Russian version

     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.

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.

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.

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April, 26
Alexander Gnedin, Sergei Kerov. Combinatorial and probabilistic properties of Fibonacci solitaire

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.

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April, 19
Gayane Yu.Panina. The structure of the virtual polytope group with respect to the cylinder subgroups filtration

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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