Title: Spectral invariants and Birkhoff Billiards
Consider a smooth, bounded, strictly convex domain in the plane. There are two games one can play. The classical one is billiards, in which a billiard ball orbits around the domain and reflects elastically at the boundary. The “quantum” analogue involves the study of wave propagation in the domain and understanding the frequencies at which such waves oscillate. In this talk, we discuss recent progress on the inverse spectral problem of determining a billiard table from its Laplace spectrum. In particular, we introduce a new class of spectral invariants for a generic class of billiard tables obtained from an explicit Hadamard-Riesz type parametrix for the wave propagator, microlocally near geodesic loops of small rotation number. These same techniques also allow us to prove (infinitesimal) Robin spectral rigidity of the ellipse, when both the boundary and boundary conditions are allowed to deform simultaneously. Finally, we mention ongoing work together with Vadim Kaloshin to cancel singularities in the wave trace for special types of domains.
For more information on the Zoom meeting please contact dmitry.jakobson [at] mcgill.ca