We all know what data is: bits of information of which in this age of Big Data we have lots of. You might also know what topology is: the study of shapes that considers two shapes to be the same if you can deform one into the other without tearing them or gluing things together.

But what is topological data analysis? And how might it help to understand proteins or diseases such as cancer? We find out with Heather Harrington a mathematician we met at the European Congress of Mathematics (ECM) this summer. Heather tells us how topological data analysis can produce a so-called barcode for a given data set which gives deep insights into its structure. Below are a couple of images illustrating a barcode.

We attended the ECM with kind support of the London Mathematical Society (LMS). Heather gave the LMS lecture at the ECM.

You might also want to listen to more episodes of our Euromaths series which reports on the ECM.

Circles drawn around 20 points in the plane. If the radius r is less than r0, the circles are small enough to not overlap (left). Once the radius exceeds r0, but is smaller than r1, the circles overlap and together form a ring-like structure (middle). One the radius is larger than r1 the circles join up in the centre of this ring-like structure. What you see now is a single blob without a hole.

The barcode captures this information. For r < r0 there are 20 red lines indicating there are twenty connected components without holes. For r0 < r < r1 there is one green line indicating there is one connected component with one hole (the colours red and green differentiate between no hole and one hole). For r > r1 there is one red line indicating there is one connected component without a hole.

This content was produced with kind support from the London Mathematical Society.

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.