Seminar talk, 4 November 2026

Speaker: Jacob Kryczka

Title: Singularity categories of differential equations

Abstract:
I will explain a categorical approach to the singularity theory of partial differential equations, starting from the classical geometry of jet spaces by studying a class of PDES whose typical smoothness and regularity hypothesis are weakened slightly. The defining PDE manifolds will have the simplest singularity structures, which are called quasi smooth or derived local complete intersections. Basic examples and geometric motivations will be given. We will then explain how to associate with each quasi smooth PDE, a category which measures how far the PDE is from being smooth. Time permitting we will show how to extract new categorical and analytic invariants of the PDE, via their periodic cyclic homology. We will investigate how these structures behave under prolongation and time permitting, will connect them to more classical notions like the singularity type of a PDE (e.g FOLD-type) and the geometric theory of solution singularities.