## Matthew Wales is a PhD student at the Department of Pure Mathematics and Mathematical Statistics. Here he tells us about the joys of pure mathematics, how it feels to produce your very own mathematical result, and the importance of coffee in the world of maths.

**In my research I study graph theory, which is an area of combinatorics. ** Graphs are essentially like networks: there are some points and there are connections between these points.

My PhD involves looking at conditions which force a graph to have what we call minors. To describe what that is, suppose we've got four points with each pair of points connected. As a substructure this graph contains a triangle, because if we look at any three points, these are pairwise connected. Such a triangle is what is called a subgraph. We can extend that to the notion of a minor. If we have a square, then that doesn't contain a triangle. But if we squeeze two of the points in the square together, then that does give you a triangle. This operation of squeezing points together is what we call a *graph minor*.

"I like working in the Faculty because there is a really nice collaborative atmosphere." Matthew Wales

Graph minors have been studied in relation to many different problems. The original motivation was the *four colour theorem*, which says that to colour a map so that no two adjacent countries have the same colour you need at most four colours. Proving that was a very hard problem in graph theory. The result was eventually proven in the 1970s, but they had to use a computer for that proof. The methods used in that proof used graph minors pretty heavily. I have been looking at the more abstract notion of graph minors which grew out of that. Many of the famous conjectures in my area, such as *Hadwiger's* conjecture, are generalisations of the four colour theorem.

**I felt attracted to this area because as an undergraduate I really enjoyed the more abstract courses.** In Part II of my degree here at Cambridge. We had a course on graph theory and a course on representation theory. Thinking about these more abstract mathematical structures really appealed to me.

Then in Part III I took courses related to both of these; a course in algebra and a course in combinatorics. In particular there was a course given by Andrew Thomason, who is now my supervisor, on extremal graph theory. The problems involved here, where we are looking for certain substructures, really grabbed my attention and I thought, that's what I want to do for my PhD.

**I like working in the Faculty because there is a really nice collaborative atmosphere. ** Usually you have three PhD students in an office who are all working on broadly similar topics. There's someone else who started with my supervisor at the same time as me, and we were able to bounce ideas off each other. That's really good because sometimes with maths the act of explaining what you are trying to do will help you realise what the key problems are and get a better understanding, so maybe you can solve your problems.

There are also coffee breaks and weekly seminars. The seminars are a chance to meet other people, you have tea and cake afterwards, and it's a good chance to see what other people are working on.

**On a typical day, we start by getting a cup of tea and looking at the latest pre-prints online.** There's an online server where people put their latest research. Each day there will be about 15 papers. Obviously you can't look at all of them, but you can get a feel for what people around the world are doing. And maybe one of these papers will draw you in and you can look at it later on.

The morning is also a good time to get a fresh look at something you were working on the day before. Sometimes with maths, having a break allows you to come back with different ideas. Then I go off to have coffee. It's a chance to relax and get some caffeine. They say a mathematician is a machine that turns coffee into theorems, and that's definitely true here at the Centre for Mathematical Sciences.

In the afternoon it's a good time to sit down and start writing things down. Often it's easier to work out proofs on paper, but when it comes to communicating with other people it's good to have them typed out. This is also a good way of seeing if you could have proved something in a more efficient way: the writing process helps you cut down to what is actually essential.

After that maybe there is a mini seminar, where with other PhD students we have a look at a particular paper each week. That means you get a deep dive and learn some new techniques. Then it's time to go home, cook, and maybe get some exercise to burn off some energy.

**One of my favourite mathematical moments was when everything came together for the first result that was my own. ** It was joint work with my supervisor, but it was the first theorem I had written down myself. This was something where I was the only person in the world who knew it. It's quite an exciting feeling to be the only person in the world who knows something!

**To someone wanting to do a PhD I'd say that it can be intimidating, but that it is a very good experience. ** You get a great feel for doing independent research. But it's very important that you think you're the right person for it: it's very independent. As an undergraduate you have lectures, problem sheets, group work, and so on. But when it comes to a PhD you have to be self-motivated. Your supervisor can give you broad directions, but you still have to work out your problems yourself. You also need to make sure that the topic you have chosen is something you'd be willing to think about for three years.

Also, doing a PhD doesn't close off other options. It doesn't shoehorn you into becoming an academic, because you can also move into industry. After finishing my PhD I will go off and become a researcher in finance.