Fundamental Mathematics Today
Conference dedicated to the 10th anniversary of the Independent University of Moscow
Moscow, December 26—29, 2001

In December 2001, the Independent University of Moscow celebrated its 10th anniversary and held the scientific Conference dedicated to this event. The Conference was supported by the Russian Foundation for Basic Research, the European Mathematical Union, and the Steklov Mathematical Institute of the Russian Academy of Sciences.

The main goal of the Conference was to present to the current students and graduates of the University a panorama of modern mathematics, stressing on the areas well established at the University. These areas cover a large part of mathematics, including, first of all, geometry, topology, algebra, algebraic geometry, dynamical systems, singularity theory, mathematical physics. This goal determined the circle of persons invited to give talks at the Conference. This circle includes

• the founders of the IUM;
• former and current IUM professors;
• persons who played an important role in the scientific life of the IUM, not being formally its professors;
• former and current IUM students;
• a small number of young researchers from all over the world who applied for participation.

The scientific schedule of the Conference included 17 plenary talks and 44 sectional talks as well as the IUM President's Yu. S. Ilyashenko opening speach also published in the present issue.

The Conference was open for the public, and in total more than 300 participants took part in the Conference.

Plenary talks

V. I. Arnold (IUM, Steklov Mathematical Institute RAS [Here and below RAS stands for Russian Academy of Sciences] & Université Dophine, Paris, France) Astroidal geometry and hyperbolic polynomials. The talk was devoted to a description of the first examples from the new ``Hessian geometry'' invented by the author. This geometry is similar to the Lagrangian and Legendrian geometry developed by Arnold in early seventies, and presumably it can be generalized to the qualitative theory of differential equations of order greater than one.

A. A. Bolibrukh (Steklov Mathematical Institute RAS) Semistable Vector Bundles with Connections and the Riemann—Hilbert Problem. The problem of constructing a semistable vector bundle over a punctured compact Riemann surface admitting a connection with logarithmic singularities at the punctures and given monodromy representation was considered. This problem generalizes the classical Riemann—Hilbert problem. Sufficient conditions for existence of solutions to this problem were described.

V. M. Buchstaber (Steklov Mathematical Institute RAS) Symmetric powers as algebraic variety. Embeddings of symmetric powers of algebraic varieties into projective spaces are considered. The results are generalized to the case of symmetric powers of arbitrary Hausdorff spaces. Applications to algebraic geometry, theory of Abelian functions and theory of integrable systems are studied.

P. Cartier (ENS, Paris) Cosmos and atom: a revival of Hermann Weyl's ideas about gauge invariance. The stationary model of the space introduced by Einstein was soon after substituted by the expanding model due to Fridman. Cartier revives the stationary model by adding to it an additional field, which he calls the Hubble field.

P. Deligne (Institute for Advanced Studies, USA) Tensor Categories. The talk concerned the study of general tensor categories with associative and commutative tensor product satisfying the following growth requirement: there is a self dual generator $X$ such that the length of the $N$th tensor power of $X$ grows at most polynomially in $N$.

B. A. Dubrovin (SISSA, Italy) From integrable systems to Gromov—Witten invariants. Conjectures relating the geometry of the Delinge—Mumford compactifications of moduli spaces of algebraic curves and integrable hierarchies of evolutionary partial differential equations were discussed.

L. D. Faddeev (Peterburg Division of the Mathematical Intsitute RAS, Peterburg) Problem of mass for the quantum Yang—Mills theory: what is it. The modern state of the central notion in quantum physics, the mass, was described.

B. L. Feigin (IUM & Landau Institute for Theoretical Physics, RAS) Vertex algebras and representations of small quantum group. The talk was devoted to representations of some infinite dimensional Lie algebras arising in physics.

V. Ginzburg (University of Chicago, USA) Symplectic reflection algebras. A symplectic reflection algebra is the algebra of deformations of a pair (a symplectic space, a group of its symplectomorphisms generated by reflections). The relations between symplectic reflection algebras and resolutions of the quotient singularities of the space modulo the group action were described.

A. Katok (Pennsylvania State University, USA & IUM) Nonuniform Hyperbolicity and Rigidity. The talk concerned actions of higher-rank abelian groups on compact manifolds. The geometric rigidity with respect to this action was studied.

A. A. Kirillov (Pennsylvania State University, USA) Representations and orbits of triangular groups. The structure of the space of orbits of the coadjoint action of the goup of upper triangular matrices still remains an open problem. Recent progress in solving this problem and new conjectures were discussed.

I. M. Krichever (Landau Institute for Theoretical Physics, RAS & Columbia University, USA) Isospectral and isomondromy equations on algebraic curves. The space of meromorphic connections on a vector bundle over an algebraic curve carries a natural symplectic structure. The talk was devoted to the description of the Hamiltonian nature of the classical isomonodromy equations and their generalizations with respect to this symplectic structure.

R. MacPherson (Institute for Advanced Studies, USA) Intersection homology and torus actions. Under some assumptions, a torus action on a topological space leads to a graph, called the moment graph of the action. An easy algorithmic way to read the intersection homology of the space from the moment graph was suggested in the talk.

S. P. Novikov (Steklov Mathematical Institute RAS, Landau Insitute for Theoretical Physics RAS, Lomonosov State University, Moscow & Maryland University, USA) Discrete systems and integrabilty. The talk was devoted to the problem ``what is the right discrete model of a continuous system''. Possible discretizations for dynamical systems with soliton-type hidden symmetries were discussed.

V. A. Vassiliev (IUM & Steklov Mathematical Institute RAS) Combinatorial formulas for cohomology of spaces of knots. Finite order cohomology of spaces of knots can be expressed through cycles of finite codimension in these spaces. These cycles admit a combinatorial description in terms of diagrams close to now classical chord diagrams invented by the author, which describe finite order zero cohomology.

A. M. Vershik (Peterburg Division of Mathematical Institute RAS) Metric classification of functions of several arguments, random matrix and space of Polish spaces, The space of degenerate random tensors is closely related to the classification problem for functions of several arguments. In some special cases (say, for metric spaces with measures) the space of all classes has a reasonable description.

E. B. Vinberg (Lomonosov State University, Moscow) The dual horospherical Radon transform for polynomials. The main result described in the talk is the computation of the constant factor which appears in the dual Radon transform. The latter acts from the ring of polynomials over the space of horospheres (orbits of the maximal unipotent subgroups) of a symmetric space $X=G/K$ of a Lie group $G$ to the ring of polynomials over the symmetric space $X$ itself.

Section A: Geometry & Topology

S. Barannikov (ENS, Paris, France), Semi-infinte variations of Hodge structures and integrable hierarchies, Yu. M. Burman (IUM) Whitney index and Laplace integrals, S. V. Duzhin (PDMI, Peterburg & IUM) Decomposable skew-symmetric functions, P. G. Grinevich (Landau Institute for Theoretical Physics), S. P. Novikov Toplological charge of the finite-gap Sine-Gordon solutions, J.-M. Kantor (Université Paris VI, Paris) Recent works on lattice polytopes, O. Karpenkov (IUM & Lomonosov State University, Moscow) Energy of a knot: variational principles, M. E. Kazarian (IUM & Steklov Mathematical Institute RAS) Multisingularities and cobordisms, S. K. Lando (IUM & NIISI RAS) The Hurwitz problem and the geometry of spaces of meromorphic functions, S. Shadrin (IUM & Lomonosov State University, Moscow) Topological classification of unitary functions of arbitrary genus, A. Skopenkov (IUM) The Whitehead link, the Hudson—Habegger invariant and classification of embeddings $S^1\times S^3\to R^7$, V. Timorin (University of Toronto, Canada) Four dimensional geometry of circles.

The subject of the talks varied from the classical geometry (J.-M. Kantor, V. Timorin) and topology (A. Skopenkov), to the theory of moduli of complex algebraic curves and related problems of mathematical physics (P. G. Grinevich, S. P. Novikov, S. K. Lando, S. Shadrin).

Section B: Algebra, Algebraic Geometry, & Number Theory

S. Arkhipov (IUM) A geometric realization of the category of modules over the small quantum group, A. Braverman (Harvard University, USA) Singularities of loop schemes and automorphic $L$-functions, A. L. Gorodentsev (IUM & ITEP) Abelian Lagrangian Algebraic Geometry and ALAG-quantization, A. Ya.Helemskii (Lomonosov State University, Moscow) Wedderbern-type theorems for operator algebras: traditional and ``quantized'' homological approaches, L. Katzarkov (UCI, USA) Algebro Geometric methods in symplectic geometry, A. G. Khovanski (IUM & University of Toronto, Canada) Parshin's symbols, toric geometry and product of the roots of a system of equations, V. Lunts (Indiana State University) Motivic measures and stable birational geometry (joint work with M. Larsen), A. Skorobogatov (Imperial College, London, UK) Rational points on certain Kummer surfaces, M. A. Tsfasman (IUM & IITP RAS) Asymptotic properties of global fields, M. Verbitsky (IUM) Proof of Mukai conjecture, F. L. Zak (IUM) Order, rank, and class of projective algebraic varieties.

Most of the talks were concentrated around modern problems of algebraic geometry. Even the talk of M. A. Tsfasman concerning a classical number theory problem had a strong algebraic geometry flavour. The only exception was the talk of A. Ya. Khelemskii concerning generalizations of structure theorems for semi-simple finite dimensional algebras to the inifinite-dimensional case.

Section C: Dynamical Systems & the Theory of Singularities

M. Blank (IUM & IITP RAS) Dynamical spectrum for random maps and Ulam conjecture, A. Eskin, A. Zorich (IUM & Université Rennes I, France) Billiards in rectangular polygons, W. Faris (University of Arizona, Tucson, Arizona, USA) Ornstein—Uhlenbeck and renormalization semigroups, A. Gorodetskii (IUM & Lomonosov State University, Moscow) Stable nonhyperbolic properties of dynamical systems and smooth realization of skew products, S. M. Gusein-Zade (IUM & Lomonosov State University, Moscow) Indices of 1-forms on an isolated complete intersection singularity, Yu. S. Ilyashenko (IUM & Steklov Mathematical Institute RAS) Centennial History of Hilbert's 16th problem, D. Kaledin (IUM) Symplectic singularities in algebraic geometry, K. Khanin (Newton Mathematical Institute, Cambridge, UK) Rigidity in one-dimensional dynamics, S. Maksimenko (Insitute of Mathematics, Kiev, Ukrain) Smooth shifts along trajectories of flows, G. Olshanskii (IITP RAS) Gelfand—Tsetlin schemes, measures of hypergeometric type, and point processes, E. Polulyakh (Insitute of Mathematics, Kiev, Ukrain) Dynamical systems which admit periodical decompositions.

The variety of subjects of the talks in this section is a tiny reflection of the diversity of dynamical systems and singularity theories. The talks concerned random dynamical systems (Blank, Gorodetski), billiards and Teichmüller spaces (Eskin—Zorich), flows in the plane and maps of the circle (Ilyashenko, Khanin), renormalization groups and measure theory (Faris, Olshanskii), index of vector fields on singular varieties (Gusein-Zade), and other topics.

Section D: Mathematical Physics

A. V. Alexeevskii, S. M. Natanzon (IUM, Lomonosov State University, Moscow & ITEP) Klein topological field theory and Hurwitz numbers of complex and real algebraic curves, A. A. Belavin, A. V. Odesskii, R. A. Usmanov (IUM & Landau Institute for Theoretical Physics RAS) New relations in the algebra of the Baxter $Q$-operators, L. Chekhov (Steklov Mathematical Institute RAS) Commutative and noncommutative Teichmüller spaces via graphs, V. A. Gordin (Hydrometeorological Centre of Russia, Moscow) Eigen-value Functional Optimization in Mathematical Physics and Stability Problems, S. Loktev (IUM) Coinvariants of one-dimensional lattice VOAs, A. N. Rudakov (IUM & University of Tronheim, Norway) Representations of infinitedimensional graded Lie $s$-algebras with $sl(3)\times sl(2)\times gl(1)$ as the zero degree component, I. M. Shchepochkina (IUM) Classification of the real simple Lie superalgebras of vector fields (joint work with Dimitry Leites), A. G. Sergeev (IUM & Steklov Mathematical Institute RAS) Seiberg—Witten equations and Abrikosov strings, O. K. Sheinman (IUM & Steklov Mathematical Institute RAS) The second order casimirs for the affine Krichever—Novikov algebras, S. Shlosman (Université Marseille, France & IITP RAS) Convex envelope of the Wiener loop, M. A. Shubin (IUM & North Eastern University, Boston, USA) Capacities in spectral theory of magnetic Schrödinger operators.

A large group of talks (A. A. Belavin et al., S. Loktev, A. N. Rudakov, I. M. Schepochkina, O. K. Sheinman) was devoted to the study of applications of Lie algebras in mathematical physics. Another large group (A. V. Alexeevskii and S. M. Natanzon, L. Chekhov, A. G. Sergeev) described topological aspects of physical applications.

Yu. S. Ilyashenko, S. K. Lando, A. B. Sossinski, O. K. Sheinman