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.

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.

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.

(sum_{k=1}ⁿ a_k b_k )² ≤ (sum_{k=1}ⁿ a_k²) (sum_{k=1}ⁿ b_k²)

ABSTRACT: I shall talk about Eilenberg-Zilber and Kunneth formulas for cocyclic and cyclic modules. Then we will apply it to the special case of Connes-Moscovici cocylic module for Hopf algebras. This will be an introduction for my main goal which is cup product and coproduct for Hopf cyllic cohomology of Hopf algebras.

Abstract: This talk is a survey of K.-T. Chen's dissertation work (as expounded in a 1951 paper of the same name) in which he demonstrated the utility of his elementary theory of integration in free groups, in informing about subgroups of these groups. This integration provides the classical framework for the non-commutative Fourier transform developed by Kapranov, which will be the subject of upcoming seminar talks.

Our plan is to give an overview of equivariant cohomology for torus actions as it is understood after Goresky, Kottwitz and MacPherson.