Abstract: I will discuss how A-infinity algebras and other structures 'up to homotopy' can be used to compute equivariant cohomology and, more generally, the cohomology of fibre bundles. The resulting constructions lead to an 'up to homotopy' version of Koszul duality as described by Goresky-Kottwitz-MacPherson. As application, I will express the integral cohomology of smooth, non-compact toric varieties purely in terms of fan data.

Abstract: Using heat equation methods, the index of an elliptic operator can be computed by a local formula. In this series of lectures, we will review the necessary analysis for defining the index of an elliptic operator, and derive a local formula for the index.

Abstract: According to the Kushnirenko theorem the number of solutions in (C^*)ⁿ of a generic system of equations P_1=...=P_n=0 with given Newton polyhedra Delta(P_1)=...=Delta(P_n)=Delta equals to n! V(Delta), where V(Delta) --- n-dimensional volume of Delta. I will present an elementary proof of this theorem using the famous Hilbert theorem on degree of projective variety. If time permits I will present a simple proof of the Hilbert theorem.