The exclusive Interview of the Day: Timothy Gowers

Sir Timothy Gowers is a research mathematician. He achieved a perfect score at the IMO, and went on to receive the Fields Medal for research connecting the fields of functional analysis and combinatorics. As well as being a Royal Society Research Professor at the University of Cambridge, he is a Fellow of Trinity College, Cambridge, and currently holds the Rouse Ball chair. In October he will also be assuming the title chaire de combinatoire at the College de France.

He delivered an online lecture for IMO 2020, speaking about “Graphs with few triangles and how to get rid of them”. After the talk he was interviewed by Parth Shimpi, a mathematics undergrad at the University of Cambridge.

Did you ever make a conscious effort towards winning a Fields medal? 

My advice would be, don’t go into it with the aim of winning a medal- that is ridiculous. You have to love the subject. Even if you do extremely good research, there is a luck element to awards. Every young mathematician dreams of the Fields medal like a child aims to become an astronaut, but not very many people do it. As with many things in life, aiming directly at that is a mistake and you should aim at getting the most beautiful mathematical results that you can, without considering other factors like recognition. For a lucky few it works out.

You might make decisions to maximise your chances of winning awards- for instance by thinking this area has more recognition than the other. That in my opinion is a big mistake. To maximise your chances, the best thing to do is to pursue the area that you like the best, because that’s where you will do the best work. From my own experience, I worked in an area considered rather ‘unfashionable’- the geometry of Banach spaces, but I liked it and it paid off. Over time, areas change in their fashionability as well- combinatorics was extremely unfashionable for a long time but now that has changed. So don’t be motivated by other factors and just work hard at the maths you like.

How hard is research mathematics? 

When you start with research you can think that there have been lots of people who have tried this question, what chance do I have of solving it? If that is actually the case and your problem has remained open for a long time, then your chances of success are very small at least to start with. But there are plenty of interesting problems around that have not received that much attention, new questions spawn every time someone solves a problem.  The more you do research the more you find that when you work on one question, you have ideas that are applicable to other questions. This happens more and more as you progress, and if you are lucky you will realise that you have an insight that solves a question that has been unsolved for long. So I would say that famous problems are difficult to solve- but its good to think about them. Not necessarily to solve them, but to get a feel of what the underlying difficulties are, in case one day you realise one of those hurdles can be overcome by some development in mathematics!

It is hard, but it is not impossible. It is a probabilistic process: if you are looking to do something interesting you will not find success with 90% of the problems you try but it is really the remaining 10% that you are looking for.

Do you have a ‘flow state’ when solving problems, when everything suddenly seems to click? How do you get there? 

Avoiding distractions certainly helps. Not just your email or social media, but it is very tempting when solving a problem to reach a stage where you are stuck and then you distract yourself by thinking about other problems. Sometimes its good and helps you get back to the problem with a fresh mind, but you must not do this too quickly and give the problem its time.

I have occasionally had moments in my research when everything happens a lot more easily than before, but I don’t know how much of it is necessarily due to some meditative state in my brain. You just keep trying and hitting dead ends, until everything feels easier because it actually is easier. Because you did all the difficult frustrating parts at an earlier stage. It is like solving a jigsaw puzzle, finding pieces is very hard at the start but gets easier as you reach the end.

What advice do you have for the future generation of mathematicians?

I have two things to say, and they go slightly against each other but that’s the way it is. When you go to university, work hard to master the so-called ‘boring stuff’, theory like linear algebra and group theory. This will involve simply learning proofs that other people have come up with, which is nowhere near as exciting as solving IMO problems. But learning these techniques will open the doors to a vast new pool of problems- you gain the ability to formulate the problems as well as the tools to solve them.

On the other hand, you should not forget about problem solving and focus only on learning theory. If you participated in the IMO you clearly have good problem solving skills, and these become important again as you start doing research. If you lose sight of it completely, it will harm you in the long run. You should also however realise that research problems are very different from IMO problems. For starters, you know that IMO problems have been framed so that you can realistically solve them in an hour, while research problems may well be unsolvable. Time scale varies as well- IMO problems should be solvable in a few hours while you can spend months thinking about the same research problem. However, when you have a project- an open problem, it often breaks up into smaller problems that you can solve over an afternoon, and your IMO skills will help here.

(Transcribed by Parth Shimpi)