‘Maths Teaches Modesty above All’

This year winner of the respected Fields medal Andrey Okounkov became the new head of HSE’s International Laboratory of Representation Theory and Mathematical Physics. In an interview for the HSE News website, he talked about what is a good mathematical education, why mathematicians are more free than other academics and what to tell people who find figures difficult.

From economists to mathematics

I studied, and then taught at MGU’s economics school, which was headed at various times by Leonid Grigoriev, Yaroslav Kuzminov, Alexandr Auzan. It was an amazing time, and place – I met my spouse there and made some of my very closest friends.

My interest in maths grew, and I transferred to the Mathematics and Mechanics Department, and since then I would advise people not to be scared of making decisions like this. I graduated from the Mathematics and Mechanics Department in 1993, and worked at the Moscow Independent University, and the RAS Institute for Information Transmission Problems. In 1995 I defended my doctoral thesis and the following year I was invited to join the Chicago University post-doc programme. It was, by the way, the only invitation of that kind I had received, so my family and I decided to go, and I began my voyage around American universities.

I worked in Berkeley, and Princeton, and now I teach and am doing research at Columbia University. I’ve really been very lucky: everywhere I have studied and worked I’ve had wonderful teachers and colleagues. I have worked with Alexander Kirillov, Grigori Olshanski, David Kazhdan, Viktor Ginzburg, Rahul Pandharipande….

Post-doc is a prophylactic to going round in circles

The post-doc, the first 2-3 years after defending your dissertation, is a special and in my view very important stage in life. Naturally, you take your first steps under the supervision of your Academic Supervisor within a particular scientific discipline and world view. But then you need to go out into the big outside world, meet people who live, work and think differently. This helps you see yourself and your subject in a broader context, expand your horizons and discover new ideas and goals for yourself. You need to find your goal. There’s nothing bad about striving to take the next step in solving problems that your teacher, and his teacher before him have worked on, but you shouldn’t do that out of inertia.

In HSE I was astonished by the number of young people I could have interesting conversations about mathematics with. They remind me of the enthusiastic students of the early days of the Mathematics and Mechanics Department I met in my youth, and that’s wonderful

Of course, we work with our heads and computers so you don’t need to travel to do that. But, in spite of the development of all imaginable means of information technology and communication, nothing stimulates the mind like actually talking to another person. If you get the chance to learn from different people, it is worth taking it.

From representation theory to algebraic geometry or there and back

Surprisingly, (it’s either luck or the limits of human consciousness), now I’m coming back to ideas that stirred my imagination back in my youth in Moscow. I was studying representation theory, and, influenced by my trip to Chicago, I got involved with algebraic geometry in particular, enumerative geometry, which seemed to me at the time to be something fundamentally different. But 15 years later, it now seems that these two spheres are not actually that far from each other. Now at HSE I’m working on the theory of representation in several geometric incarnations. The question of which is the basis and which an application depends on a person’s training: for me representation theory is the basis and algebraic geometry an important application. 

My history at HSE

I began working at HSE this year as Academic Supervisor at the International Laboratory of Representation Theory and Mathematical Physics. The lab was created by Boris Feigin. He is one of the greatest minds in the field. He has been a colossal influence on me throughout my career and I’m very happy to be with him under one roof.

HSE has brought together some very strong mathematicians. A lot of my colleagues from the Independent Moscow University are here. A couple of years ago my close friend and colleague Igor Krichever came to work here and another important thing — there are a lot of talented young people keen to learn here. In 2013 I was at the summer school organised by Fedor Bogomolov’s International Laboratory for Algebraic Geometry and I was astonished by the number of young people I could have interesting conversations about mathematics with. They remind me of the enthusiastic students of the early days of the Mathematics and Mechanics Department I met in my youth, and that’s wonderful.

Mathematics is an indispensable subject for self-development 

On the one hand, as in any area, there are more able and less able people, so teaching mathematics is a selective skill, like music for example. Only music works directly on our emotions. To experience a similar emotional impact from mathematics, you need to get deep into it for a long time.

On the other hand, I’m convinced that all of us need mathematics. How to teach it to people who specialise in other subjects  — economists, physicists, historians — is another question. Even with all these differences, I think that there is one important thing that we must try to get through to all students, whatever their subject. (And it is the people who find mathematics hard we need explain it to). Mathematics is not only multiplying big numbers or transforming formulas in trigonometry (that’s just a small part of it).   

The value of mathematics for those who use it is that it enables us to see structures, a particular way of thinking about a problem. In any task, the most important thing is how you think about it, whether you understand what is important and what doesn’t matter, what is trivial and what is significant. Mathematics allows us not to get bogged down in irrelevant details and to see what matters.

There are so many tasks in nature and we can only solve a tiny number of them and when we can, we rejoice. In fact scientific discovery is happiness, you solved a problem — that’s good, you didn’t solve it — it’s ok

For example, if we apply it to our work in the Laboratory on enumerative geometry (and the latest mathematical physics),then it is the source of all the endless tables, long numbers, and we are trying to find organizing principles that make it possible to reduce all these numbers to a finite number of profound words. I.e., using representative theory to try to find the organizing principles that underpin all this. Occasionally we succeed but more often we don’t. 

My favourite answer to the question of what is the most important thing mathematics teaches us. For me, most of all it teaches modesty. There are so many tasks in nature and we can only solve a tiny number of them and when we can, we rejoice. In fact scientific discovery is happiness, you solved a problem — that’s good, you didn’t solve it — it’s ok.

Fields Medal

I was given the Fields Medal in 2006 for my achievement in bringing together the theory of probability, representation theory and algebraic geometry. That’s a rather fuzzy explanation. For me it is recognition not so much of my own achievements as support in studying one and the same thing — connecting problems in enumerative geometry with those in probability theory. It is about a number of problems in enumerative geometry, which in their physical sense came from symmetrical string theory and gauge theory, can be reworked mathematically as a problem of probability regarding how a random surface or a random curve  behaves. This relates to the works I co-authored with Nikita Nekrasov, Rahul Pandharipande, Rick Kenyon, and others. I was very glad that this link between two very distant mathematical disciplines turned out to be so useful to each.

How a good mathematical education differs from a bad one

A good mathematical education gives a person the chance to learn to distinguish those things you need to take with you on your professional journey through life from the things which don’t seem entirely useless and look pretty but you don’t need to drag around with you as you can always buy them in a shop nearby (search the internet). Mathematics is a kind of journey where superfluous baggage doesn’t help.

There’s a lot of mathematics but it isn’t everything

If you are washing the dishes and thinking about how to solve a particular problem is that mathematics or not? In terms of focussing your mind, yes it is. And there is a lot of it in my life but it doesn’t occupy 100% of my time. There’s family, children, travel, sport — just like other people. Most of the year I live with my family in New York, but for a few months I live in Russia and give (and attend) lectures at HSE and the Institute for Information Transmission Problems.

I think that mathematicians have it easier than other academics and scientists. For a start, mathematicians around the world have more or less the same ideas about on the criteria of truth and beauty in our science. From conventional traction to working out relationships, the ‘dirt repellent’ fabric of mathematics is very useful. The joy that comes with mathematical discoveries is not dependent on prizes, rewards or likes on Facebook.

On the other hand, we are not experimental physicists who need 30 years, billions of dollars, and a massive concentration of minds, to open up new horizons in science. Financing and managing those kinds of projects turns any scientist into a manager. Fortunately it’s easier for us: our problems can be solved by small groups of like-minded people, and can wait until we’ve ‘grown’ enough to deal with them. That is why I think mathematics is an extremely attractive career.

Ludmila Mezentseva, HSE News website