December 27, 2005
S.K.Lando S.K.Lando (Moscow). Algebraic geometry proof of Witten's conjecture.

December 22, 2005
V Dome Uchenykh
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.

December 13, 2005
A.Kusraev A.G.Kusraev (Vladikavkaz). Analysis, algebra, and logic in operator theory.
December 6, 2005
S.V.Duzhin. Rasmussen's invariant.
November 29, 2005
V.S.Kulikov (Moscow). "Dif=Def" problems.
November 22, 2005
Joint meeting with the mathematics Section of Dom Uchenykh
Urgent problems of school mathematical education
Speakers: A.M.Abramov (Moscow), O.A.Ivanov, M.Ya.Pratussevich, S.E.Rukshin
October 18, 2005
V.Ya.Eiderman (Moscow). Cartan type estimates for the Cauchy potential.
September 13, 2005
A.Suslin A.A.Suslin. Motivic cohomologies and the Bloch-Kato conjecture.

June 14, 2005
S.M.Natanzon (Moscow). Topological field theories.
May 24, 2005
Joint meeting of the society and the General PDMI Seminar
S.V.Buyalo. Embedding and nonembeddability theorems in asymptotic geometry.
May 5, 2005
A.S.Khoroshkin (Moscow). Koszul duality for operads.
April 27, 2005
Joint meeting of the society and the conference Analytical methods in number theory, probability, and statistics dedicated to Yu.V.Linnik's 90th anniversary
Reminiscences about academician Yu.V.Linnik
Talks by A.N.Andrianov, K.Kubilius (Vilnius), L.P.Linnik, V.A.Pliss, I.V.Romanovsky, A.L.Rukhin (Baltimore), T.Tonkov (Sofia), M.Jutila (Turku), A.A.Zinger have been given.
April 19, 2005
A. Bruno (Moscow). Power geometry as new mathematics.
April 5, 2005
P. Zograf. The Witten-Kontsevitch theory: from two-dimensional topological gravitation to random trees.
March 18, 2005
Mathematical Lectorium for Students
Yu. Lovyagin. Leibnitz's infinitesimal calculus in terms of nonstandard analysis.
March 1, 2005
B.Z.Moroz (Max-Planck-Institut für Mathematik, Bonn). Diophantine equations (an attempt of a survey).
January 18, 2005
A meeting dedicated to organizational matters.
  1. 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).
  2. 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.)
  3. Elections of the new Bodies of the Society.
N.A.Shanin N.A.Shanin has been elected an Honorary Member of the Society.


December 14, 2004
Alexey Borodin (Caltech and Clay Mathematical Institute, USA). Discrete Painleve equations in probability.
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.

December 7, 2004
Aims and values of school mathematical education, and evaluation of its results.
Speakers: M.I.Bashmakov, A.I.Plotkin, V.I.Ryzhik
November 30, 2004
A.M.Vershik. Universality and randomness in geometry, combinatorics, and analysis.
November 9, 2004
A.V. Zorich (Rennes). Irrational windings of flat surfaces, Teichmüller's geodesic flow, and "time machine".
October 19, 2004
A talk of the Society Prize winner for 2003
A.N.Zinoviev. Explicit reciprocity laws in local class field theory.
September 14, 2004
A talk of the Society Prize winner for 2002
Anna Erschler (Lille). Poisson boundary for random walks on groops.
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.

April 20, 2004
A.D.Bruno (Moscow). A new generalization of the continued fraction.
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.

March 30, 2004
N.A.Shanin N.A.Shanin. A version of analysis that does not use the notion of continuum.

March 25, 2004
Mathematical Lectorium for Students
S.K.Godunov (Novosibirsk). Guaranteed exactness in spectral problems .
March 16, 2004
A meeting dedicated to the memory of academician O.A.Ladyzhenskaya (1922--2004)
Speakers: N.N.Uraltseva, M.Sh.Birman, G.A.Seregin, A.V.Fursikov (Moscow), N.M.Ivochkina, A.L.Skubachevskii (Moscow), E.A.Tropp, A.M.Vershik. A video film has been shown.
The meeting has approved the decision to give the Society "Young Mathematician" Prise for 2003 to A.N.Zinoviev.
February 24, 2004
A meeting dedicated to the memory of G.I.Natanson (1930--2003)
Speakers: V.Babich, I.Daugavet, V.Fainshmidt, V.Havin, B.Makarov, Ya. Natanson, I.V.Nedzvetskaya, V.Odinets, A.Podkorytov, M.Skopina, V.Videnskii, O.Vinogradov, V.Zhuk.


November 25, 2003
A.N. Kolmogorov, John von Neumann: mathematical genii of the XXth century. On the occasion of the 100th anniversaries of A.N.K. and J.v.N.
Speakers: V.M.Tikhomirov (Moscow), A.M.Vershik, M.A.Semenov-Tyan-Shanskii.
October 21, 2003
G.Yu.Panina. Hyperbolic virtual polytopes and a uniqueness hypothesis for convex surfaces
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.

October 7, 2003
V.A.Vassiliev (Moscow). Cohomolgies of the space of knots and their combinatorial formulas
September 19, 2003
Mathematical Lectorium for Students
Yu.V.Matiyasevich. Tenth Hilbert problem: what one can and can not do with Diophantine equations
September 9, 2003
I.B. Fesenko (Nottingham). Noncommutative geometry, nonstandard mathematics, and the theory of elliptic curves with "real multiplication".
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.

June 24, 2003
Ya.M. Eliashberg (Stanford University). Introduction to symplectic topology: from Rolle's theorem to Floer homology.
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.

May 27, 2003
V.A.Timorin (Moscow). An algebra constracted via the volume polynomial of a simple polyhedron.
April 15, 2003
Laurent Lafforgue (IHES, France, a 2002 Fields Prize laureate).
Pavings of polyhedra, glueing of Schubert cells and compactification of configuration spaces.
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).

April 8, 2003
Urgent problems of teaching mathematics at school
Speakers: M.I.Bashmakov, A.L.Semenov (Moscow), V.A.Ryzhik, M.Ya.Pratusevich.
The desision of the meeting (in Russian).
March 25, 2003
Joint meeting of the Society and the PDMI General Seminar
E. Hirsch. Semialgebraic proofs
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.

February 18, 2003
S.Yu.Pilyugin. Orbit shadowing.


November 28, 2002
Mathematical Lectorium for Students
M.A.Lifshits. "Star dust" and probability.
Problems related to the emergence of cosmic bodies from the star dust under the influence of gravitation forces and stochastisity will be discussed.

November 6, 2002
Mathematical lectorium for students
S. Lando (Moscow). What is a tangent function.
November 5, 2002
S. Lando (Moscow). Invariants of knots and graphs.
October 29, 2002
A meeting dedicated to the 200th anniversary of the outstanding Norwegian mathematician N.H.Abel
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.

October 15, 2002
D. Siersma (Netherlands). Polynomial functions and their behaviour at infinity.
We consider polynomials as functions from Cn 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.

October 8, 2002
A meeting with G.M.Zuckerman, a representative of the "Mir" editors.
September 24, 2002
Gerrit van Dijk (Leiden). Generalized Gelfand pairs: a survey.
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 L2 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 L2(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.

July 11, 2002
Joint Meeting of the Society and the General PDMI Mathematics Seminar
M.Z.Solomyak (Israel). On spectral properties of the Laplacian on metric graphs.
June 22, 2002
Joint meeting of the Society and the Russian-German Meeting dedicated to the 90th Anniversary of A.D.Aleksandrov
Reminiscences about A. D. Aleksandrov
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.

June 18, 2002
V.A.Lifschitz (Austin, USA). Stable models of logic programs.
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.

May 27, 2002
Heiner Zieschang (Bochum). Minimal 3-dimensional manifolds.
April 23, 2002
Discussion of the approaching school reform
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.

April 17, 2002
Mathematical lectorium for students
V.M.Babich. The notion of function in the making.
April 8, 2002
A joint meeting with the General Mathematics Seminar of the PDMI
Yu. S. Il'yashenko (Moscow). Infinitesimal Hilbert's 16th problem.
April 2, 2002
A. I. Neishtadt (Moscow). Dragging out of the stability loss in the dynamical bifurcations.
February 26, 2002
S. G. Kryzhevich. A sharpening of some classical results in stability theory.
January 22, 2002
V. Korepin (Stony Brook, USA). Quantum computers.
Principles of quantum computing, quantum teleportation, Deutsch's algorithm were considered


December 4, 2001
A.V.Malyshev (Rybinsk). Cell structure of the space of real polynomials.
November 27, 2001
A.V.Malyutin. Normal forms of the braid group.
October 31, 2001
Mathematical Lectorium for Students
A.I. Nazarov. Symmetry and asymmetry of the solutions of extremal problems.
October 27, 2001
Mathematical Lectorium for Students
S.V.Duzhin. On points, straight lines and curves
October 9, 2001
A meeting dedicated to organizational matters.
 1. Reports of the president, the treasurer, and the auditing commission.
 2. Elections of the bodies of the Society.
 3. Short communications of the Society Prize laureates for 2001: S.G.Kryzhevich, A.V.Malyutin.
October 2, 2001
Grigorchuk R.I. (Moscow). Random walks on groups and the Atiyah conjecture about L2 Betti numbers.
September 24-28, 2001
Session dedicated to the 200th anniversary of M. V. Ostrogradski, organized with the participation of the St. Petersburg Mathematical Society.

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.

May 22, 2001
G.L.Litvinov, V.P.Maslov (Moscow). Idempotent Mathematics and Mathematical Physics
April 3, 2001
V.A.Malyshev (Moscow). Discrete two-dimensional quantum gravitation
Mathematically, the topic of the talk is a number of simply formulated combinatorial and probabilistic problems about growing graphs on surfaces.

February 15, 2001
Mathematical Lectorium for Students
Sergey Fomin (St. Petersburg & U. of Michigan). Criterion of complete positivity.
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.

February 12, 2001
A joint meeting of the Society and the Mathematical Section of "Dom Uchenykh"
Can we succeed in saving traditions of Leningrad-St. Petersburg mathematical schools?
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.