Computer Science/Discrete Mathematics Seminar

I will discuss the wonderful theory of intersection homology, introduced by Goresky and MacPherson, and its connections to the combinatorics of convex polytopes and cellular spaces. I will mention results and questions about "toric g-vectors", important invariants of polytopes, with particular emphasis on the combinatorially motivated search for a suitable ring structure—or a substitute.

I will then turn to the search for traces of intersection homology in Stanley–Reisner rings and exterior face rings of triangulated spaces. Inspired by the many algebraic manifestations of ordinary Betti numbers for manifolds, we seek descriptions of intersection homology that do not require a chosen stratification.

Finally, I will discuss possible extensions of intersection homology to multiperversities and the prospect of obtaining new topological invariants for singular spaces.

This lecture continues discussions from the pleasant informal 1995 seminar here at the IAS with Bob MacPherson, Mark Goresky, Tom Braden, and a few others. I will try to make it self-contained and easygoing.