The mathematical area of topology is all about figuring out what truly defines a shape. Famously, topologists consider a coffee cup to be the same as a doughnut because one can be turned into the other without cutting or gluing — what defines and relates these two shapes for a topologist is that they have a single hole.

As you might imagine, if you have ever tried to drink coffee out of a doughnut, topology has traditionally been part of pure mathematics. Topological data analysis (TDA), however, opens up a world of applications by applying ideas from topology to vast data sets, helping us to understand their "shape" and draw out important features.

In this episode of Maths on the Move we talk to algebraic topologist Michael Hill about some of the fascinating uses of topological data analysis — from understanding breast cancer to making sure that voting is fair.

We talked to Michael after he gave a brilliant Rothschild lecture at the Isaac Newton Institute for Mathematical Sciences (INI) in Cambridge. He was at the INI to attend the research programme Equivariant homotopy theory in context.

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

Also, here is an image illustrating the intuition behind topological data analysis. As discs drawn around a bunch of points arranged in a circle increase in radius, they eventually overlap to form a ring, and later overlap to form a single blob.

This podcast 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.