Pyotr Konstantinovich Rashevsky
(1907 - 1983)

 

      

Dear Colleague,

 

    We invite you to a meeting of the Moscow Mathematical Society, dedicated to the centenary of the birth of the outstanding mathematician, Professor Pyotr Konstantinovich RASHEVSKY, who was the head of the Department of Differential Geometry at the Faculty of Mechanics and Mathematicsof Moscow State University.  The meeting will take place on Tuesday, November 27, 2007, in Room 1624 (16th floor) of the Main Building of Moscow State University, starting at 6:00 PM.30 m

 

                      Agenda:

 

1. G.L. Litvinov, A.T. Fomenko. "In Memory of Petr Konstantinovich"

   Rashevsky".

 

2. E.B. Vinberg. "On the works of P.K. Rashevsky on the theory of Lie groups".

 

3. O.V. Manturov.  "Seminars by Pyotr Konstantinovich Rashevsky.

   Participants and Results."

 

4. D.B. Persits. "Pyotr Konstantinovich Rashevsky in my life".

 

   If you do not have a one-time pass to the Main Building of Moscow State University, you can apply for it either to the Board of the Moscow Mathematical Society or to the Department of Differential Geometry and Applications, tel. 939-39-40 (Valentina Nikolaevna Firsova).

 

                       Sincerely,

                       A.T. Fomenko

   Academician of the Russian Academy of Sciences, Head of the Department of Differential Geometry

       and Applications of the Faculty of Mechanics and Mathematics at Moscow State University

 

 

 

 

==================================================================

 

 

 

 

 

G.L. Litvinov. A.T. Fomenko

"IN MEMORY OF PETR KONSTANTINOVICH RASHEVSKY"

 

Presentation at the meeting of the Moscow Mathematical Society,

dedicated to the 100th anniversary of the birth of P.K. Rashevsky on November 27, 2007.

 

Key points of A.T. Fomenko's presentation

 

   a) P.K. Rashevsky - an outstanding mathematician, author of many remarkable works, widely known for his fundamental research in the fields of Riemannian geometry and tensor analysis, the theory of groups and Lie algebras, and the theory of their representations.(In particular, he obtained important results in the problem of describing tensors that admit a given invariance group, in the geometric theory of differential equations, solved Cartan's problem, investigated the structure of the set of spherical functions on homogeneous spaces, studied the associative enveloping algebra of Lie algebras and their infinite-dimensional representations, the topological properties of automorphismsLie groups).Rashevsky made significant contributions to several fundamental areas of modern geometry and significantly influenced their development.

 

b) P.K. Rashevsky For a long time, he headed the Department of Differential Geometry at the Faculty of Mechanics and Mathematics at Moscow State University (1964–1983). After his death, the department was temporarily merged with another department and was re-established in 1992 under the name "Department of DifferentialGeometry and Applications" (Head A.T. Fomenko). This department is actively developing, largely thanks to the problems and ideas that were posed and articulated by P.K. Rashevsky.

 

    c) P.K. Rashevsky was interested in a wide variety of problems in modern geometry.He had an exceptionally well-developed mathematical intuition, and he was very successful in setting tasks to his students.  For many years, the well-known seminar "Tensor Analysis and Its Applications" "Tensor Analysis and Its Applications" was held for many years.He was a focal point not only only for Moscow geometers, but also for many colleagues from other cities.The research conducted by various scholars within the framework of this seminar formed the content of the well-known periodical series of works (under the same name).These works (albeit not as regularly) are still published today.

 

   I will list the main research areas of today's Department of Differential Geometry and Applications, inspired by Rashevsky:

 

   1.Rashevsky devoted much attention to the study of the geometry and topology of Lie groups and homogeneous spaces.Several scientific disciplines have emerged from this.For example, the study of subgroups of Lie groups, non-homologous to zero in the enveloping Lie group, and the calculation of Poincaré polynomials of homogeneous spaces (for more details, see the presentation by Professor O.V. Manturov). Fomenko's description of totally geodesic surfaces, which realize non-trivial homology cycles in that realize non-trivial homology cycles insymmetric spaces (including including in the Lie groups).In particular, a description was obtained a description of completely geodesic and topologically non-trivial spheres in symmetric spaces.

 

   2.Geometric variational problems, including including multidimensional ones.It was P.K. Rashevsky who directed once directed my interests toward the study of globally minimal surfaces in Riemannian manifolds.This is where the theory I developed with my students originated:in particular, the proof of the existence of globally minimal surfaces in each class of spectral bordisms of a Riemannian manifold.And also - in each class of extraordinary homologies and cohomologies of a manifold (spectral bordisms are a special case).These ideas were developed in the works of Professors Dao Chong Thi and Le Hong Van. Subsequently, Professors A.O. Ivanov and A.A. Tuzhilin achieved significant results in the Steiner problem - the classification of one-dimensional minimal networks (with fixed ends (i.e.branching geodesics) or without boundary points at all) on two-dimensional surfaces, in particular, on a plane, on a torus, and on surfaces of constant negative curvature.Although Rashevsky himself did not work on one-dimensional did not work on one-dimensional branching geodesics himself, he was very interested in this new approach.

 

   3.Rashevsky often talked to me about the general properties of groups and algebras of Lie.He was interested in properties that hold simultaneously for all Lie algebras and for a sufficiently broad class (for example, the existence of a Poisson bracket on any finite-dimensional Lie algebra), non-degenerate on the orbits of the coadjoint action of the group on its own coalgebra. To a large extent, it was these ideas of Rashevskywere subsequently revived in the hypothesis formulated by A.S. Mishchenko and A.T. Fomenko, according to which any finite-dimensional Lie algebra possesses a complete commutative set of independent polynomials, i.e.in involution (with respect to the Poisson bracket).In other words, their number must be equal to half the sum of the dimension of the algebra and its index (index = dimension of the annihilator of a general position covector).Such sets of polynomials generate completely integrable systems of differential equations in the Liouville sense.In the case of semisimple algebras L, the hypothesis was proven by Mishchenko and Fomenko themselves, and in the general case (relatively recently) by S.T. Sadetov.In a sense, this means that on any finite-dimensional algebra L, there exists a polynomial analogue of Darboux coordinates (the classical Darbu's theorem states that in the neighborhood of each point on a symplectic manifold, the symplectic form can always be reduced to a constant canonical form with respect to locally existing Darboux coordinates).

 

   4.Rashevsky devoted a lot of attention to geometry in mathematical physics.It is no coincidence that in his famous book "Riemannian Geometry and Tensor Analysis," a significant portion is devoted to the theory of relativity and spinors (spinor representations of the orthogonal group).Today, the department continues to actively pursue the scientific field led by Professors V.L. Golo and A.I. Shafarevich, "Mathematical Physics, Geometry, and Topology."Studies on the geometry and topology of complex protein molecules stood out separately: molecules:Professors V.L. Golo, A.O. Ivanov, and A.A. Tuzhilin with their students, in collaboration with the Faculty of Biology at Moscow State University (the laboratory of Professor K.V. Shaitan).Another area of focus:  "Differential Equations in Geometric Problems of Celestial Mechanics and Mathematical Physics" is being developed by Associate Professor E.A. Kudryavtseva.

 

   5.Rashevsky was very interested in smooth functions, whose singularities fill non-degenerate submanifolds (Morse functions are a special case where these submanifolds are individual points).This topic was discussed repeatedly at the Rashevsky seminar.After some time, these ideas of his thoughts served as one of the foundations of the theory developed by Fomenko and his students, and, primarily, A.V. Bolsinov and A.A. Oshemkov, the theory of classification of integrable Hamiltonian systems with two degrees of freedom.It turned out that such systems are classified with an accuracy up to Liouville equivalence (as well as with accuracy up to continuous and smooth trajectory equivalence) by certain graphs whose vertices are the so-called"atoms", and some numerical labels some numerical labels are placed.Such invariants are more or less algorithmically calculated for a variety of specific mechanical and physical systems.By comparing these invariants, it was possible, for example, to discover unexpected topological trajectory isomorphisms between some known dynamical systems (e.g., the Jacobi system and the Euler system).I also managed to prove the topological (and smooth) non-equivalence of some well-known systems.

 

   6.Rashevsky himself did not have time to formulate his thoughts on certain aspects of algebraic topology and algebraic geometry in the form of published works.  However, he was constantly interested in topology and kept abreast of all the new noteworthy results.As a result, a favorable atmosphere has developed within the department for the development of this field.It is  currently undergoing active development within the Department of Differential Geometry.These are the students of Professor Yu.P. Solovyov:Associate Professor F.Yu. Popelensky and Assistant Professor I.M. Nikonov, as well as Professor A.B. Skopenkov and Associate Professor A.B. Zheglov.

 

   7.During Rashevsky's lifetime, computational geometry began its rapid development.Although Pyotr Konstantinovich himself Konstantinovich himself did not use a computer, but he followed this field with interest. field with interest.He was interested in mathematical modeling of physical processes.Once, I addressed the Academic Council of the Faculty of Mechanics and Mathematics with proposals on the integration of computational methods into geometry teaching.Today, computational geometry is a major and important topic in the department, developing under the the guidance of Associate Professor G.V. Nosovsky and Assistant Professor D.P. Ilyutko.A textbook on computational geometry has been written, and a special course is being taught.

 

   8.Rashevsky was also concerned with the foundations of mathematics.For example, he and I discussed non-standard mathematical analysis.  Rashevsky pondered extensively on the properties of very large natural numbers and contemplated the idea of a branching natural number series.This image emerged during the discussion of possible algorithms for comparing large numbers without "counting them down to zero."In his view, large numbers are fuzzy objects with fuzzy boundaries.A certain new algebra of such objects is emerging.Some ideas were published by Rashevsky in a note in the Russian Mathematical Surveys, which at the time elicited mixed reactions from some mathematicians.

    Rashevsky was a man of exceptional scientific integrity.If he didn't like a piece of work like a piece of work or found it flawed, he would clearly state so, regardless of who was involved.He was principled in matters of scientific and public morality.Once, as the chairman of the admissions committee at the Faculty of Mechanics and Mathematics, I categorically refused to comply with a behind-the-scenes request that came "from the very top" to improve an applicant's grade.  As a result, I got myself into trouble, but I never regretted such actions.

 

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Students and colleagues who participated in P.K. Rashevsky's seminar:

 

Akivis M.A. (Max Aizekovich)

Biktashev R.A.

Bogaevsky A.N.

Borovsky Yu.V. (Yuri Vladimirovich)

Gurevich G.B. (Grigory Borisovich)

Wagner V.V. (Victor Vladimirovich)

Vasiliev A.M. (Anatoly Mikhailovich)

Weinstein A.G.

Vartolomeev V.V. (Vitaly Viktorovich)

Vinberg E.B. (Ernest Borisovich)

Vishnevsky V.V.

Dao Chong Thi

Doan Quynh

Evtushik L.E. (Leonid Evgenievich)

Efremovich V.A. (Vadim Arsenievich)

Kamyshansky N.R.

Kantor I.L. (Isaiah Lvovich)

Komrakov B.P. (Boris Petrovich)

Konyushikhina T.N. (Tatyana...)

Kruchkovich G.I. (Georgy Ionovich)

Kushner G.F. (Guriy Fedorovich)

Liber A.E.

Litvinov G.L. (Grigory Lazarevich)

Lopshits A.M. (Abram Mironovich)

Mishchenko A.S. (Alexander Sergeevich)

Manturov O.V.  (Oleg Vasilyevich)

Norden A.P. (Alexander Petrovich)

Persits D.B. (David Borisovich)

Petrov E.E. (Evgeny Evgenievich)

Petrova V.T. (Vera Timofeevna)

Platonov S.S. (Sergey Sergeevich)

Pyasetsky V.S. (Vladimir Semenovich)

Rosenfeld B.A. (Boris Abramovich)

Sabinin L.V.  (Lev Vasilyevich)

Sabitov I.Kh. (Idzhat Khakovich)

Samborsky S.N. (Sergey Nikolayevich)

Semyanisty V.I. (Vladimir ...)

Sirota A.I. (Alexander Isaakovich)

Skopets I.M.

Solovyov Yu.P. (Yuri Petrovich)

Solodovnikov A.S. (Alexander Samuilovich)

Trofimov V.V. (Valery Vladimirovich)

Fedenko A.S. (Anatoly Semenovich)

Fomenko A.T. (Anatoly Timofeevich)

Shapiro Ya.L.

Shirokov A.P. (Alexander Petrovich)

Shpiz G.B. (Grigory Borisovich)

Elashvili A.G. (Alexander Grigorievich)

 

 

 

 

 
A.T. Fomenko, O.V. Manturov. V.A. Vasiliev




E.B. Vinberg, A.T. Fomenko




I.A. Shishmarev, A.T. Fomenko




G.L. Litvinov




E.B. Vinberg




O.V. Manturov




D.B. Persits







=================================================================

 

 

                         To the Dean

                          Faculty of Mechanics and Mathematics, Moscow State University

                          Corresponding Member of the Russian Academy of Sciences, Professor

                                  O.B. LUPANOVA

                        from the Head of the Department of Differential

                         Geometry and Applications Member of the Russian Academy of Sciences

                                   A.T. FOMENKO

 

                   Dear Oleg Borisovich!

 

     For many years, at the Faculty of Mechanics and Mathematics Faculty of Moscow State University, at the Department of Differential Geometry, under the supervision of Professor P.K. Rashevsky, Head of the Department, a seminar on vector and tensor analysis with applications to geometry, mechanics, and physics was held.The seminar was founded many years ago by Professor V.F. Kagan.The Department of Differential Geometry regularly published the Proceedings of this seminar.In recent years, the seminary has not been operating, and the Proceedings have not been published.

 

     In the current year, 1998, the Department of Differential Geometry and Applications has resumed the seminar and plans to to resume the publication of its Proceedings.The seminar is very popular, and its Many mathematicians attend its sessions, and not only those from Moscow State University.As in previous years, it once again became not only a university seminar but also a Moscow-wide geometry seminar.

 

    The Department requests:

    1) Name the seminar after P.K. Rashevsky.

    2) Approve the seminar board consisting of:

         Academician A.T. Fomenko (Chairperson),

         Professor V.V. Trofimov (Co-Chair),

 

         Professor A.V. Bolsinov,

         Professor N.P. Dolbilin,

         Professor L.E. Evtushik,

         Professor G.L. Litvinov

         Professor O.V. Manturov,

         Professor A.S. Mishchenko,

         Professor A.L. Onishchik,

         Professor I.Kh. Sabitov,

         Professor Ryshkov S.S.

 

     3) Appoint Prof. A.T. Fomenko and

        Prof. V.V. Trofimova.

 

     4) Resume the publication of the Proceedings of the Seminar at the Faculty of Mechanics and Mathematics of Moscow State University under its former title "Proceedings of the Seminar on Vector and Tensor Analysis with Their Applications to Geometry, Mechanics and Physics".

 

         With the utmost respect,

                                              A.T. Fomenko

 

                                            June 19, 1998

 

 

Approved at the meeting of the Academic Council

Mechanics and Mathematics

of the Faculty on June 26, 1998.