Event Type:

Geometry-Topology Seminar

Date/Time:

Monday, May 24, 2021 - 12:00 to 12:50

Location:

Zoom. Please contact Christine Escher for a link.

Guest Speaker:

Institution:

University of Utah

Abstract:

We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology and introduce new invariants to study these equivalence classes. These new invariants are as simple but more discerning than existing topological invariants, such as persistence barcodes and Reeb graphs. We give a method to relate any two Morse-Smale vector fields on the sphere by a sequence of fundamental moves by considering graph-equivalent Morse functions. We also explore the combinatorially rich world of height-equivalent Morse functions, considered as height functions of embedded spheres in 3-dimensional Euclidean space. Their level set invariant, a poset generated by nested disks and annuli from level sets, gives insight into the moduli space of Morse functions sharing the same persistence barcode. We also provide a visualization system that allows mathematicians to explore the complex configuration spaces of Morse functions, their gradients, and their associated Morse-Smale complexes.

This talk is a result of a two-year collaborative project formed to better understand how persistence interacts with Morse functions on surfaces. This is joint work with Jānis Lazovskis, Mike Catanzaro, Justin Curry, Brittany Fasy, Greg Malen, Hans Riess, and Matt Zabka. The visualization system is developed by Youjia Zhou.

Host: