Yan Soibelman

Homological mirror symmetry and torus fibrations

In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya category of a Calabi-Yau manifold and the derived category of coherent sheaves on the dual Calabi-Yau manifold). Our point of view on the origin of torus fibrations is based on the standard differential-geometric picture of collapsing Riemannian manifolds as well as analogous considerations for Conformal Field Theories. It seems to give a description of mirror manifolds much more transparent than the one in terms of D-branes. Also we make an attempt to prove the homological mirror conjecture using the torus fibrations. In the case of abelian varieties, and for a large class of Lagrangian submanifolds, we obtain an identification of Massey products on the symplectic and holomorphic sides. Tools used in the proof are of a mixed origin: not so classical Morse theory, homological perturbation theory and non-archimedean analysis.

Algebras of functions on compact quantum groups, Schubert cells and quantum tori

AbstractThe structures of Poisson Lie groups on a simple compact group are parametrized by pairs (a, u), wherea∈R,

Affine Structures and Non-Archimedean Analytic Spaces

u \in \Lambda ^2 \mathfrak{h}_R

Mirror symmetry and noncommutative geometry of A∞-categories

, and

Homological mirror symmetry, deformation quantization and noncommutative geometry

\mathfrak{h}_R

Quantum p-adic Spaces and Quantum p-adic Groups

is a real Cartan subalgebra of complexification of Lie algebra of the group in question. In the present article the description of the symplectic leaves for all pairs (a,u) is given. Also, the corresponding quantized algebras of functions are constructed and their irreducible representations are described. In the course of investigation Schubert cells and quantum tori appear. At the end of the article the quantum analog of the Weyl group is constructed and some of its applications, among them the formula for the universalR-matrix, are given.