Not offered in 2017-2018

Laurentiu Maxim, Fall 2016:

The intersection homology of Goresky-MacPherson is a homology theory well-suited for the study of singular spaces. I will first introduce intersection homology in the geometric way, i.e. using chains that meet the strata of a singular space in a controlled way, and I will prove the basic properties of this theory, e.g. that it satisfies Poincare Duality (while the usual homology does not). I will also characterize the intersection (co)homology groups in terms of sheaves (using a description due to P. Deligne). This brings the perverse sheaves in the picture. I will discuss the formalism behind perverse sheaves and describe various applications to Singularity theory.