Ergodic properties of boundary actions.

Classification of measurable functions in several variables and random equidistributed matrices.

Anosov diffeomorphisms on the quotients of nilpotent groups by discrete subgroups.

Methods of noncommutative computer algebra.

Quantum groups, Barnes functions, q-deformation Toda chains, and modular duality.

Branching groups and their applications.

Pseudorandom finite sequences

ASM matrices and calculation of the statistical sum for the 6-vertex model

Characters and representations of the Heisenberg group over the algebraic closure of a finite field

Measures on laminations related to rational mappings

Knitting theory (or coding the action of braid groups on simple loops of punctured disks)

Youth Day:

Impressions of some conferences in USA

Unique $\beta$-expansions, one-dimensional "mappings with holes", and digital channels

Combinatorial models of classyfying spaces in piecewise-linear topology

A brief review of the theory of classifying spaces, the construction of the exact BPL model and the Gauss map for combinatorial manifolds.

Asymptotical lower bound for the dimension of the space of Vassiliev-Goussarov invariants

The talk is about the lower bound for the number of independent knot invariants of finite type obtained in 1996 by S.Chmutov and S.Duzhin and improved in 1997 by O.Dasbach. Currently the best bound is exp(C \sqrt(n)) for any constant C < pi * \sqrt(2/3). The proofs consist in an explicit construction of a big family of independent elements in the space of uni-trivalent graphs, distinguished by a special linear mapping into the space of multivariate polynomials.References:

S.V.Chmutov, S.V.Duzhin. A lower bound for the number of Vassiliev knot invariants. "Topology and its Applications" 92 (1999) 201-223.

O.Dasbach. On the Combinatorial Structure of Primitive Vassiliev Invariant III - A Lower Bound, Communications in Contemporary Mathematics, Vol. 2, No. 4, 2000, pp. 579-590.

Estimation of some parameters for random walks on amenable groups

- For symmetric random walks on finitely generated groups, examples of calculation of drift and entropy, and some inequalities relating these functions with the word function of growth of the group will be considered. Some new examples of zero and positive entropy will be given.

About workshops in Marseille and Vienne

Combinatorial structures associated with finite type knot invariants

- Due to the theorem of M.Kontsevich, the study of many properties of finite type knot invariants can be reduced to the study of graded algebras generated by combinatorial objects such as chord diagrams, Feynmann diagrams, 3-1-valent graphs etc. The talk will contain a survey of these algebras and an explanation of their relation to knot invariants. In particular, I will mention the upper and lower bounds for the number of independent finite type knot invariants obtained by combinatorial techniques.

Spherical designs and reflection groops

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