Isaac Ren: Morse theory of distance functions between algebraic hypersurfaces

Abstract

We study the generalization of Morse theory to non-smooth Lipschitz functions such as distance functions between subsets of Euclidean space (i.e., distance functions from a subset, restricted to another subset). After defining appropriate notions of subdifferentials and critical points, we restrict our scope to continuous selections, as presented by Agrachev, Pallaschke, and Scholtes, where we have a notion of nondegenerate critical points. This leads us to Morse-like results, which we present.

We then turn to the algebraic setting, where we compare our critical points with bottleneck and Euclidean distance degrees. We show, for generic hypersurfaces of degrees greater or equal to \(4\), that the critical points of the distance function from one surface to another are nondegenerate, and that their cardinality has a finite upper bound, which we make explicit.

This is joint work with Andrea Guidolin, Antonio Lerario, and Martina Scolamiero.