Softness and stiffness of diffurs

Mikhail Entov , a professor at the Faculty of Mathematics of the Technion (Israel), commented on the award of the Royal Academy of Sciences of the Sweden Krowford Academy of Sciences for 2016 in mathematics. This high award was awarded to professor at the University of Stanford (USA) to Yakov Eliashberg for "the development of contact and sympyed topologies and breakthrough discoveries in the field of strict and soft methods [solving differential equations and inequalities] (Rigidity and Fexibility Phenomena)."

Yakov Matveevich Eliashberg was born in 1946 in Leningrad. He graduated from the Mathematics and Mechanical Department of Leningrad University, studied with Vladimir Rokhlin. During this period, he was also under the strong mathematical influence of Mikhail Gromov, who was slightly older than him (when Eliashberg was a student, Gromov was a graduate student). In 1972, he defended the brilliant candidate dissertation on differential topology, but because of his Jewish origin, he could not find work in Leningrad and moved to Syktyvkar, where he worked at the University of Syktyvkar until 1979.

In 1979 he returned to Leningrad and filed on departure from the USSR. For about 7 years he was in “Refusal”, then he left for the USA. During the years of “refusal” he worked as a programmer, continuing to engage in mathematics in his free time. Since 1989, he is a professor at Stanford University. Pupils grown by him in Stanford are a whole school in sympyes and contact topology. In 1986, 1998 and 2006, Yakov Eliashberg was an invited speaker at the international mathematical congresses (in 2006 - with a plenary report; in 1986, the Soviet authorities did not allow him to go to Congress). He is a member of the US National Academy of Sciences, laureate of numerous scientific awards.

In the late 1960s-early 1970s, M. Gromov developed a very deep geometric approach to the study of general differential equations and inequalities. In very general terms, Gromovsky approach is as follows. Each differential equation/inequality has a certain “simple” version - the solution of this simple version is called a formal solution to the original equation/inequality.

Each real solution (if any) of the initial differential equation/inequality immediately gives formal immediately - this is simple. But is it possible to make a present from a formal solution (if any) and how - the question is very non -trivial. For some types of tasks from any formal solution (if any at all), the present can be made; Then, in the terminology introduced by Thunder, they say that the task is soft (Soft or Fexible). And for some problems, the presence of a formal solution does not guarantee the existence of the present; Then, according to Gromov, the task is called hard (Hard or Rigid).

For example, it is easy to build a “formal differentiated function” on a circle that does not have critical points, while there are no real functions with this property - this is a simple example of a tough task. On the other hand, in the sensational work of the late 1950s, Stephen Smeil discovered and proved the property of softness for a very important class of display, the so-called. diving areas. Then this theorem was summarized by various authors, but decisive progress was achieved in the works of Gromov and Eliashberg.

Accordingly, in order to show that the task is soft, it is necessary to show how to build real solutions from formal ones, and in order to show that it is tough, we must prove that there are no real solutions, although there are formal ones. Hence the terms Fexibility Methods and Rigidity Methods . Evidence both in the other direction is usually very difficult, and formal solutions, as a rule, are very difficult to find. And most importantly, it often happens that it is initially incomprehensible, a tough task or soft, that is, which of these two properties to prove; And in order to guess it, you need a very powerful geometric intuition.

Eliashberg's style is that he has this intuition is super -powerful and deep, and on the basis of this intuition, he knows how to make very unexpected moves in solving geometric problems. Eliashberg’s main mathematical contribution is to study the softness/stiffness of tasks arising in the areas of mathematics called sympy and contact topology.

Symptical topology studies mathematical objects of even dimension that arise during a mathematical description of mechanical systems. Contact topology-the “twin sister” System-studies related objects of an odd dimension. Both of these areas are rooted in the work of Henri Poincare, who was a pioneer in the study of mechanical systems using geometric and topological methods.

In the 1960s, Vladimir Arnold reformulated the mathematical language of classical mechanics in modern geometric terms and, under the influence of Poincar's work, formulated several very deep hypotheses about the mathematical objects that arise.

In the 1970s, M. Gromov and J. Eliashberg realized that Arnold’s hypotheses can be reformed as a question whether certain tasks are gestured or soft, and that the answer to this question would show whether the behavior of geometric objects arising in classical mechanics is something fundamentally new and special (if the task is tough), or, on the contrary, these objects are actually not distinguished from others from others similar geometric objects (if the task is soft).

In the late 1970s, while still in Syktyvkar, Eliashberg came up with the first evidence that in the dimension of 2 the task of Arnold is tough. In the 1980s, other works appeared (including the most important work of Gromov and another work of Eliashberg), showing that there are different other strict tasks associated with geometric objects arising in classical mechanics, from which these objects behave in their own special way, not similar to other similar geometric objects.

Thus, the area of ​​mathematics, which studies these objects, received the “right to independence” - it was called a symptian topology. In the second half of the 1980s, the symptical topology began to develop rapidly and, surprisingly, it soon turned out that this area of ​​mathematics, which originally arose from classical mechanics, is of greater importance for the theory of strings in modern theoretical physics.

Also, in the late 1980s, Eliashberg wrote a number of fundamental works, which showed that the odd “twin sister” of the symptical topology-the so-called contact topology-has the right to exist as a separate area of ​​mathematics. Among other things, Eliashberg, in a sense, spent a clear line between soft and hard tasks in three -dimensional contact topology. These works of Eliashberg determined the direction of the entire further development of contact topology.

Since the late 1980s, Yakov Eliashberg has written many important and brilliant works on sympy and contact topology, as well as differential topology and multidimensional comprehensive analysis, showing the rigidity or softness of certain tasks and creating new powerful tools for studying the objects that occur in these areas. He is actively working today - his work in recent years is absolutely outstanding.

In personal terms, he is an exceptionally kind person who is always generously divided by his ideas and helps everyone who needs it.

We express gratitude to the Academician of the Russian Academy of Sciences to the professor of the Higher Professional Education V. A. Vasiliev for help in the preparation of the material.

See also:
Page J. Eliashberg on the site of Stanford University:
http://mathematics.stanford.edu/people/name/yakov-eliashberg/