New Minicourse by Leonid Polterovich (Tel Aviv University)

January 5, 2023

Minicourse "Courant, Bezout, and persistence"

Meeting Tuesday and Thursday from 1-2pm in Eckhart 202 from 2/7-3/2

Course Overview:

 I'll discuss an approach to study oscillation of functions based on persistence modules and barcodes, an area of algebraic topology
 originating in data analysis. This technique has a number of applications to spectral geometry, a discipline dealing, in particular, with geometric features of eigenfunctions  of the Laplace-Beltrami operator on Riemannian manifolds. The first one  is an extension of Courant's nodal domain theorem to linear combinations  of eigenfunctions, a problem with long history, with a caveat being that  the direct generalization is known to be false. The second application is a version  of Bezout's theorem  in the context of eigenfunctions, supporting an intuition, due to H.Donnelly  and C.Fefferman, that eigenfunctions behave as polynomials whose degree is comparable to the square  root of the eigenvalue. The necessary preliminaries from topological persistence and spectral geometry will be  explained. The lectures are based on a paper "Coarse nodal count and topological persistence"  with Lev Buhovsky, Jordan Payette, Iosif Polterovich, Egor Shelukhin, and Vukašin Stojisavljević, arXiv:2206.06347.

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