I am mathematician, 1933 w.b. Member of European Academy (2015),Priciple researcher at St.Petersburg branch of Mathematical Instituteof Russian Academy of Sciences.
American Mathematical Society Translations: Series 2, 1996
This first of a two-volume collection is a celebration of the scientific heritage of F. A. Berezi... more This first of a two-volume collection is a celebration of the scientific heritage of F. A. Berezin (1931-1980). Before his untimely death, Berezin had an important influence on physics and mathematics, discovering new ideas in mathematical physics, representation theory, analysis, geometry, and other areas of mathematics. His crowning achievements were the introduction of a new notion of deformation quantization, and Grassmannian analysis ("supermathematics"). Collected here are papers by his many of his colleagues and others who worked in related areas, representing a wide spectrum of topic
It is probably difficult to find areas of XIXth Century mathematics more remote from each other t... more It is probably difficult to find areas of XIXth Century mathematics more remote from each other than algebra, the foundation of mathematics, and probability theory, a semi-applied area perceived from the time of its emergence as an almost experimental science. We note, in passing, that in the works of P. L. Chebyshev’s students, A. A. Markov and A. M. Lyapunov, many assertions of probability theory (for example, the central limit theorem) were proved with complete rigour in great generality. Nevertheless, it is no accident that one of the famous problems, proposed by D. Hilbert involved axiomatization of mechanics and axiomatization of probability theory: at that time one could not assume that these areas were fully mathematicized. In the first half of the XXth Century the works of A. N. Kolmogorov, S. N. Bernstein, von Mises et al. created the foundations of probability theory, which were unconditionally accepted by the mathematical community, and all doubts about whether or not this was mathematics were removed. However, probability theory has retained a certain isolation until now. It is difficult to explain this rationally. Certainly, a number of its methods are specific to that science and were difficult to understand even 20 years ago. For example, specialists on differential equations were for a long time unable to assimilate techniques of stochastic calculus, although results obtained by probabilistic methods in the theory of equations competed successfully with theorems obtained by classical methods.
In this paper first the Laplace operator on a completely nonholonomic Riemannian manifold is defi... more In this paper first the Laplace operator on a completely nonholonomic Riemannian manifold is defined in an invariant manner and its properties are considered. The method presented for studying it, as well as for the study of other hypoelliptic operators, involves the use of the geometry of nonholonomic manifolds. The nonholonomic metric (Carnot-Carathéodory metric), the Carathéodory measure, and hypoharmonic functions are defined. A theorem on the comparison of the spectra is proved and the connection is established between the bases of eigenfunctions of the ordinary and nonholonomic Laplacians. Conjectures are formulated on the principal term of the spectral asymptotic expansion of the nonholonomic Laplacian, on the structure of the wave fronts, and on the propagation of singularities.
Asymptotic calculations are applied to study the degrees of certain sequences of characters of sy... more Asymptotic calculations are applied to study the degrees of certain sequences of characters of symmetric groups. Starting with a given partition $\mu$, we deduce several skew diagrams which are related to $\mu$. To each such skew diagram there corresponds the product of its hook numbers. By asymptotic methods we obtain some unexpected arithmetic properties between these products. The authors do not know "finite", nonasymptotic proofs of these results. The problem appeared in the study of the hook formula for various kinds of Young diagrams. The proofs are based on properties of shifted Schur functions, due to Okounkov and Olshanski. The theory of these functions arose from the asymptotic theory of Vershik and Kerov of the representations of the symmetric groups.
Connections between the Robinson–Schensted–Knuth algorithm, random infinite Young tableaux, and c... more Connections between the Robinson–Schensted–Knuth algorithm, random infinite Young tableaux, and central indecomposable measures are investigated. A generalization of the RSK algorithm leads to a combinatorial interpretation of extended Schur functions. Applications are given to Ulam’s problem on longest increasing subsequences and to a law of large numbers for representations. An analogous theory for other graphs is discussed.
In this paper we continue the line started in ~,2,3] and formulate a ~umber of problems of the re... more In this paper we continue the line started in ~,2,3] and formulate a ~umber of problems of the real complexity theory and combinatorial optimization as the problems investigating the properties of sequences of semialgebraic sets, their singularities, boundaries etc. The main idea is to apply the methods of real analysis and singularity theory to the study of spaces of problems and to treat the main notions of complexity theory (P, NP, NP-completeness and others) as the complexity of different spectra of semialgebraic sets. An important role is played by a new class of examples, introduced by the author and his colleagues quite recently, i.e. the group, algebraic and geometric examples such as optimization on finite groups and their orbits in representations. The term "configurational topology' is introduced in connection with the study of the topology of configuration spaces and ~{n~v's theorem on universality of configurations spaces or the spaces of convex polyhedrons from a topological point of view. Configurational topology studies topological spaces as configuration spaces, spaces of optimization problems etc. A more detailed description of the whole spectrum of these questions is to be published soon (see also [2,3,7] ).
American Mathematical Society Translations: Series 2, 1996
This first of a two-volume collection is a celebration of the scientific heritage of F. A. Berezi... more This first of a two-volume collection is a celebration of the scientific heritage of F. A. Berezin (1931-1980). Before his untimely death, Berezin had an important influence on physics and mathematics, discovering new ideas in mathematical physics, representation theory, analysis, geometry, and other areas of mathematics. His crowning achievements were the introduction of a new notion of deformation quantization, and Grassmannian analysis ("supermathematics"). Collected here are papers by his many of his colleagues and others who worked in related areas, representing a wide spectrum of topic
It is probably difficult to find areas of XIXth Century mathematics more remote from each other t... more It is probably difficult to find areas of XIXth Century mathematics more remote from each other than algebra, the foundation of mathematics, and probability theory, a semi-applied area perceived from the time of its emergence as an almost experimental science. We note, in passing, that in the works of P. L. Chebyshev’s students, A. A. Markov and A. M. Lyapunov, many assertions of probability theory (for example, the central limit theorem) were proved with complete rigour in great generality. Nevertheless, it is no accident that one of the famous problems, proposed by D. Hilbert involved axiomatization of mechanics and axiomatization of probability theory: at that time one could not assume that these areas were fully mathematicized. In the first half of the XXth Century the works of A. N. Kolmogorov, S. N. Bernstein, von Mises et al. created the foundations of probability theory, which were unconditionally accepted by the mathematical community, and all doubts about whether or not this was mathematics were removed. However, probability theory has retained a certain isolation until now. It is difficult to explain this rationally. Certainly, a number of its methods are specific to that science and were difficult to understand even 20 years ago. For example, specialists on differential equations were for a long time unable to assimilate techniques of stochastic calculus, although results obtained by probabilistic methods in the theory of equations competed successfully with theorems obtained by classical methods.
In this paper first the Laplace operator on a completely nonholonomic Riemannian manifold is defi... more In this paper first the Laplace operator on a completely nonholonomic Riemannian manifold is defined in an invariant manner and its properties are considered. The method presented for studying it, as well as for the study of other hypoelliptic operators, involves the use of the geometry of nonholonomic manifolds. The nonholonomic metric (Carnot-Carathéodory metric), the Carathéodory measure, and hypoharmonic functions are defined. A theorem on the comparison of the spectra is proved and the connection is established between the bases of eigenfunctions of the ordinary and nonholonomic Laplacians. Conjectures are formulated on the principal term of the spectral asymptotic expansion of the nonholonomic Laplacian, on the structure of the wave fronts, and on the propagation of singularities.
Asymptotic calculations are applied to study the degrees of certain sequences of characters of sy... more Asymptotic calculations are applied to study the degrees of certain sequences of characters of symmetric groups. Starting with a given partition $\mu$, we deduce several skew diagrams which are related to $\mu$. To each such skew diagram there corresponds the product of its hook numbers. By asymptotic methods we obtain some unexpected arithmetic properties between these products. The authors do not know "finite", nonasymptotic proofs of these results. The problem appeared in the study of the hook formula for various kinds of Young diagrams. The proofs are based on properties of shifted Schur functions, due to Okounkov and Olshanski. The theory of these functions arose from the asymptotic theory of Vershik and Kerov of the representations of the symmetric groups.
Connections between the Robinson–Schensted–Knuth algorithm, random infinite Young tableaux, and c... more Connections between the Robinson–Schensted–Knuth algorithm, random infinite Young tableaux, and central indecomposable measures are investigated. A generalization of the RSK algorithm leads to a combinatorial interpretation of extended Schur functions. Applications are given to Ulam’s problem on longest increasing subsequences and to a law of large numbers for representations. An analogous theory for other graphs is discussed.
In this paper we continue the line started in ~,2,3] and formulate a ~umber of problems of the re... more In this paper we continue the line started in ~,2,3] and formulate a ~umber of problems of the real complexity theory and combinatorial optimization as the problems investigating the properties of sequences of semialgebraic sets, their singularities, boundaries etc. The main idea is to apply the methods of real analysis and singularity theory to the study of spaces of problems and to treat the main notions of complexity theory (P, NP, NP-completeness and others) as the complexity of different spectra of semialgebraic sets. An important role is played by a new class of examples, introduced by the author and his colleagues quite recently, i.e. the group, algebraic and geometric examples such as optimization on finite groups and their orbits in representations. The term "configurational topology' is introduced in connection with the study of the topology of configuration spaces and ~{n~v's theorem on universality of configurations spaces or the spaces of convex polyhedrons from a topological point of view. Configurational topology studies topological spaces as configuration spaces, spaces of optimization problems etc. A more detailed description of the whole spectrum of these questions is to be published soon (see also [2,3,7] ).
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