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.