Bukhshtaber's ring of simple polyhedra

Percolation and black noise

We will prove the conjecture of Tsirelson, that 2D critical percolation is a noise in the scaling limit. Thus we construct the first example of a black noise in the plane.

The result is a joint work with Oded Schramm and has two parts:

First we propose a new approach to constructing the scaling limit of percolation in the plane: a random collection of rectangles which are crossed. We show that for many percolation models there is tightness and hence subsequential scaling limits, and that for critical site percolation on triangular lattice the limit is unique.

The second part proves that (in the scaling limit) when a domain is cut by a smooth curve, percolation configuration in it can be reconstructed from configurations in two halves (i.e. there is no information "stored" on the curve). On the lattice it means that resampling the configuration in a small neighborhood of the curve and slightly perturbing it off the curve does not change the crossing properties much.

Graded theory of Frobenius n-homomorphisms and its topological applications

Pre-Markov partitions for hyperbolic automorphisms of the two-dimensional torus and pseudo-Anosov diffeomorphisms of surfaces

Problem for students

Proof of the Vershik-Kerov conjectue on the entropy of the Plancherel measure

Asymptotic facility location problems

On the Gaudin model

Quantum group symmetries of integrable spin chains (Birman-Wenzl-Murakami (BMW) algebra)

Integrable open spin chains related to the quantum affine algebras U_q(o(3)) and U_q(A_2^(2)) are considered. Hamiltonians (and other integrals) belong to the Birman-Wenzl-Murakami algebra (BMW). The symmetry algebra U_(o(3)) and the BMW-algebra centralize each other in the spin chain representation space. Consequently, the multiplet structure of the energy spectra is obtained.

Enumeration of plane curves via floor diagrams

(joint work with G.Mikhalkin)

Absolutely nonfree group actions and their characters

Symbolic dynamics and affine convex geometry

On a two-dimensional generalization of Szemeredi's theorem

The combinatorics of the universal Groebner basis and multidimensional Young diagrams

The structural stability of vector fields with shadowing properties corresponding to some classes of reparametrizations

(joint work with S.Yu.Pilyugin)

Probabilistic models related to three-dimensional Young diagrams

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