Geometric Analysis and Topology Seminar

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A nonlinear spectrum on closed manifolds

Speaker: Christos Mantoulidis, Rice University

Location: Warren Weaver Hall 202

Videoconference link: https://nyu.zoom.us/s/99465062960

Date: Wednesday, November 3, 2021, 11 a.m.

Synopsis:

Abstract: The p-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace--Beltrami operator, which was defined by Gromov in the 1980s and corresponds to areas of a certain min-max sequence of hypersurfaces. By a recent theorem of Liokumovich--Marques--Neves, the p-widths obey a Weyl law, just like the eigenvalues do. However, even though eigenvalues are explicitly computable for many manifolds, there had previously not been any >= 2-dimensional manifold for which all the p-widths are known. In recent joint work with Otis Chodosh, we found all p-widths on the round 2-sphere and thus the previously unknown Liokumovich--Marques--Neves Weyl law constant in dimension 2. Our work combines Lusternik--Schnirelmann theory, integrable PDE, and phase transition techniques.