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The sound of a singularity

Banff International Research Station for Mathematical Innovation and Discovery

Analytically computing the spectrum of the Laplacian is impossible for all but a handful of classical examples. Consequently, it can be tricky business to determine which geometric features are spectrally determined; such features are known as geometric spectral invariants. Weyl demonstrated in 1912 that the area of a planar domain is a geometric spectral invariant. In the 1950s, Pleijel proved that the n-1 dimensional volume of a smoothly bounded n-dimensional Riemannian manifold is a geometric spectral invariant. Kac, and McKean & Singer independently proved in the 1960s that the Euler characteristic is a geometric spectral invariant for smoothly bounded domains and surfaces. At the same time, Kac popularized the isospectral problem for planar domains in his article, ``Can one hear the shape of a drum'' Colloquially, one says that one can ``hear'' spectral invariants. In this talk I will not only discuss my work with many collaborators (Rafe Mazzeo, Zhiqin Lu, Clara Aldana, Klaus Kirsten, David Sher, Medet Nursultanov) but also highlight the works of many other colleagues, who share a similar interest in ``hearing singularities.''

Mathematics; Global analysis, analysis on manifolds; Dynamical systems and ergodic theory; Global analysis

video/mp4

Author affiliation: Chalmers University and the University of Gothenburg

2019-10-13

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/71898

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/