Abstract: Regarding the space of Hermitian matrices as an N^2 dimensional real vector space with nondegenerate bilinear form given by the trace, we may apply Feynman's theorem to compute matrix integrals. First I will show how to evaluate such integrals by a sum over compact oriented surfaces with boundary, and then I will use this expansion to prove a theorem of t'Hooft which states that in the limit for large N of such integrals the sum only depends on the contribution of planar connected fat graphs.

Abstract: After defining equivariant (co)homology for torus actions, I will present an equivariant version of Poincaré-Alexander-Lefschetz duality and relate it to an old result of Duflot.

Then I will turn to syzygies in equivariant cohomology. Syzygies are modules (over a polynomial ring) that interpolate between torsion-free and free modules. I will recall how syzygies are related to equivariant homology, the Atiyah-Bredon sequence and the equivariant Poincaré pairing. For actions on manifolds I will then give a "geometric criterion" that characterizes such syzygies in terms of the orbit space together with its stratification by orbit dimension. At the end I will discuss the existence of "maximal" syzygies for compact orientable manifolds. Here an interesting connection with singularities of real algebraic varieties appears.

This is joint work with Chris Allday and Volker Puppe.