# In conversation with... Ian McIntosh

Third year maths student Molly spoke to Dr Ian McIntosh about why he enjoys teaching and researching maths.

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Why is it interesting to be studying mathematics at the moment?

I think it's always been interesting to study mathematics, simply because it's such a rich subject of depth, complexity and beauty. But there are two main reasons I believe studying mathematics now is very valuable.

The first is that mathematics is finding more and more applications, partly because of the digital age, but also because mathematical tools are being used in more fields of study. We often hear about 'the algorithm' - well, that's a mathematical algorithm.

The second reason is that studying mathematics develops a very acute awareness of sound reasoning and logical consistency. A true mathematician has an almost allergic reaction to faulty reasoning. In the current age, where we are overwhelmed with conflicting opinions, well-developed skills of logical critical analysis are increasingly important.

How does mathematics relate to the world we live in?

Beyond the standard use of numbers, mathematics plays a huge role in the way our civilisation functions. There's only room to give a few examples, and I'll try to touch on a diverse range of applications.

Secure financial transactions over the internet require safe encryption. The standard public key encryption, called RSA, is based on number theory and properties of prime numbers.

Mathematical modelling is used to predict infection transmission numbers during the current pandemic: governments around the world have been informed by mathematicians. The development of any vaccine requires clinical tests and the data from those tests has to be properly analysed by medical statisticians.

Quantum computing is a brand new field which requires mathematics as well as physics. And I haven't said anything about the huge role mathematical modelling plays in predicting climate change effects. Mathematical models have been used for weather prediction since the first computers made it possible.

There is an interesting "local" aspect to this last example. One of the pioneers of numerical weather prediction was Lewis Fry Richardson, who attended Bootham School, York, as a boy. Our termly open lectures are named after him.

Which mathematical formula or result do you find most exciting?

I have always loved partial differential equations. When I was an undergraduate I loved learning about the Navier-Stokes equations, which are the equations governing fluid flow, and being taught how to find some (simple) solutions.

In my current field of research there are many different partial differential equations, but they all come from one source: the Gauss-Codazzi-Ricci equations. These govern the shape that surfaces take in space. New results about solutions to these equations, or different ways of looking at these equations, always excite me.

When you were studying at university, did any mathematicians inspire or influence you?

When I was an undergraduate my brother was doing a PhD in mathematical oceanography, so I met a lot of PhD students and they were always fascinating to talk to.

I was really looking forward to learning the mathematics of General Relativity, which was taught in a 4th year module. As a teenager I had seen interviews with Roger Penrose and I was fascinated by the way he talked about the curvature of space-time. That got me interested in geometry, because I wanted to know what 'curvature' meant, and I started to read text books on surface theory.

You can't read about surface theory without learning about Gauss. He was one of the most influential mathematicians of the 19th Century, whose development of Gaussian curvature is the basis of pretty much every modern course on classical surface theory.

Could you describe your current research and why you find this area interesting?

I study minimal surfaces. These are surfaces which use the least area possible to take their shape - like a soap bubble in a loop of wire. The shape they take depends on the geometry of the space they are in. That could be 'flat' Euclidean space, or have its own curvature. I'm currently very interested in minimal surfaces which sit in a space of negative curvature (sometimes called a hyperbolic space). It fascinates me because I like surfaces (I can visualise them) and I like using clever methods to find solutions to partial differential equations.

The surfaces I study have infinite extent, but enough symmetry that they can be determined by looking at just a small piece (like a tile being repeated over and over to cover an infinite floor). The method for studying these uses a clever theorem, proved in the 1980s, which says that they can all be written down using an advanced type of geometry which involves complex-valued functions ("holomorphic vector bundles over compact Riemann surfaces" is the technical name). There is a rich higher dimensional geometry involved in understanding families of surfaces of this type.

Is there any question or problem relating to your current research that you would particularly love to solve?

The clever theorem mentioned above has a drawback: it cannot say when the surface produced has a nice tiling. The problem I would like to solve is: how can you know when the tiling is going to be nice? My collaborator and I have managed to do this for some very specific cases, but we'd really like to have a more widely applicable method.

What do you enjoy about teaching mathematics?

Teaching mathematics always reminds me of how important undergraduate-level mathematics is to all my work, including research. The ideas don't become boring with time, because they hold the central core of all further understanding, so I just appreciate them more and more. Talking about these ideas with students and trying to provide insight when someone is stuck is very enjoyable: I like seeing students' faces light up when they finally 'get it' after struggling with some problem or concept.

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