Moduli Spaces of Morse Functions for Persistence

Explore a comprehensive lecture on the interaction between persistence and Morse functions on surfaces, focusing on the sphere. Delve into the decomposition of Morse-Smale functions into cells and discover how these functions are related through three elementary moves based on cancellation of critical value pairs. Investigate the various ways Morse-Smale functions on S^2 factor through embeddings into R^3, uncovering the nesting poset of level sets invariant. Gain insights into the inverse problem of persistence by examining the barcode of the height function in relation to the nesting poset. Learn about topics such as equivalences, elementary deformations, maps between postsets, and embeddings. This 55-minute talk, presented by Jānis Lazovskis, covers the results of a two-year collaborative project and includes contributions from Mike Catanzaro, Justin Curry, Brittany Fasy, Greg Malen, Hans Riess, Bei Wang, and Matt Zabka.

Introduction
Motivation
Background
Equivalences
Different embeddings
Herzomerodion
Elementary deformations
Nesting postsets
Maps between postsets
Extensions
Embeddings
Regular values