Fields Laureate Andrey Okounkov on Where to Study Math

Columbus University Professor Andrey Okounkov is the Academic Supervisor of HSE’s International Laboratory of Representation Theory and Mathematical Physics. In an interview with Afisha, he discusses how mathematics education differs in Russia and the U.S., where his insight comes from, and whether it is true that mathematicians are in fact ‘strange individuals.’

Andrey Okounkov is a laureate of the Fields Medal, the most prestigious award in mathematics and often compared to the Nobel Prize, which is not offered to mathematicians. The Fields Medal is awarded once every four years to no more than four mathematicians under the age of 40. Professor Okounkov received the Medal 'for his contributions bridging probability, representation theory and algebraic geometry' in 2006 alongside Grigori Perelman, who declined the award. There have only been nine Soviet or Russian Fields Medalists, the most recent of which was Stanislav Smirnov, who received the Medal in 2010.

— People's first reaction to the word 'mathematics' is often associated with boredom. Why do you think that is? Is the school curriculum to blame?

— I can't say I've encountered this frequently in Russia, as it’s actually more in the United States that this is the most common reflex reaction to the word mathematics. Here, schoolteachers, math enthusiasts, and parents have instilled in a lot of people at least a respect towards mathematics, if not a love for it. And as for the idea that basic mathematics in school is usually boring – most people didn’t like singing lessons as well, but this didn’t stop them from developing a love for music.

— Yet it’s still difficult to imagine that after lessons in trigonometry and solving inequalities, someone would want to devote their life to this.

— I myself went to a completely ordinary school near where I lived, and I decided to devote my career to mathematics much later after I had studied for a couple of years in Moscow State University’s Economics Faculty. In my day, a lot of people wanted to go into professions somehow connected with the hard sciences, at least engineering. I don’t know how it is today, but it’s clear that a certain push is needed from teachers, friends, and parents. It’s never the case that a person sits and thinks, ‘hmm, I’ll take up Sanskrit…it has to be interesting…’ There are a ton of popular books with interesting tasks and without the inequalities that you talked about. Russia has a long history of mathematical puzzles; they were really popular back in the 19^th century. But returning to the school curriculum – for mathematics, like for music, technique is important, as is the associated training.

What impresses me the most is the fact that students in HSE’s Faculty of Mathematics have preserved the spirit that flowed through the walls of the MSU Faculty of Mechanics and Mathematics when I studied there

— A year ago, you became the head of a laboratory  at the Higher School of Economics in Moscow and began teaching in Russia. How have Russian students changed in your opinion? What motivates them to study math and get an education in something that doesn’t promise obvious career prospects?

— What impresses me the most is the fact that students in HSE’s Faculty of Mathematics have preserved the spirit that flowed through the walls of the MSU Faculty of Mechanics and Mathematics when I studied there. In today’s globalized world, one’s job is determined by nonmathematical factors such as family interests. It’s not at all surprising that HSE graduates are now going to graduate schools in the West, while Western professors are coming to Moscow to teach. What’s important and what miraculously remains in Moscow is the mathematical spirit I mentioned. It unites our students in their desire to get to the root of things.

— How true is the stereotype that mathematics students in the U.S. are not as strong as their Russian counterparts?

— Really, there are so many people who get a higher education in America that the word ‘student’ has become a very blurred concept. There are thousands of universities there, among which are Princeton and Harvard, but then there is, say, the University of Nowhere, which is made up of students interested in nothing but American football. The U.S. has a lot of students who view their education as four years of a constant party, but there are also those who want to gain knowledge. And by the way, the situation is similar in Russia. So if we’re talking about those who go to school to actually gain knowledge, I don’t think they have enough of a real educational foundation. American students majoring in mathematics have to take a lot of different classes like ‘history of literature,’ for example, and well below half of their time goes towards math. Conversely, at the MSU Faculty of Mechanics and Mathematics, for example, mathematics courses account for just under 90% of the curriculum. If I’m not mistaken, you can get a mathematics degree at Princeton by taking 10 semester-long courses, whereas in Russia, this is how many classes students take in their first year. As a result, before graduate school, people are generally at a lower level than our students who come from good universities. But then, students – those who get in – sort of catch up in graduate school.

— Is it true that mathematicians can be separated into the categories of those who prefer solving technically complicated tasks and those who like to create theories?

— That’s generally how it is. It is usually clear which people like what. Those who like solving tasks are occasionally prepared to build a complex logical construction from the blocks they know about, which is probably comparable to building a bridge across a river using the materials you have on hand. And those who like creating theories, they prefer building mathematical factories whose products can generally be used to build both a bridge and a lot of other things. There’s a famous story about the Soviet mathematicians Andrey Kolmogorov, Israel Gelfand, and Vladimir Arnold. Once at a birthday party, Arnold gave a speech saying that there are mathematicians like Andrey who, when discovering a new country, would first find its mountains and start to clime them one by one. Then there is Israel Gelfand, who would start planning – an airport will go here and a food warehouse here. I must say, both were terribly offended.

— And does this change with time and experience? For those who climb the mountain, do you gradually become an architect of new theories?

— You know, mathematics has an athletic element to it, as people compete to see who the first to prove something is. And this is a critical part for a lot of people. It’s not important what the proof is, it’s just important to be first. People rush to climb that mountain, jumping over the streams, charging through the bushes, and they end up being the first to the top. But you realize with time that it wasn’t at all necessary to cross the chasm or dive into the bushes. You can just go around things on the same pass and have a beautiful view while doing so. As you get older, you not only want to find a working argument, but also crystallize what’s important while getting rid of what’s not. Proof gives us a clear understanding that a certain statement is correct. But the correctness of a statement is not intrinsically valuable. There are many levels to understanding a mathematical phenomenon, and proof is not only notthe only one, but it’s just one of the first.

Too strong a belief in logical thinking can do you a disservice. People miss key elements and certain concepts without a clear definition because they have a hunch

— What do you think about the commonplace idea that after 40, you can’t achieve anything serious in math because your intellect isn’t as strong?

— If Gordie Howe played hockey until he was 52, then a mathematician can certainly carry on after 40.

— Math is very rational, but new discoveries can’t happen without so-called aha moments…

— Of course. More often than not, breakthrough moments are linked to the sudden realisation that, just like in a detective film, some calculation or some object or task is not at all what the film’s audience had been imagining. This differs considerably from detective films in that the hitherto hidden essence of something always ends up being your friend, as if a Sherlock Holmes in disguise stood in front of you each time with a finished solution to the problem. At the same time, mathematics does not consist of just breakthroughs and ‘aha’ moments. It’s mostly technical work – you have tools, you go to the household goods store and there you are – dig a hole, put a frame there, and build the foundation. And when a building’s already there, it turns out that you still have to put electrical outlets in it so nothing short circuits.

— Scientific strokes of genius have a very romantic feel to them – like the story of the apple falling on Newton's head. How do these ‘eureka’ moments actually occur?

— They can come in anyway possible. But perhaps experts on higher nervous activity are able to explain why it’s sometimes good to stop what you’re doing and go run, for example. There’s a saying attributed to Picasso: ‘Inspiration exists, but it has to find us working.’ But this isn’t exactly correct in my experience. That’s like someone sneezing writing a treatise about a cold. It’s important to have certain things in mind, then they’ll stew, and if you’re really lucky they might turn into insight. Anything can give this process a push. If a person is listening to a paper on a completely abstract topic, for example, and he hears a certain word combination that the presenter uses. The listener might not even be following the paper in particular, but certain neurons start to race in his head, and suddenly he has an understanding and the solution to his own task. Generally speaking, the main trigger is communication. Humans are social beings with social brains.

— It sometimes seems that mathematics has some sort of internal feature that causes mathematicians themselves to change and be extremely different from everyone else. Paul Erdős moved around from place to place his whole life and never had a permanent address; Alexander Grothendieck turned into a hermit, practically closing himself off from the outside world; and Grigori Perelman turned down a million-dollar award. Is mathematics to blame?

— Well, the people you just mentioned stand out first and foremost by their huge talent, and in Perelman’s case, this is true even within the mathematical community. As concerns how mathematics can affect a person overall, I’m not sure there’s some sort of general mechanism for everyone. Math trains the brain, of course, and mathematicians are trained to make many logical moves instantaneously. And this is why mathematicians have the impression that they are capable of instantly getting to the heart of things, though this impression is quite often misleading. This is actually an interesting phenomenon. Many mathematicians try to do something in sciences other than math, and this always ends in a mixture of success and failure. Too strong a belief in logical thinking can do you a disservice. People miss key elements and certain concepts without a clear definition because they have a hunch.

— What is going on in contemporary mathematics? Is it true that it has become so complex that scholars no longer understand what their colleagues are doing?

— Mathematics is complex and becoming more complex. This is not because we have wandered into some jungle, but this is simply a reflection of the complexity of our world. Any person who looks at the world around him for a few minutes will agree that any theory capable of accurately describing the observed world must be a very rich theory indeed. How do we deal with this complexity? It’s clear that we have to let computers do everything they’re capable of doing. We should also create groups of researchers that supplement and support one another so they can split this complexity among themselves. I certainly don’t believe that the fact that science is becoming more complex will lead to a crisis. On the contrary, I think we are living in math’s golden age.