Abstracts for MPI-Oberseminar

I will explain how to use polyhedral complexes of constant
curvature and  hyperbolic orbifolds
in order to construct some interesting complex-projective varieties.
Among other things,I will show how to construct projective varieties
with normal crossing singularities which have prescribed
finitely-presented fundamental groups. This is partly joint work with
Janos Kollar.

I will talk about the volume conjecture that would relate the colored
Jones polynomial of a knot to the hyperbolic volume of the knot
complement.
I will also describe generalizations of the conjecture.

Local Whittaker function is a special funciton on a group G over local
field F, defined as a matrix element in a principal series representation of G(F). In
the mid 70's Langlands conjectured, and Shintani and Casselman-Shalika proved, the
explicit formula, expressing the p-adic Whittaker function as a character of the dual
group G^. My talk is devoted to various aspects of Archimedean counterpart of the
Langlands-Shintani-Casselman-Shalika (LSCS) formula, discovered recently in
collaboration with A.Gerasimov and D.Lebedev.

We study the Donaldson-Thomas invariants of the 2-dimensional stable sheaves in a smooth projective threefold. The DT invariants are defined via integrating over the virtual fundamental class when it exists. When the threefold is a K3 surface fibration we express the DT invariants of sheaves supported on the fibers in terms of the the Euler characteristics of the Hilbert scheme of points on the K3 surface and the Noether-Lefschetz numbers of the fibration. Using this we prove the modularity of the DT invariants which was predicted in string theory.