University of Georgia

Mathematics Department Colloquium

Misha Grinberg

Massachusetts Institute of Technology

February 7, 2000 (Monday) at 3:00 pm

Self-Indexing and Perverse Sheaves

Abstract: A self-indexing Morse function f on a smooth manifold is a function whose index at every critical point p is equal to f(p). The existence of self-indexing functions was first used by Smale in his proof of the Poincare Conjecture in dimensions > 4. In this talk, we will discuss the existence of self-indexing (real-valued) functions on a smooth complex algebraic variety X with a fixed algebraic stratification. Here the notions of a Morse function, a critical point, and the index must all be understood in the stratified sense. To show that X admits many self-indexing functions, we examine the ascending and descending sets for a suitable gradient-like vector field in the neighborhood of a critical point. The motivation for this work comes from the theory of (middle perversity) perverse sheaves on X. We will show how the idea of self-indexing naturally leads one to this theory.