The aim of this seminar is to understand the recent work [KS08] of Kontsevich and Soibelman concerning stability conditions on derived categories of Calabi--Yau 3-folds and wall crossing formulas for counting invariants of stable objects.

Here is a rough sketch of the expected structure. Let X be a Calabi--Yau 3-fold and D its derived category. T. Bridgeland constructed the space Stab(D) of stability conditions on D. It is a complex manifold locally modelled on Hom(K(D),C), where K(D)=K(X) is the K-theory of X and C denotes the complex numbers. The Euler characteristic defines a bilinear form on K(D) which is skew-symmetric (since X is CY3, using Serre Duality) and nondegenerate. This defines a holomorphic symplectic structure on Stab(D). There is a Lagrangian submanifold L of Stab(D) which is identified (by classical mirror symmetry) with the moduli space M of complex deformations of the mirror Y of X (together with the choice of a holomorphic volume form). Following Donagi and Markman [DM94] one considers the total space U of the universal family of intermediate Jacobians over M. They show that U admits a hyperkahler structure. In one of the complex structures, the torus fibration U --> M=L admits a construction of the type introduced in [KS04], where standard fibrations are glued along real codimension 1 walls via certain symplectomorphisms. In this setting the walls are given by the chamber structure on Stab(D) and the associated automorphisms are determined by counting invariants.

Some effort will be made to make talks accessible to graduate students. Everybody is welcome to contribute talks.