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Explore advanced mathematical concepts in this research seminar from the Spectral Geometry in the Clouds series, focusing on marked length spectral invariants of Birkhoff billiard tables and compactness of isospectral sets. Delve into how the marked length spectrum encodes lengths of action-minimizing orbits with specific rational rotation numbers in planar billiard tables. Learn about Mather's beta function, a continuous extension of renormalized lengths in strictly convex tables, and discover how the algebraic structure of its Taylor coefficients proves C infinity compactness of marked length isospectral sets. Examine the connection between these findings and the Laplace spectral results of Melrose, Osgood, Phillips and Sarnak in this hour-long mathematical exploration presented by Amir Vig from the University of Michigan.
