Jacob Sinai. Jewish traditions and exact sciences. Awards and prizes

Y. G. Sinai was born into a family with rich cultural traditions. His parents were researchers in the field of medicine, and his grandfather was V.F. Kagan, one of the first mathematicians in Russia who worked in the field of non-Euclidean and differential geometry. He studied at the Faculty of Mechanics and Mathematics of Moscow State University, graduating in 1957. Student of A. N. Kolmogorov. Candidate of Sciences (1960), Doctor of Sciences (1964). Since 1960 he worked at Moscow State University, since 1971 - professor. He also worked as a senior (1962), chief (1986) researcher at the Institute of Theoretical Physics. L. D. Landau. Since 1993 - Professor at Princeton University. Member of the American Mathematical Society.

Sinai's main works lie in the field of both mathematics and mathematical physics, especially in the close interweaving of probability theory, the theory of dynamical systems, ergodic theory, and others mathematical problems statistical physics. He was among the first to find the ability to calculate entropy for a wide class of dynamical systems (the so-called “Kolmogorov-Sinai entropy”). Great importance have his work on geodesic flows on surfaces of negative curvature, where he proved that shifts along the trajectories of a geodesic flow generate random processes that have the strongest possible properties of stochasticity and, among other things, satisfy the central limit theorem probability theory.

Among his students, the most famous is G. A. Margulis.

Winner of the Poincare Prize (2009), the International Dobrushin Prize (2009), the Wolf Prize (1996/7). Awarded the Boltzmann Medal (1986).

In 2009 he was elected a foreign member of the British Royal Society.

Books in Russian

Sinai Ya. G. Theory of phase transitions: rigorous results - M.: Nauka, 1980
Kornfeld I. P., Sinai Ya. G., Fomin S. V. Ergodic theory - M.: Nauka, 1980
Sinai Ya. G. Course on probability theory. Part 1, Part 2 - M.: Moscow State University Publishing House, 1985

Professor of the Faculty of Mechanics and Mathematics of Moscow State University and the Department of Mathematics of the National Research University Higher School of Economics, Professor of Cornell University (USA), Vice-President of the Moscow Mathematical Society, Rector of the Moscow Mathematical Society Independent University Yuliy Ilyashenko.

Yakov Grigorievich Sinai
Abel Prize laureate 2014

On March 26 in Oslo, the President of the Norwegian Academy of Sciences announced the name of the winner of the Abel Prize for 2014, an analogue of the Nobel Prize in mathematics. It was an outstanding scientist representing Russia and the USA, Yakov Grigorievich Sinai. This prize is named after the mathematician . The Norwegian Academy of Sciences and Letters selects its laureate by a committee of five leading international mathematicians. Since 2003, winners of this prize have been those scientists whose work is of extraordinary depth and has had a significant impact on this field of science. Yakov Grigorievich Sinai received it “for his fundamental contribution to the study of dynamical systems, ergodic theory and mathematical physics.”

Kolmogorov School

— So why is Yakov Sinai recognized as the laureate of the most prestigious prize in the field of mathematics?

— Yakov Grigorievich is one of the most famous students. In turn, Andrei Nikolaevich is a student of the founder of the Moscow mathematical school. Kolmogorov is one of the most remarkable not only mathematicians, but also scientists of the twentieth century. He grew his own huge school, in which, in addition to Sinai, many academicians and professors became famous. I will name only one of them - . Sinai, in turn, created a school, about which I will say a few words later.

Andrei Nikolaevich Kolmogorov made fundamental contributions to various fields of mathematics. His works on probability theory and dynamical systems are especially famous. Yakov Grigorievich has been working all his life at the intersection of these two areas with mathematical physics.

Probability theory and dynamic systems theory.

— What do these two sciences do?

— Probability theory studies random events. For example, you toss a coin and randomly get heads or tails. One of the main results of probability theory is the law large numbers, proved by Kolmogorov. It lies in the fact that on average the number of heads or tails will be the same for a large number of trials. What I said is not strict mathematical formulation. One of Kolmogorov's main achievements was that he gave an exact mathematical meaning to this naive statement, and then proved what happened.

The theory of differential equations or dynamical systems at first glance deals with opposite problems. She studies the so-called deterministic, completely predictable processes. was the first to understand that differential equations describe most processes occurring in nature over time. For example, the flight of planets, as well as the movement of molecules. Using the theory of differential equations he created, Newton described the rotation of the planets around the Sun and, in particular, proved the laws previously discovered experimentally. For example, the fact that all planets move around the Sun in flat orbits shaped like an ellipse.

At the end of the 18th century, mathematicians began to understand that differential equations have the so-called property of uniqueness of solutions. If we know at some point in time the state of the process (for example, the position of the planet and its speed), then we can predict at an infinite time in the future, and also reconstruct at an infinite time in the past the fate of this planet, its flight, trajectory.

P.S.

Abel Prize ceremony Yakov Grigorievich Sinai took place on May 20, 2014.
The award ceremony took place in the atrium of the University of Oslo (Aula), at the Faculty of Law, where from 1947 to 1989 Nobel Prize peace. The amount of this award is about a million dollars.

Yakov Grigorievich Sinai(born September 21, 1935, Moscow, USSR) - Soviet and American mathematician, full member RAS (December 7, 1991), laureate of several prestigious awards, including the Abel Prize (2014).

Biography

Y. G. Sinai was born into a family of medical scientists. Grandson of V.F. Kagan, one of the first mathematicians in Russia who worked in the field of non-Euclidean and differential geometry. Father - Lieutenant Colonel of the Medical Service, Doctor of Medical Sciences Grigory Yakovlevich Sinai (1902-1952), head of the Department of Microbiology of the 3rd Moscow medical institute, since 1945, professor of the department of microbiology and virology of the 2nd Moscow State Medical Institute, editor of the fundamental manual “Microbiological methods of research in infectious diseases” (1940, 1949), author of the monographs “Tularemia” (1940) and “A short guide to fighting the plague” (1941 ). Mother - Nadezhda Veniaminovna Kagan (1900-1938), senior researcher at the Institute of Experimental Medicine named after. M. Gorky; was developing a goat vaccine against spring-summer encephalitis; together with laboratory assistant N. Ya. Utkina, she died as a result of infection with a drug of the encephalitis virus, the properties of which she was studying. Brother - mechanic G.I. Barenblatt.

He studied at the Faculty of Mechanics and Mathematics of Moscow State University, graduating in 1957. In 1956, he married his fellow student Elena Bentsionovna Vul, daughter famous physicist Benzion Moiseevich Vul.

Student of A. N. Kolmogorov. Candidate of Sciences (1960), Doctor of Sciences (1964). Since 1960 he worked at Moscow State University, since 1971 - professor. He also worked as a senior (1962) and then chief (1986) researcher at the Institute of Theoretical Physics. L. D. Landau. Since 1993 - Professor at Princeton University.

Scientific interests

The main works lie in the field of both mathematics and mathematical physics, especially in the close intertwining of probability theory, the theory of dynamical systems, ergodic theory and other mathematical problems of statistical physics. He was among the first to find the ability to calculate entropy for a wide class of dynamic systems (the so-called “Kolmogorov-Sinai entropy”). Of great importance are his works on geodesic flows on surfaces of negative curvature, where he proved that shifts along the trajectories of a geodesic flow generate random processes that have the strongest possible properties of stochasticity and, among other things, satisfy the central limit theorem of probability theory. A large series of works is devoted to the theory of scattering billiards - “Sinai billiards”. The works of Ya. G. Sinai in the field of the theory of phase transitions, quantum chaos, dynamic properties of the Burgers equation, and one-dimensional dynamics are well known.

Among his students, the most famous is G. A. Margulis.

In 2009 he was elected a foreign member of the British Royal Society. Member of the US National Academy of Sciences. Since 2012 he has been a full member of the American Mathematical Society.

Awards and prizes

Boltzmann Medal (1986)
A. A. Markov Prize of the USSR Academy of Sciences (1989)
Solomon Lefschetz Memorial Lecture (1989)
Danny Heineman Prize in Mathematical Physics (1990)
Dirac Medal (1992)
Wolf Prize in Mathematics (1996/1997)
Lecture by Yu. Moser (2001)
Nemmers Prize in Mathematics (2002)
Kolmogorov Medal (2007)
Poincaré Prize (2009)
International Dobrushin Prize (2009)
Steele Prize (2013)
Abel Prize (2014)
Marcel Grossmann Prize (2015)

Proceedings

Sinai Ya. G. Theory of phase transitions: rigorous results. - M.: Nauka, 1980.
Kornfeld I. P., Sinai Ya. G., Fomin S. V. Ergodic theory. - M.: Nauka, 1980.
Sinai Ya. G. Course on probability theory. Part 1 - M.: Moscow State University Publishing House, 1985.
Sinai Ya. G. Course on probability theory. Part 2 - M.: Moscow State University Publishing House, 1986.
Sinai Ya. G. Contemporary issues ergodic theory. - M.: Fizmatgiz, 1995.
Yakov G. Sinai. Selecta. Volume I: Ergodic Theory and Dynamical Systems, Springer, 2010.
Yakov G. Sinai. Selecta. Volume II: Probability Theory, Statistical Mechanics, Mathematics Physics and Mathematical Fluid Dynamics, Springer, 2010.

Multicomponent random systems / IPPI AS USSR; resp. ed. R. L. Dobrushin, Ya. G. Sinai. - M.: Nauka, 1978. - 324 p.
Strange attractors: collection of articles / trans. from English edited by Ya. G. Sinaya, L. P. Shilnikova. - M.: Mir, 1981. - 253 p.
Sailer E. Gauge theories: connections with constructive quantum theory fields and statistical mechanics / trans. from English V.V. Anshelevich, E.I. Dinaburg; Ed. Ya. G. Sinaya. - M.: Mir, 1985. - 222 p.
Neumann J. von. Selected works on functional analysis. In 2 vols. / Ed. A. M. Vershika, A. N. Kolmogorov and Ya. G. Sinaya. - M.: Nauka, 1987.
Fractals in physics: Proceedings of the VI international symposium. Per. from English / Ed. Y. G. Sinaya and I. M. Khalatnikova. - M.: Mir, 1988. - 670 p.
Mathematical events of the twentieth century. Collection of articles / Ed. Yu. S. Osipov, A. A. Bolibrukh, Ya. G. Sinai. - M.: Fazis, 2003, - 548 p.

Sources

Ilyashenko Yu. S. Ya. G. Sinai, winner of the Abel Prize // Mathematical education. - 2015. - Issue. 19 (third episode). - P. 40-51.
Raussen M., Skau K. Interview with J. G. Sinai, Abel laureate 2014 // Mathematical education. - 2015. - Issue. 19 (third episode). - P. 52-69.

On March 26 in Oslo, the President of the Norwegian Academy of Sciences announced the name of the winner of the Abel Prize for 2014 - an analogue of the Nobel Prize in mathematics. It was an outstanding scientist representing Russia and the USA, Yakov Grigorievich Sinai. This prize is named after the mathematician Niels Henrik Abel. The Norwegian Academy of Sciences and Letters selects its laureate by a committee of five leading international mathematicians. Since 2003, winners of this prize have been those scientists whose work is of extraordinary depth and has had a significant impact on this field of science. Yakov Grigorievich Sinai received it “for his fundamental contribution to the study of dynamical systems, ergodic theory and mathematical physics.”

Kolmogorov School

- So why is Yakov Sinai recognized as the laureate of the most prestigious prize in the field of mathematics?

Yakov Grigorievich is one of the most famous students of Andrei Nikolaevich Kolmogorov. In turn, Andrei Nikolaevich is a student of the founder of the Moscow mathematical school, Nikolai Nikolaevich Luzin. Kolmogorov is one of the most remarkable not only mathematicians, but also scientists of the twentieth century. He grew his own huge school, in which, in addition to Sinai, many academicians and professors became famous. I will name only one of them - Vladimir Igorevich Arnold. Sinai, in turn, created a school, about which I will say a few words later.

Probability theory and dynamical systems theory

What do these two sciences do?

Probability theory studies random events. For example, you toss a coin and randomly get heads or tails. One of the main results of probability theory is the law of large numbers, proven by Kolmogorov. It lies in the fact that on average the number of heads or tails will be the same for a large number of trials. What I said is not a strict mathematical formulation. One of Kolmogorov's main achievements was that he gave an exact mathematical meaning to this naive statement, and then proved what happened.

The theory of differential equations or dynamical systems at first glance deals with opposite problems. She studies the so-called deterministic, completely predictable processes. Newton was the first to understand that differential equations describe most processes that occur in nature over time. For example, the flight of planets, as well as the movement of molecules. Using the theory of differential equations he created, Newton described the rotation of the planets around the Sun and, in particular, proved Kepler’s laws that had previously been discovered experimentally. For example, the fact that all planets move around the Sun in flat orbits shaped like an ellipse.

Moreover, Laplace realized that the same principle of determinism applies not only to the movement of planets, but also to the movement of microscopic objects. For example, molecules.

So this is a universal property?

Yes. This is the universal property of uniqueness. And in his treatise on the theory of probability, Laplace wrote: “A mind that would know for any at this moment all the forces that animate nature, and the relative position of all of it components, if, in addition, it turned out to be extensive enough to subject these data to analysis, it would embrace in one formula the movement of the greatest bodies of the Universe on an equal basis with the movements of the lightest atoms; there would be nothing left that would be unreliable for him, and the future, as well as the past, would appear before his eyes.”

This is much more than a mathematical result. This is a philosophy that comprehends the development of the entire Universe around us. Philosophy, despite Laplace's pathos, is rather dull. It lies in the fact that we live in a world in which everything is predicted. If some great mind knew the initial velocities and positions of all molecules and all other bodies in the Universe, he would calmly predict the past and reconstruct the future.

But he doesn't know.

But he doesn't know. And most importantly, the subsequent development of science refuted this philosophy. Yakov Grigorievich Sinai works in this area.

In the 19th century, it seemed that there were no more opposing branches of mathematics than differential equations and probability theory. But the development of mathematics in the twentieth century showed that these are two closely intertwined areas. And Sinai made a decisive contribution to the understanding of these connections. However, more on this a little later.

Before moving on to the story of these connections, which are studied by the so-called ergodic theory, I want to talk about some of Sinai’s youthful works.

Sinai's early works

- Richard Feynman writes that the variety of natural laws is not depressingly vast. This happens because different laws are described by the same mathematical formulas. The same can be said about differential equations. At first glance, the variety of differential equations seems completely endless. But there is an approach that allows many differential equations to be considered the same. Roughly speaking, such equations are obtained from each other by replacing coordinates, and thus, despite their external differences, they have deep internal similarity and almost identity. The question arises: how do you know whether two differential equations are the same or different? To answer this question, mathematicians study so-called invariants. These are properties of differential equations that do not change when we make coordinate changes. If we saw two differential equations that are not similar in appearance, and the invariant that we discovered is calculated for them and takes different meanings, this means that no changes in coordinates can transform one equation into another.

In addition to differential equations, there are also mappings. These are something like functions. The function matches one number to another, and the display matches one point to another. Well, for example, in school they study mappings of the plane - rotations, translations, extensions. And you can study much more complex mappings of the plane, for example, take a straight line complex numbers: z = x+iy and consider the mappings p(z) = z 2 or p(z)= z 2 +C. Dynamical systems study not only differential equations, but also iterations of mappings. Writing the iterative square of the mapping p is the same as taking the mapping p and applying it not to z, but to the image of the point z under the action of the mapping p: P 2 (z) = P(P(z)). A good exercise is to write down what polynomial and what degree it will produce. Dynamical systems consider the mapping p applied k times and examine what happens to the point: p k (z), k=1,2... as k goes to infinity. This was such a small example that shows that the theory of dynamical systems deals not only with differential equations, but also with mappings and their iterations.

In mapping theory, the so-called Bernoulli shift is very popular. The Bernoulli shift can be understood as a mathematical formalization of the history of coin tossing. We flip a coin and record the results of heads and tails.

Now imagine that we are not throwing a coin, but, say, a six-sided die. And it falls on one of the six faces. We are recording the history of these throws. Looking at the resulting sequences, it is easy to come up with a mapping (the so-called one-position shift mappings), which I will not describe in detail; it is called the Bernoulli shift.

For a long time there was a question about whether these are different or identical dynamical systems: Bernoulli shifts in a sequence of two and six symbols. Andrei Nikolaevich Kolmogorov came up with an invariant called “entropy” and which made it possible to prove that these two dynamic systems are different. In other words, the Bernoulli shift for a sequence of two symbols and of six symbols (Kolmogorov had three symbols instead of six) are different, nonequivalent dynamical systems.

Young Yakov Sinai, when he was a graduate student of Kolmogorov, took an active part in the development of the theory of a new invariant, and this invariant entered the theory of dynamical systems and literally permeated it through and through under the name “Kolmogorov-Sinai entropy.”

This, in fact, was the first major cycle of works by Yakov Grigorievich Sinai?

Absolutely right. The invariant was introduced by Kolmogorov, and they developed it together.

Ergodic theory

- The next important cycle of Sinai’s works relates to the so-called ergodic theory. Here it is worth taking a step back again and telling where ergodic theory came from.

From Laplace's point of view, the movement of the molecules of the air around us is described by differential equations. Here we are sitting in a room and breathing air whose behavior represents a solution differential equation in space with a very, very large number of coordinates. Question: why do we breathe homogeneous air? Why is the pressure in the upper right corner of the room and in the opposite, lower left corner of the room the same? After all, the molecules in the lower right corner do not know at all what is happening in the upper left. Why do they behave the same?

The Austrian physicist Boltzmann at the end of the 19th century tried to comprehend this issue and came up with the so-called ergodic theory. He proposed that solutions to very complex differential equations behave probabilistically. In geometric language, this assumption looks like this. The solution to a differential equation is a description of the movement of a point in space. This space can have a lot of coordinates or, as they say, a very large dimension. It is called phase space. Boltzmann suggested that if we wait long enough, the solution to a complex differential equation will have time to visit all regions of phase space. For example, the entire collection of molecules in a room is represented by one point in phase space of colossal dimension. This point will visit all parts of the phase space and will visit each part with a frequency proportional to its size. You can imagine the following illustration: there is a volume in space (relatively speaking, a room), and one point moves very, very quickly there, which, of course, occupies a certain position at each moment of time. But after enough time, she will have time to visit every cubic decimeter of the room. And if we give her even more time, she will have time to visit every cubic centimeter. If even longer - in every cubic millimeter. And so on…

Ergodic theory formalizes exactly what this statement means, which I formulated intuitively. And turns it into a theorem. However, Boltzmann formulated only concepts and hypotheses. He did not prove a single theorem in ergodic theory.

That is, he was just getting started.

Yes. He was something of a visionary.

Ergodic theory was formalized in the 1930s by Birhoff and von Neumann, who were the first to formulate neat theorems and prove them under certain conditions. It turned out that they are valid not for any dynamic system, but for a dynamic system that preserves the so-called phase volume. We can compare the movement of points under the action of a differential equation to the movement of molecules under the action of a gas flow or the movement of water particles under the action of a hydrodynamic flow. So, gas is compressible, but water is not. Dynamic systems that conserve volume are similar to the flow of water rather than the flow of gas. It was for such dynamical systems that Birhoff and von Neumann proved the ergodic theorem. This theorem builds a bridge between probability theory and dynamical systems.

Here is a thought experiment from probability theory. Coins are thrown randomly onto a table with large and small plates.

Probability theory states that after large number throwing the number of coins on each plate will be proportional to its area. But here is what the theory of dynamical systems says: a point moving under the action of an ergodic differential equation visits each section of phase space with the same frequency as a randomly thrown “coin” would fall there.

In order for a dynamical system to have the property of ergodicity, it is necessary to impose conditions that are very difficult to verify. Not all dynamic systems have ergodic behavior, that is, the ability to visit any corner of the phase space. Question: is it true that gas dynamics systems have this property?

The young Jacob Sinai took up this problem. At the same time, a glorious generation of scientists was studying the theory of dynamical systems - Anosov and Arnold in Russia, Smale in the USA. Smale came to Russia and had a very strong influence on our scientists. He also wrote about the strong influence they had on him. In particular, one of the problems posed by Smale was to prove (whatever that means) the structural stability of geodesic flow on a manifold of negative curvature (too many jargon here to explain this to the uninitiated). I mentioned this problem because, while thinking about it, Dmitry Viktorovich Anosov created the theory of so-called hyperbolic dynamic systems. The geodesic flow discussed is one of the important, but far from the only example of a hyperbolic system.

Does this have anything to do with terrestrial geodesy?

Only because the geodesic line on the surface is the shortest line. And geodesy deals with measuring distances on earth and, in particular, drawing the shortest paths between two points. Therefore, the connection is direct, but it takes too long to explain.

Yakov Sinai was the first to apply the methods of hyperbolic theory to the Boltzmann hypothesis and to problems of gas dynamics. He greatly advanced the proof of Boltzmann's ergodic hypothesis (his followers are now working on it, and this problem, which has not been completely solved, has now been studied very deeply). It is now called the ergodic Boltzmann-Sinai hypothesis.

Not all dynamic systems are like water flow and maintain phase volume. There are many dynamic systems that are similar to the raging wind carrying clouds of dust, or the movement of dispersed matter in the Universe. This dispersed substance can, over time, form clusters, group and form figures much smaller than the space in which the movement began. Initially, matter uniformly dispersed throughout the Universe can form very dense clusters, and we can talk about the mass different parts of these clusters.

This picture illustrates what mathematicians call the limiting invariant measure for a dynamical system. One of the most famous and also intensively studied measures is the so-called Sinai-Ruelle-Bowen measure. Yakov Grigorievich was one of the three creators of this concept, and it is also central to the theory of dynamic systems.

Common Faith modern mathematicians is that most dynamic systems exhibit both deterministic and probabilistic behavior. Deterministic behavior controls the output of all particles to that set, that accumulation of matter on which the Sinai-Ruelle-Bowen measure is concentrated. This cluster is called an attractor. And the theory of probability controls the movement along this accumulation of matter - along the attractor.

Turbulence problem

The word "turbulence" may seem unfamiliar. However, almost everyone who flew on airplanes heard it. Remember calm voice flight attendant: “Our plane entered a zone of turbulence. Fasten seat belts". This means that the plane has entered a zone of air vortices that swirl, collide and interfere with the flight. Turbulent fluid flow looks approximately the same.

Recently, Yakov Grigorievich has put a lot of effort into studying mathematical hydrodynamics.

Fluid flow is described by the so-called Navier-Stokes equation. This is a partial differential equation. His research was named by the Clay Institute as one of the seven leading problems of the 21st century, and it is one of the so-called millennium prize problems, for the solution of which a million-dollar prize has been announced. The problem is this: start with the Navier-Stokes equations, fairly compact partial differential equations, and use them to explain completely mysterious phenomenon turbulence, which also in some sense contradicts the deterministic philosophy of Laplace. Namely, let’s imagine the following experiment: let’s take a liquid in a vessel and slowly accelerate it. For example, a vessel may be a gap between two cylinders containing liquid. One cylinder is stationary, and the other begins to spin slowly, accelerating to a very high speed. This process can be described by a differential equation, but only in an infinite-dimensional space. In accordance with the theory of existence and uniqueness, the same solution to a differential equation is studied in two experiments carried out in identical mode. And thus the same picture should be observed. Meanwhile, at first this hypothesis is indeed confirmed (that is, in two experiments the development of the flow is approximately the same: there are neat jets that are easy to trace and describe), but then small vortices appear, chaos begins, and two flow patterns in two almost identical experiments completely different from each other.

How to explain this? The hypothesis formulated by academician Arnold was that the Navier-Stokes equation is an infinite-dimensional hyperbolic system (as you can see, everything is connected in the theory of dynamical systems). This hypothesis has not yet been proven. One of the key questions relates to the equation describing the movement of a fluid without viscosity (the movement of the so-called ideal fluid). This is a simplified version of the Navier-Stokes equation, called the Euler equation. The question is: is it true that solutions of the Euler equation, in a certain sense, go to infinity in a finite time?

Yakov Grigorievich answered a similar question. If we continue solving the Euler equation into the complex region, then singularities arise there. This is a recent result, and it also has not only a mathematical, but also a physical and philosophical interpretation. It must be emphasized that Yakov Grigorievich has worked closely with physicists all his life. We can talk about this for a long time.

Tradition of giving

- You said that Sinai is the creator of his own mathematical school...

Like his teacher, Andrei Nikolaevich Kolmogorov, Yakov Grigorievich is the creator of an absolutely wonderful school. Many of his students created their own schools and became professors at various universities. I will name only one of them - Fields laureate G.A. Margulis. Sinai is an outstanding teacher. It retains the inherent Russian math school the principle of giving, coming from his teacher Kolmogorov.

That is, Yakov Grigorievich Sinai gives his ideas to his students?

Yes. The teacher generously gives his ideas to the students.

In a situation where Western scientists usually publish joint articles with their students, and this is fair (the statement of the problem and the idea of the solution are often the decisive contribution), the Russian tradition is to give this statement and the initial impetus to the student. And Sinai is a very generous giver.

It is known that the Sinai summer seminar has been taking place in Moscow for many years.

Absolutely right. When Sinai comes to Russia in the spring and summer, his seminar works intensively. Although recently Yakov Grigorievich has been mainly raising students at Princeton University, where he is a professor. Princeton's mathematics department is one of the world's great mathematics departments, home to many Fields Laureates. And Sinai occupies an honorable place in this mathematical guard.

The material was prepared with the support of the Moscow Department of Education.

P.S. The ceremonial presentation of the Abel Prize will take place on May 20: the award ceremony takes place in the atrium of the University of Oslo (Aula), at the Faculty of Law, where the Nobel Peace Prize was awarded from 1947 to 1989. The amount of this award is about a million dollars.

Interviewed by Natalya Ivanova-Gladilshchikova
"Gazeta.Ru"

Yakov Grigorievich Sinai(born September 21, Moscow, USSR) - Soviet and American mathematician, full member of the Russian Academy of Sciences (December 7, 1991), winner of a number of prestigious awards, including the Abel Prize (2014).

Biography

Y. G. Sinai was born into a family of medical scientists. Grandson of V.F. Kagan - one of the first mathematicians in Russia who worked in the field of non-Euclidean and differential geometry. Father - Lieutenant Colonel of the Medical Service, Doctor of Medical Sciences Grigory Yakovlevich Sinai (1902-1952), head of the Department of Microbiology, since 1945, Professor of the Department of Microbiology and Virology of the 2nd Moscow State Medical Institute, editor of the fundamental manual “Microbiological Research Methods for Infectious Diseases” (1940, 1949 ), author of the monographs “Tularemia” (1940) and “A Brief Guide to Fighting Plague” (1941). Mother - Nadezhda Veniaminovna Kagan (1900-1938), senior researcher at the Institute of Experimental Medicine named after. M. Gorky; was developing a goat vaccine against spring-summer encephalitis; together with laboratory assistant N. Ya. Utkina, she died as a result of infection with a drug of the encephalitis virus, the properties of which she was studying. Brother - mechanic G.I. Barenblatt.

Scientific interests

The main works lie in the fields of both mathematics and mathematical physics, especially in the close interweaving of probability theory, the theory of dynamical systems, ergodic theory and other mathematical problems of statistical physics. He was among the first to find the ability to calculate entropy for a wide class of dynamic systems (the so-called “Kolmogorov-Sinai entropy”). Of great importance are his works on geodesic flows on surfaces of negative curvature, where he proved that shifts along the trajectories of a geodesic flow generate random processes that have the strongest possible properties of stochasticity and, among other things, satisfy the central limit theorem of probability theory. A large series of works is devoted to the theory of scattering billiards - “Sinai billiards” ( English). The works of Ya. G. Sinai in the field of the theory of phase transitions, quantum chaos, dynamic properties of the Burgers equation, and one-dimensional dynamics are well known.

Among his students, the most famous is G. A. Margulis.

Awards and prizes

Proceedings

Sinai Ya. G. Theory of phase transitions: rigorous results. - M.: Science, 1980.
Kornfeld I. P., Sinai Ya. G., Fomin S. V. Ergodic theory. - M.: Science, 1980.
Sinai Ya. G. Probability theory course. Part 1 - M.: Moscow State University Publishing House, 1985.
Sinai Ya. G. Probability theory course. Part 2 - M.: Moscow State University Publishing House, 1986.
Sinai Ya. G. Modern problems of ergodic theory. - M.: Fizmatgiz, 1995.
Yakov G. Sinai. Selecta. Volume I: Ergodic Theory and Dynamical Systems, Springer, 2010.
Yakov G. Sinai. Selecta. Volume II: Probability Theory, Statistical Mechanics, Mathematics Physics and Mathematical Fluid Dynamics, Springer, 2010.

Publication Editor

Multicomponent random systems / ; resp. ed. R. L. Dobrushin, Ya. G. Sinai. - M.: Nauka, 1978. - 324 p.
Strange attractors: collection of articles / trans. from English edited by Y. G. Sinaya, L. P. Shilnikova. - M.: Mir, 1981. - 253 p.
Seiler E. Gauge theories: connections with constructive quantum field theory and statistical mechanics / trans. from English V.V. Anshelevich, E.I. Dinaburg; Ed. Y. G. Sinaya. - M.: Mir, 1985. - 222 p.
Neumann J. von. Selected works on functional analysis. In 2 vols. / Ed. A. M. Vershika, A. N. Kolmogorov and Ya. G. Sinaya. - M.: Nauka, 1987.
Fractals in physics: Proceedings of the VI international symposium. Per. from English / Ed. Y. G. Sinaya and I. M. Khalatnikova. - M.: Mir, 1988. - 670 p.
Mathematical events of the twentieth century. Collection of articles / Ed. Yu. S. Osipov, A. A. Bolibrukh, Ya. G. Sinai. - M.: Fazis, 2003, - 548 p.

Sources

Ilyashenko Yu. S.// Mathematical education. - 2015. - Vol. 19 (third episode). - pp. 40-51.
Raussen M., Skau K.// Mathematical education. - 2015. - Vol. 19 (third episode). - pp. 52-69.

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An excerpt characterizing Sinai, Yakov Grigorievich

- Saw.
“Tomorrow, they say, the Preobrazhensky people will treat them.”
- No, Lazarev is so lucky! 10 francs life pension.
- That's the hat, guys! - shouted the Transfiguration man, putting on the shaggy Frenchman’s hat.
- It’s a miracle, how good, lovely!
-Have you heard the review? - the guards officer said to the other. The third day was Napoleon, France, bravoure; [Napoleon, France, courage;] yesterday Alexandre, Russie, grandeur; [Alexander, Russia, greatness;] one day our sovereign gives feedback, and the next day Napoleon. Tomorrow the Emperor will send George to the bravest of the French guards. It's impossible! I must answer in kind.
Boris and his friend Zhilinsky also came to watch the Transfiguration banquet. Returning back, Boris noticed Rostov, who was standing at the corner of the house.
- Rostov! Hello; “We haven’t even met,” he told him, and could not resist asking him what had happened to him: Rostov’s face was so strangely gloomy and upset.
“Nothing, nothing,” answered Rostov.
-Will you come in?
- Yes, I’ll come in.
Rostov stood at the corner for a long time, looking at the feasters from afar. A painful work was going on in his mind, which he could not complete. Terrible doubts arose in my soul. Then he remembered Denisov with his changed expression, with his humility, and the whole hospital with these severed arms and legs, with this dirt and disease. It seemed to him so vividly that he could now smell this hospital smell of a dead body that he looked around to understand where this smell could come from. Then he remembered this smug Bonaparte with his white hand, who was now the emperor, whom Emperor Alexander loves and respects. What are the torn off arms, legs, and killed people for? Then he remembered the awarded Lazarev and Denisov, punished and unforgiven. He caught himself having such strange thoughts that he was frightened by them.
The smell of food from the Preobrazhentsev and hunger brought him out of this state: he had to eat something before leaving. He went to the hotel he had seen in the morning. At the hotel he found so many people, officers, just like him, who had arrived in civilian dress, that he had to force himself to have dinner. Two officers from the same division joined him. The conversation naturally turned to peace. The officers and comrades of Rostov, like most of the army, were dissatisfied with the peace concluded after Friedland. They said that if they had held out any longer, Napoleon would have disappeared, that he had no crackers or ammunition in his troops. Nikolai ate in silence and mostly drank. He drank one or two bottles of wine. The internal work that arose in him, not being resolved, still tormented him. He was afraid to indulge in his thoughts and could not leave them. Suddenly, at the words of one of the officers that it was offensive to look at the French, Rostov began to shout with vehemence, which was not justified in any way, and therefore greatly surprised the officers.
– And how can you judge what would be better! - he shouted with his face suddenly flushed with blood. - How can you judge the actions of the sovereign, what right do we have to reason?! We cannot understand either the goals or the actions of the sovereign!
“Yes, I didn’t say a word about the sovereign,” the officer justified himself, unable to explain his temper otherwise than by the fact that Rostov was drunk.
But Rostov did not listen.
“We are not diplomatic officials, but we are soldiers and nothing more,” he continued. “They tell us to die—that’s how we die.” And if they punish, it means he is guilty; It's not for us to judge. It pleases the sovereign emperor to recognize Bonaparte as emperor and enter into an alliance with him—that means it must be so. Otherwise, if we began to judge and reason about everything, there would be nothing sacred left. This way we will say that there is no God, there is nothing,” Nikolai shouted, hitting the table, very inappropriately, according to the concepts of his interlocutors, but very consistently in the course of his thoughts.
“Our job is to do our duty, to hack and not think, that’s all,” he concluded.
“And drink,” said one of the officers, who did not want to quarrel.
“Yes, and drink,” Nikolai picked up. - Hey, you! Another bottle! - he shouted.

In 1808, Emperor Alexander traveled to Erfurt for a new meeting with Emperor Napoleon, and in high society in St. Petersburg there was a lot of talk about the greatness of this solemn meeting.
In 1809, the closeness of the two rulers of the world, as Napoleon and Alexander were called, reached the point that when Napoleon declared war on Austria that year, the Russian corps went abroad to assist their former enemy Bonaparte against their former ally, the Austrian emperor; to the point that in high society they talked about the possibility of a marriage between Napoleon and one of the sisters of Emperor Alexander. But, apart from external political considerations, at this time the attention of Russian society was especially keenly drawn to the internal transformations that were being carried out at that time in all parts of public administration.
Life meanwhile real life people with their own essential interests of health, illness, work, leisure, with their interests of thought, science, poetry, music, love, friendship, hatred, passions, proceeded as always independently and outside of political affinity or enmity with Napoleon Bonaparte, and outside of all possible transformations.
Prince Andrei lived in the village without a break for two years. All those enterprises on estates that Pierre started and did not bring to any result, constantly moving from one thing to another, all these enterprises, without showing them to anyone and without noticeable labor, were carried out by Prince Andrei.
He had in highest degree that practical tenacity that Pierre lacked, which, without scope or effort on his part, gave movement to the matter.
One of his estates of three hundred peasant souls was transferred to free cultivators (this was one of the first examples in Russia); in others, corvee was replaced by quitrent. In Bogucharovo, a learned grandmother was written out to his account to help mothers in labor, and for a salary the priest taught the children of peasants and courtyard servants to read and write.
Prince Andrei spent half of his time in Bald Mountains with his father and son, who was still with the nannies; the other half of the time in the Bogucharov monastery, as his father called his village. Despite the indifference he showed Pierre to all the external events of the world, he diligently followed them, received many books, and to his surprise he noticed when fresh people came to him or his father from St. Petersburg, from the very whirlpool of life, that these people, in knowledge of everything that happens in the external and domestic policy, far behind him, who was sitting in the village without a break.
In addition to classes on names, in addition to general studies of reading a wide variety of books, Prince Andrei was at this time engaged in a critical analysis of our last two unfortunate campaigns and drawing up a project to change our military regulations and regulations.
In the spring of 1809, Prince Andrei went to the Ryazan estates of his son, whom he was guardian.
Warmed by the spring sun, he sat in the stroller, looking at the first grass, the first birch leaves and the first clouds of white spring clouds scattering across the bright blue sky. He didn’t think about anything, but looked around cheerfully and meaninglessly.
We passed the carriage on which he had spoken with Pierre a year ago. We passed a dirty village, threshing floors, greenery, a descent with remaining snow near the bridge, an ascent through washed-out clay, stripes of stubble and green bushes here and there, and entered a birch forest on both sides of the road. It was almost hot in the forest; you couldn’t hear the wind. The birch tree, all dotted with green sticky leaves, did not move, and from under last year’s leaves, lifting them, the first green grass and purple flowers crawled out. The small spruce trees scattered here and there along the birch forest with their coarse, eternal greenness were an unpleasant reminder of winter. The horses snorted as they rode into the forest and began to fog up.
The footman Peter said something to the coachman, the coachman answered in the affirmative. But apparently Peter had little sympathy for the coachman: he turned on the box to the master.
- Your Excellency, how easy it is! – he said, smiling respectfully.
- What!
- Easy, your Excellency.
"What he says?" thought Prince Andrei. “Yes, that’s right about spring,” he thought, looking around. And everything is already green... how soon! And the birch, and the bird cherry, and the alder are already starting... But the oak is not noticeable. Yes, here it is, the oak tree.”
There was an oak tree on the edge of the road. Probably ten times older than the birches that made up the forest, it was ten times thicker and twice as tall as each birch. It was a huge oak tree, two girths wide, with branches that had been broken off for a long time and with broken bark overgrown with old sores. With his huge, clumsy, asymmetrically splayed, gnarled hands and fingers, he stood like an old, angry and contemptuous freak between the smiling birches. Only he alone did not want to submit to the charm of spring and did not want to see either spring or the sun.