Category: Russian Federation

Translartion. Region: Russians Fedetion –

Source: State University Higher School of Economics – State University Higher School of Economics –

Sergey Kuksin

© Higher School of Economics

Since March of this year Faculty of Mathematics, National Research University Higher School of Economics The leading Russian mathematician, Doctor of Physical and Mathematical Sciences, Honorary Professor of the University of Edinburgh, National Professor of the PRC and laureate began working Prizes named after A.M. Lyapunova Sergey Kuksin. In an interview with Vyshka.Glavnoe, he spoke about the role of personality in mathematics, KAM theory, and why analysis is so relevant today.

— Sergey Borisovich, what brought you to the HSE?

— A year ago, I received a mega-grant from the Russian government, started working at RUDN and visiting Moscow for seminars, and when the grant ended, I was offered a job at HSE. I know many people at the HSE Mathematics Department. Mathematicians, they are all connected one way or another. True, given that mathematics is divided into three large parts: algebra, geometry and analysis. What is algebra, it is clear, right? For example, it is addition of fractions, square trinomial, “x plus y squared” and so on. Geometry, as we all remember well, is plane geometry, problems on construction and all that sort of thing. Well, and analysis is when there are functions and graphs.

– And which one of them are you?

– I am analysis.

— Have you been here before?

— Of course, I have. HSE is a very good place: smart students, a strong faculty. Many years ago, I even gave a short course of lectures at the local mathematics department. But that experience was not very successful. The thing is that the HSE department was organized by big algebra enthusiasts and was focused on algebra, so the students were not very impressed. They simply did not understand why they needed it. And that is wrong. Everyone needs to know analysis. Analysis is also probability theory, which is very relevant now, since it is closely related to such topics as artificial intelligence, machine translation, and pattern recognition. By the way, the then management understood this well when they invited me to give the course. But in mathematics, in order to get something moving, you have to make serious efforts. And it seems that this is happening now — the expansion of the profile of the mathematics department. That is partly why they invited me.

— Will you teach or work as a researcher?

— First of all, I will work as a researcher. One of my main tasks is to participate in the creation of a seminar with the preliminary title “Dynamics, Analysis and Probability”. I would like it to be a seminar of the highest level, with the involvement of good speakers who motivate students to develop in this area. This is not easy, but it is possible, especially since the impetus to develop the analysis component comes from the faculty management. In particular, from the dean Alexandra Skripchenko. By the way, she recently defended her doctoral dissertation.

— Remember the most vivid impression in your life related to mathematics.

— My parents, with whom I was very lucky, subscribed to several magazines for me. One of them was “Knowledge is Power”. Once, when I was still in high school, I read an article about mathematics. And there was a phrase in it that I still remember: “The heights of mathematics are beautiful, and it’s a pity that very few can admire them.” I wanted to admire them and, yes, I confirm: they are beautiful.

— What qualities do you need to have to become a good mathematician?

— You know, mathematics is, fortunately, a gift that manifests itself early. Or doesn’t. That is, a person already at school understands whether mathematics is for him or not. Already in high school, I couldn’t imagine that I would do anything else in this life.

— Which of your scientific achievements do you consider the most significant?

– I’ll start from afar. There was such a scientist, the largest Soviet mathematician Andrei Nikolaevich Kolmogorov. He was a completely fantastic person who made a huge contribution to mathematics. Including was the founder of Cam-theoria. This is an abbreviation composed of the first letters of the surnames of the authors: Kolmogorov, Arnold and Moser. And now let’s figure out what Cam-theoria is. Consider the solar system. For this, they usually take five main planets from Venus to Saturn. We know that each planet rotates according to the ellipse – according to the law of Kepler. This is because the sun attracts it. But besides, the planets interact with each other. Therefore, their movement – the Kepler movement – is gradually distorted. And there is a relatively simple equation that describes how the planet interacts with the sun. But when we also take into account the interaction of the planets with each other, then small disturbances and interaction are added to the main equation. Due to these interactions, the orbit of the planets begin to gradually deform. The question that Isaac Newton still raised is what will happen to this ellipse, for example, after a million years? After all, he can burst, and then the planet will fly away to distant galaxies. The ellipse can stretch out so much that at the point closest to the sun, the planet will fall into the sun and burn. Ellips orbits of different planets can cross, and the planets will collide. It is clear, not in the next ten thousand years, but still. It was an outstanding, wonderful task, and it was solved with the help of Cam-theoria.

– And what is the answer?

— The answer is negative: the orbits will always be close to the initial ones. The exact statement is more complicated and has significant reservations, which it is inappropriate to state now. But such a simplified story describes the essence of KAM theory quite accurately. To answer your question about my contribution to mathematics: I learned to do calculations for systems that have not five or even five hundred, but an infinite number of components. This was the topic of my doctoral dissertation — “KAM theory for partial differential equations.”

— You were awarded the Lyapunov Prize of the Russian Academy of Sciences for it?

— Yes, that’s right: for the creation and development of the Kolmogorov-Arnold-Moser theory for partial differential equations.

— What would you like to achieve while working at the Higher School of Economics?

— To participate in the development of analysis at the HSE in particular and in Moscow in general. During the Soviet Union, analysis here was very strong, but for a number of reasons it has declined significantly. Unlike, say, algebra. Which confirms the thesis about the role of personality in history, since this happened solely due to the efforts of several outstanding algebraists who never left Moscow. They were the ones who preserved the seminars and the youth in the seminars.

— Would you like to become the person in mathematics who will move analysis forward?

— I would like to participate.

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