Gil Kalai, of Reichman University and the Hebrew University of Jerusalem, lectures at the Institute for Advanced Study's Computer Science/Discrete Mathematics Seminar on intersection homology, the theory introduced by Goresky and MacPherson, and its links to the combinatorics of convex polytopes and cellular spaces. He covers toric g-vectors as invariants of polytopes and the search for a ring structure to support them, then turns to traces of intersection homology inside Stanley-Reisner rings and exterior face rings of triangulated spaces. He closes with speculative extensions to multiperversities and new topological invariants for singular spaces. Kalai frames the talk as a continuation of a 1995 informal IAS seminar with Bob MacPherson, Mark Goresky, and Tom Braden, and pitches it as self-contained for a discrete mathematics audience rather than specialists in topology alone.