Russian mathematician building bridges across mathematical fields

"Okounkov's body of work reflects an age-old idea of mathematics: that all mathematical fields are part of one big family tree and contain connections waiting to be discovered. As mathematics has become more and more specialized, this is a very difficult feat to accomplish," the University of Primorska, the lead organiser of the congress, described the Russian mathematician.

Obtaining the medal, considered the Nobel Prize in mathematics, in 2006, Okounkov connected wide-ranging areas of mathematics, while his research and collaborations have also helped solve problems in physics and chemistry. Moreover, the techniques that he and his collaborators have developed in these other fields have proved fruitful for mathematics.

The Fields Medal is considered the Nobel Prize in Mathematics. How has this recognition affected your life and especially your professional career?

I think it mainly affected it through an added sense of responsibility for mathematics, and science in general. Prize winners are, in a way, ambassadors for their field of expertise and, for instance, get a chance to have conversations like the one we are having now. And of course, it is my duty to use these opportunities to stress both the great importance and the great beauty of mathematics. On the one hand, we are talking about the importance of mathematics for the whole of modern society, which relies extremely heavily on so many math-intensive processes, math-intensive devices, and math-intensive professions. It is important to appreciate this even if one is not interacting with mathematics directly. On the other hand, I am always trying to send the message, especially to young people, that mathematics is a very beautiful and rewarding subject, and it is not only pragmatic, but also fun to study and do mathematics.

Mathematics is usually not one of the most popular school objects. What has been your personal journey into the world of mathematics from an early age? When did you really feel that math was what you wanted to do in life?

I think it varies from school to school, and also perhaps from country to country. Maybe I was particularly lucky to be growing up in Moscow, in the atmosphere of enjoyment and excitement around mathematics and other subjects, propagated not only by our teachers, but also by math circles, other outreach events, Olympiads, etc. My wife, for instance, had a very similar experience, which is why she chose to be mathematical and quantitative in her own profession, which has to do with asset managements, such as pensions, endowments, etc., and which many people do without due respect for mathematical models. The local volunteers I met here at the Congress all come across as having a genuine interest and perhaps even passion for mathematics, so I am sure there are a lot of ways in which math can be presented to our youth for what it really is. And math really is an amazingly beautiful and exciting subject.

Russia is one of the world's leading countries in mathematics, and you also work at Columbia University in the USA, which is considered to be the most advanced in this field. Which countries or regions are currently among the most propulsive in mathematics? Where is the fastest scientific development taking place?

Progress in mathematics goes hand in hand with the general growth of importance of science and technology (think about the Sputnik times), and also with the inflow of young talent in mathematics. It is obvious, that very populous, education-oriented, and technologically advanced countries like China would be the real driving force in mathematics in the very near future. And it is an important challenge for mathematics as a global profession to make sure that no part of the world misses the transition to the new digital and very mathematical world.

In your opinion, what are the key scientific challenges that modern mathematics deals with? Where are its findings most applied?

My own interests are in mathematical physics, which aims to give a very precise mathematical description to the world around us. This obviously ties with many core things in the functioning of our society, both in what concerns discovery of new objects and phenomena, and harnessing them for their practical use and engineering applications. The first thing mathematics provides is the language in which to speak of natural phenomena, and that language changes tremendously from Pythagoras, to Galileo, to Newton, to the times of Einstein, to modern times.

Pythagoras thought that everything in the world can be described by a fraction (and this is remembered in today's language when we say that a person thinks "rationally", that is, in terms of fractions). So it was a huge crack in his view of the universe when it was discovered that the diagonal of the unit square, that is, the square root of 2, is not a rational number. Since then, the notion of what is a number, or what is a geometric object, or what sort of mathematical structures should be used to describe the world around us, has seen many revolutionary transformations. By 1915, mathematicians had developed the beautiful language of the classical Riemannian geometry for Einstein to write down his famous equation of General Relativity.

By now the mathematics, hopefully, has advanced enough for the new Einstein to formulate a theory that would combine gravity with quantum fluctuations in a manageable way. I say manageable because after we develop the language, we need to work out what it really says about the behavior of physical systems. For instance, for Einstein's theory we have seen very important progress, both theoretical and numeric, on actually analyzing his equations in recent years. Perhaps the most visible event here was the theoretical prediction and subsequent experimental observation of the form of the gravitational wave from a black hole merger.

On a personal note, my wife and I are extremely proud that our elder daughter is part of the very large community of scientists who work on these problems. And our younger daughter is just starting at graduate school in 2-dimensional materials, which used to be considered a toy, and perhaps unrealistic, model in mathematical physics until extremely thin, that is essentially 2-dimensional, devices started conquering the world.

These days you have been participating at the 8th European Congress of Mathematics, which is taking place in Slovenia. Is this, for you, the first academic and professional occasion to be in contact with Slovenian mathematicians and mathematical university departments or have you also had these contacts before the 8ECM? With whom mostly?

It is my first visit to Slovenia, but I was certainly interacting with Slovenian mathematicians before on both purely scientific and also other professional matters. Obviously, there was a lot of interaction during the preparations for the 8ECM, but in addition to that, for instance, Prof. Klavdija Kutnar, the rector of the University of Primorska, and one of the main 8ECM organizers, is a very valuable member of the International Advisory Committee for ICM2022, which will be taking place in St Petersburg next year. Since the ICM came up in our discussion, I certainly hope we will be able to have many, many more in-person participants than the pandemic situation allowed for the ECM. I really hope to welcome a large Slovenian delegation to St Petersburg next year. Here, in Portorož, I made many new friends, like Prof. Franc Forstnerič, and it would be super nice to see them all again next year in St Petersburg.

And finally, the standard question: How will you remember our country after this visit?

My wife and I will certainly be longing to come visit again. Of course, we expected to see beautiful nature interwoven with the rich history of the land. We also expected a chance to enjoy things like a swim in the sea and a delicious dinner. But what really conquered our hearts was the warm, easy, and remarkably "normal" attitude of people here. We certainly come from parts of the world where people show either stress or indifference much more frequently.