1.

Presentation of new problems (resulting from a visit to USA, Canada, and England)

2.

Geometical and dynamical properties of wreath products of groups

We study the thermodynamical limit for a mean
field model describing how a closed symmetric queueing network

operates. The Markov process under consideration
is invariant under the action of certain symmetry group G in

the phase space. We prove that the quotient
process on the space of orbits of the G-action converges to the limit

deterministic dynamical system.

Impressions about recent conferences (in Nor Amberd, by A.M.Vershik, and in Katsiveli, by A.A.Lodkin), reports on students' works (A.Gorbulsky and P.Nikitin).

Boundaries of random walks on groups (coinciding with exit boundaries, stationary boundaries and almost identical with Martin boundaries) are presently found for a rather restricted class of groups. The method to be described in the talk allows one to find the boundary for groups admitting a stable normal form. Along with a new realization of the path space in Cayley graph, the method provides a geometric model of the boundary for free solvable groups, without any use of a normal form. New open problems will be presented. The students of high levels are invited.

Zeroes of polynomials, and rational transformations of algebraic curves

- The work of M.S.Livshic and his collaborators in operator theory associates
to a system of commuting nonselfadjoint operators an algebraic curve (called
the discriminant curve). This discovery leads to a very fruitful interplay
between operator theory and algebraic geometry: problems of operator theory
lead to problems of algebraic geometry and vice versa. A natural problem
in operator theory is to define properly the notion of a rational transformation
of a system of commuting nonselfadjoint operators. This arises whenever
one wants to study the algebra generated by a given system of commuting
nonselfadjoint operators. It may also allow representing the given system
of commuting nonselfadjoint operators in terms of another system which
is simpler in some sense (e.g., it contains fewer operators, or the operators
have a smaller nonhermitian rank). A related problem in algebraic geometry
is to find an image of an algebraic curve given by a determinantal representation
under a rational transformation. For the simplest case, when a system of
operators consists of a single operator and the discriminant curve is a
line, these problems were solved by N.~Kravitsky using the classical elimination
theory. Our original objective was to find an analogue of the constructions
of N.~Kravitsky in the general case. This led us to consider elimination
theory for pairs of polynomials along an algebraic curve given by a determinantal
representation. One immediate result of elimination theory along an algebraic
curve is that analogues of the classical constructions allow us to describe
an image of an algebraic curve given by a determinantal representation
under a rational transformation. This, in turn, gives a natural way to
properly define the notion of a rational transformation of a pair of commuting
nonselfadjoint operators.

- Fibonacci solitaire is a combinatorial algorithm transforming a permutation into the following objects (in the order of decreasing simplicity):

(1) an involution (a partition of cards into couples and singletons);

(2) a Motzkin path;

(3) a subset (in the set of all cards).

The rule is pretty simple: the newcoming card is compared with the highest card in the deck of previously considered cards. If the rank of the new one is lower, we put it on the top; if the rank is higher, we remove both cards. The algorithm plays for the Young-Fibonacci graph the role similar to that of Robinson-Schensted correspondence for the Young graph.

We prove that:

1) the *number* of remaining cards is distributed as the number
of odd cycles;

2) the *subset* of these cards is asymptotically similar to the
Poisson process on the interval $[0,1]$ with the density $(1-t)^{-1}$;

3) the Motzkin path is almost surely uniformly close to a parabola;

4) the involution (as a binary relation) approaches (in the limit of
large permutations) the uniform distribution in a triangle.

We also find all invariant measures for an analog of Fibonacci solitaire
acting on an infinite deck of cards.

- We define a filtration of $k$-cylinder subgroups in the group $\Cal
P^*$ of virtual polytopes: $$ \Cal P^*=Cyl_1\supset Cyl_2 \supset \dots
\supset Cyl_n. $$ We define a collection of mutually orthogonal projectors
$$ \delta_k :\Cal P^* \to Cyl_k, $$ whose sum is the identity operator.
This collection induces the following expansions: $$ \Cal P^* = \delta_1
\Cal P^* \oplus \dots \oplus \delta_n \Cal P^* $$ and $$ Cyl_k = \delta_k
\Cal P^* \oplus \dots \oplus \delta_n \Cal P^*. $$

- In the talk I will discuss an attempt to describe the set $Ext(A,B)$ of homotopy classes of extensions of a $C^*$-algebra $A$ by a $C^*$-algebra $B$ in terms of asymptotic homomorphisms. The set $Ext^{as}(A,B)$ of homotopy classes of asymptotic homomorphisms from $A$ into the Calkin algebra $Q_B$ (if $B={\cal K}$ is the algebra of compact operators, then $Q_B=B(H)/{\cal K}$) is introduced and the natural map $i:Ext(A,B)\to Ext^{as}(A,B)$ is considered. It is shown that under reasonably general conditions on $A$ the set $Ext^{as}(A,B)$ coincides with $E$-functor of Connes and Higson and the map $i$ is surjective. In particular, any asymptotic homomorphism of a suspended $C^*$-algebra into the Calkin algebra is homotopic to a genuine homomorphism, whence one gets a description of the kernel of $i$.

- Systems of generators in exponential groups can be ordered by degree of representability of group elements. The examples of free, locally free, and solvable groups will be presented.

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