livio and Nobel laureate Jack Szostak combine both these question and find fascinating answers. In this episode of Maths on the move we talk to Livio about the new book. From creating life in the lab to searching for it on distant planets, Livio gives us a riveting whistelstop tour of research into the origin of life, here on Earth an elsewhere. Is Earth exceptional is out on September 10, 2024 in the US and on September 26, 2024 in the UK. To get a taster of Livio's writing (which we think is very good) see a list of his Plus articles. We have also reviewed six of Livio's books: Galileo and the science deniers Why? What makes us curious Brilliant blunders Is god a mathematician? The equation that couldn't be solved The golden ratio

We're getting excited for the summer here but before we all head off on holidays we catch up with Marianne in Spain at the European Congress of Mathematics, and Justin and Rachel in the UK having just attended some fascinating events in London and Cambridge held by the Isaac Newton Institute of Mathematical Sciences and the Newton Gateway. Marianne told us about her recent interview with Avi Widgerson – winner of the Abel Prize in 2021 and the most recent Turing Prize. Justin told us about  how the philosophical concept of causality can help us understand AI.  And Rachel tells us about the surprising phenomena of anti-diffusion and how it links the patterns we see on Juniper, staircases in our oceans and fusion reactors of the future. We'll be back with  more podcasts in the Autumn, but here are some of our recommendations for your summer reading and listening pleasure! Articles: How to (im)prove mathematics Fractal photo finish  Chaos on the billiard table  Podcasts: Tying together black holes, quantum gravity and number theory The murmuration conjecture: finding new maths with AI From clicks to chords  Books: Collision – Stories from the Science of CERN This content was produced as part of our collaborations with the London Mathematical Society, the Isaac Newton Institute for Mathematical Sciences and the Newton Gateway to Mathematics.