Jānis Lazovskis (8/26/20): Moduli spaces of Morse functions for persistence

Title: Moduli spaces of Morse functions for persistence

Abstract: I will present the results of a two year collaborative project formed to better understand how persistence interacts with Morse functions on surfaces. Restricting to the sphere, Morse--Smale functions can be decomposed into cells. We show that all such functions are related by three elementary moves on these cells, based on (reverse) cancellation of pairs of critical values. At a finer level, we consider the different ways a Morse--Smale function on S^2 factors through embeddings into R^3, and uncover the nesting poset of level sets invariant. For persistence as the barcode of the height function, the nesting poset gives insight into the inverse problem of persistence. This is joint work with Mike Catanzaro, Justin Curry, Brittany Fasy, Greg Malen, Hans Riess, Bei Wang, Matt Zabka.