Abstracts for Seminar Algebraic Geometry (SAG)

(Joint work with Shrawan Kumar and Nicolas Ressayre) It is an interesting open problem to connect the structure constants in the cohomology ring of homogeneous spaces (in the Schubert basis), and those in the invariant theory of groups (generalizing the two appearances of Littlewood Richardson coefficients: in the cohomology of Grassmannians and the invariant theory of $GL_n$). I will talk about a generalization of a conjecture of Fulton which is a step in this direction.

Given a compact Riemann surface of genus at least two, there are two algebraic varieties attached to it: the character variety Ch, and the Hitchin moduli space M. The non-Abelian Hodge theorem asserts that they are diffeomorphic (but have different complex structures). While the rational cohomology rings H*(Ch) and H*(M) are isomorphic, the mixed Hodge structures are different and so are the weight filtrations, which therefore cannot possibly correspond via the non Abelian Hodge theorem. In recent joint work with T. Hausel (Oxford) and L.

We study flips of moduli schemes of stable torsion free sheaves on the projective plane via wall-crossing phenomena of Bridgeland stability. They are described as stratified Grassmann bundles by variation of stability of modules over certain finite dimensional algebra.