On faces of Klein's multidimensional continued fractions

Markov processes on partitions

Dynamics on the moduli spaces of quadratic differentials

The theory of billiards in rational polygons leads to the theory of Riemann surfaces and quadratic differentials. We will present, in an elementary way, the relations between these objects. Then we will give recent results on these moduli spaces.

Integrable boson model and plane partitions

Three approaches to Young tableaux

A survey of Segal's theory

Towards a proof of the Razumov-Stroganov conjecture

We shall recall the remarkable properties of the O(1) Temperley-Lieb loop model and how they led Razumov and Stroganov to formulate a conjecture relating it to Fully Packed Loops and Alternating Sign Matrices. We shall then discuss recent attempts at proving this conjecture; in particular we shall try to generalize it by introducing inhomogeneities and explain the connection with the Izergin-Korepin determinant formula for the six-vertex model.

Cellular automata and Yang-Baxter equation

A generalization of particular soliton cellular automata, described by the q to 0 limit of R-matrix, to the of general q is given.

New definition for randomness of subsequence of natural numbers and Ramsey properties of normal sequences

We give a new definition for randomness of a subset A of natural numbers as follows: A is weak-mixing (any other ergodic type property) if the point 1_A of {0,1}^{infinity} is a generic point for the weak-mixing (any other ergodic type property) system (cl{T^n 1_A},B,\mu,T) (T is the usual shift to the left) and additionally A has a positive density in natural numbers. We list some Ramsey-type results for weak-mixing (totally ergodic) sequences, for example, if A is weak-mixing, then there exist x,y in A such that x+y=square. We will remind some Ramsey-type results for partitions of N, in particular, the Rado theorem for regularity of a linear system of equations and will explain a connection between our results and some open questions of the Ramsey theory of partitions of natural numbers. Finally, we will present the following result with a proof (sketch): There exists a normal subset A of natural numbers for which the equation xy=z is not solvable inside A; this means that for any x,y,z in A xy \not= z.

Classical Lie superalgebras, their invariants and representations

Bijective and asymptotic theory of partitions

Aperiodicity of cocycles

We give conditions when a cocycle $\Phi$ does not trivially satisfy the equation $\gamma\Phi =\lambda\frac{g\circ T}{g}$, where $\lambda\in S^1$, $g$ is an $S^1$-valued function, $\gamma$ is a character, and $T$ is a measure-preserving automorphism.

Quasi-similarity of K-automorphisms and simple polymorphisms

Two new dynamical properties of the Urysohn space

Combinatorial model of the moduli space of curves

Invariants of graphs, symmetric tensors, and Hamiltonian cycles

Random trees and random triangulations of the plane

Equivariant quantization of coadjoint orbits of GL(n)

Homotopic structures and problems of modern mathematical physics

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