Vanishing cycle cohomology in geometry and representation theory

by Prof. Ben Davison (The University of Edinburgh)

Wednesday May 14, 2025, 10:00 AM → 11:00 AM Europe/Berlin

Endenicher Allee 60/1-016 - Lipschitzsaal (Mathezentrum)

Given a regular function f on a smooth d-dimensional variety X, as long as f is in some sense generic, we expect the critical locus of f to be zero-dimensional, since it can be described locally as the zero-locus of the d linearly independent partial derivatives of f. Many spaces S that arise naturally in geometry, topology and algebra are locally described as critical loci of functions that are not, however, generic. This non-genericity is actually good news, reflecting the fact that these spaces are richer and more interesting than disjoint unions of points.

One may assign a kind of "critical" Euler characteristic to such a space, which heuristically counts the critical points of a generic deformation of f. While the traditional Euler characteristic of a space can be defined as the alternating sum of the dimensions of certain cohomology groups of that space, this critical Euler characteristic can be written as the alternating sum of the dimensions of a different type of cohomology, called vanishing cycle cohomology.

I will introduce this cohomology theory, along with a related cohomology theory called BPS cohomology. This is a special finite-dimensional subspace of the vanishing cohomology, which generates it even when the critical cohomology is infinite dimensional. These cohomology theories have many applications within geometry, combinatorics and representation theory, which I will begin to describe.