S.K.Lando (Moscow). Algebraic geometry proof of Witten's conjecture. |
Joint meeting with the mathematics Section of Dom Uchenykh Discussion on the interplay between mathematics and real world. Speakers: O.Viro, A.Vershik, A.Grib, N.Shanin, N.Firsova, A.Nazarov, A.Netsvetaev et al. See video of the main events. |
- Reports of the Council (A.Vershik, president); of the Treasurer (B.Lurie); of the Editorial Board of the "Proceedings of the St. Petersburg Mathematical Society" (N.N.Ural'tseva, editor-in-chief); of the Auditing Commission (A.Nazarov).
- Discussion (amendments to the Rules concerning admission to the Society and presentation of preprints to the Society archive; dues; contents of the Society site and, in particular, its "Pantheon" section; memorial dates, etc.)
- Elections of the new Bodies of the Society.
N.A.Shanin has been elected an Honorary Member of the Society. |
Distribution functions of interest in a variety of discrete probabilistic models (like length of the longest increasing subsequence of random permutations or last passage time in directed percolation) satisfy certain recurrence relations known as discrete Painleve equations. These equations were first obtained in an algebro-geometric work on surfaces obtained by blowing up the two-dimensional projective space at 9 points. The link between probability and geometry is provided by the theory of isomonodromy transformations of linear difference equations.
Poisson boundary for a class of groups that act on rooted trees were described. First examples of groups with subexponential growth that admit random walks with nontrivial boundary were constructed. As an application, new asymptotics of intermediate growth were presented.
A new two-dimensional concept of the continued fraction is proposed that is further generalized to a three-dimensional case. The new algorithm is simple and gives the best rational approximations to a real number. At the same time, it is periodic for cubic irrationalities. Detailed examples were provided.
N.A.Shanin. A version of analysis that does not use the notion of continuum. |
We describe and discuss counterexamples to the old hypothesis: if the principal curvature radii of a smooth 3-dimensional body K are ewerywhere separated by a constant C, then K is a ball of radius C. The talk is based on papers by A.V. Pogorelov, Y. Martinez-Maure, and the speaker.
In the last years methods of operator and von Neumann algebras and more generally so called noncommutative geometry have been applied in a number of works to the study of (rather algebraic structures) of commutative number theoretical objects: e.g. left uncompleted work of Connes on Riemann's zeta function, the works of Manin and Marcolli on Arakelov geometry and modular symbols, the work of Manin on hypothetical theory of real multiplication (Manin's Alterstraum). The last work is a research project aimed to use quantum torii (or quantum degenerate elliptic curves) and quantum theta functions to construct analogues of parts of the classical theory of elliptic curves with complex multiplication. #The talk will first present some of the main structrues of and ideas in those works. Then a new approach to study the number theoretical objects, which unlike the previous, works at the level of arithmetic structures too, will be described. The approach is based on using hyper objects (e.g. hyper elliptic curves with hyper complex multiplication) and the shadow (standard path) map to descend to ordinary objects. It revives some of old ideas of Robinson and Weil. As an applications of hyperdiscretization principle, "noncommutative" spaces can be studied via covering them by hyper commutative spaces. A relation between the former and the shadow image of the latter is then given by a mapping, which could be viewed as a vast generalization of the Seiberg-Witten map in string theory.
During two decades wich passed since the emergence of symplectic topology, there were discovered many deep connections of the new science with several areas of mathematics and theoretical physics: from Hamiltonian dynamics and low-dimensional topology to algebraic geometry and the theory of integrable PDE's.
Main ideas of symplectic toology and some of the applications were discussed.
A lecture in projective geometry. When studying the compactifications of Drinfeld's moduli spaces of shtukas with level structure or (according to Faltings) local models of Shimura varieties, one is led to the problem of compactifying the quotients PGL(r) x ... x PGL(r) / PGL(r) in an equivariant way. A general method for compactifying these quotients is presented. It also applies to configuration spaces of matroids.
All the compactified schemes we obtain are endowed with a structure morphism (which is smooth when there are at most 3 factors or when the rank is 2 but not in general) over a "toric stack" whose points are the pavings of some integral polyhedron. There is an induced stratification and the strata can be described in terms of glueing of thin Schubert cells. And all the compactified schemes have at least two modular interpretations:
- classifying equivariant vector bundles on some toric varieties,
- classifying some kind of projective rational varieties with logarithmic singularities (which generalise the "minimal models of projective spaces" introduced by Faltings).
Speakers: M.I.Bashmakov, A.L.Semenov (Moscow), V.A.Ryzhik, M.Ya.Pratusevich.The desision of the meeting (in Russian).
Propositional proof complexity is a rapidly progressing area of research. The existence of polynomial-size proofs for every propositional tautology would imply NP=coNP. So far, only lower (and upper) bounds on the complexity of proofs in specific proof systems (and for specific tautologies, respectively) are known.The first part of the talk will contain an introduction to propositional proof complexity and a survey of known proof systems and results.
The second part of the talk concerns the results of the speaker (joint with Dima Grigoriev and Dimitrii V. Pasechnik). We study semialgebraic proofs, i.e., proofs that use reasoning about polynomial inequalities. For example, here is a proof of the propositional pigeon-hole principle:
\sum_{k=1}^m (\sum_{l=1}^{m-1} x_{kl}-1) + \sum_{l=1}^{m-1} ( \sum_{k=1}^m (\sum_{k\neq k'=1}^m (1-x_{kl}-x_{k'l}) x_{kl} + (x_{kl}^2-x_{kl})(m-2)) + (\sum_{k=1}^m x_{kl}-1)^2 ) = -1.
(It will be explained in the talk why this is a proof.) The proofs of this tautology in many other systems have exponential (in the number m of pigeons) length.
Problems related to the emergence of cosmic bodies from the star dust under the influence of gravitation forces and stochastisity will be discussed.
N.S.Ermolaeva. The life and work of N.Abel.
A.V.Yakovlev. Abel's theorem on algebraic equations.
V.A.Malyshev (Rybinsk). Integrals with quadratic irrationalities.
M.A.Semenov-Tyan-Shansky. Abel varieties and integrable problems.
K.V.Manuilov. Abel's theorem on additive properties of Abel integrals and Abel functions.
We consider polynomials as functions from C^{n} to C. For certain values the topological type of the fibres can change (due to affine critical points or to "singularities at infinity"). Under certain conditions one can show that the generic fibre has the toplogy of a bouquet of spheres and that there exist invariants, which detect the values, where the function is not a fibration. Moreover we study deformations of polynomials, monodromy and relate this to boundary singualrities and Arnol'd's theory of fractions.
This is a joint research with M. Tibar.
The group G=SL(2,R) of 2 x 2 matrices with determinant 1, acts on the complex upper half plane by fractional linear transformations in a multiplicity free way: the L^{2} space decomposes multiplicity free as a direct integral of irreducible spaces. This property was studied and extended by Gelfand a.o. to pairs (G,K), where G is a Lie group and K a compact subgroup. The equivalent of the upper half plane is the space G/K. Pairs (G,K) such that L^{2}(G/K) splits multiplicity free are called Gelfand pairs. The most well-known examples are given by pairs (G,K) where G is a semi-simple Lie group and K a maximal compact subgroup.
An extension of the notion of Gelfand pair for pairs (G,K) where K is a closed, non-necessarily compact subgroup of G was discussed and several examples of (generalized) Gelfand pairs were be given.
Speakers: Yu.G.Reshetnyak, G.M.Idlis, A.L.Verner, Yu.F.Borisov, N.A.Shanin, S.S.Kutateladze, V.N.Berestovskii, A.I.Nazarov, A.M.Vershik.
A logic program is a set of symbolic expressions called "rules." Stable models of a logic program are defined as the fixpoints of an anti-monotone operator on sets of atomic symbols that is associated with this program. The concept of a stable model was originally introduced for describing the behavior of the programming system PROLOG. In recent years, it led to the development of a new approach to solving combinatorial search problems. We show how some concepts of graph theory can be represented in terms of stable models.
The problems that could be a result of the contraversial school reform were discussed.
Speakers: A.M.Abramov (Moscow), B.M.Makarov, M.I.Bashmakov, V.A.Ryzhik, N.N.Udal'tsova, A.L.Verner, A.M.Vershik, Yu.V.Matiyasevich, A.A.Lodkin. See our forum.
Principles of quantum computing, quantum teleportation, Deutsch's algorithm were considered
September 25, 2001
A joint meeting of the Society and the History of Mathematics and Mechanics section
of the conference.
1. I. Lopatukhina . An outline of Ostrogradski's human
and scientific biography
2. V. Tikhomirov (Moscow). M.Ostrogradski and variational calculus.
3. Yu. Aleshkov . Ostrogradski's methods in
mathematical physics.
4. L. Brylevskaya . Myth about Ostrogradski:
truth and invention.
Mathematically, the topic of the talk is a number of simply formulated combinatorial and probabilistic problems about growing graphs on surfaces.
The topic is matrices with all minors positive. The description of the set of such matrices, criteria of complete positivity, an answer of the question how can one quickly detect this property, relation to combinatorics of pseudo-lines in the plane, and applications, some open problems.
Current dramatic situation with mathematics (and science in general) in our country raises this question. A positive solution, impossible as it might seem to many of us, is to be found anyway. We open a discussion of the problem on our site.