“Universal sound” of anti-de Sitter manifold. Sophus Lie Days. Cornell, USA, 10 October 2013.

In music instruments, shorter strings produce a higer pitch than longer strings. Thinner strings produce a higer pitch than thicker strings. Similarly, in compact Riemann surfaces, any nonzero spectrum of the Laplacian varies as a function of the Teichmller space. In Riemannian geometry, the eigenvalues of the Laplacian of a compact manifold has a rich information on the original manifold, and some aspect of spectral geometry is well-known by the article ''Can one hear the shape of a drum?” (M. Kac, 1966).

How about the feature beyond Riemannian geometry? I plan to talk about a strange phenomenon in anti de Sitter manifolds that there exist countably many stable spectra of the Laplacian that do not vary under the deformation of geometric structures. The proof uses the geometry of discrete groups, partial differential equations, integral geometry (e.g. the idea of CT scan), and various ideas coming from Lie groups.

The talk will be aimed at a general mathematical audience, and a main part will be accessible to undergradutes with basic knowledge of advanced calculus, linear algebra, and (a little of) topology.

The non-expository part of the talk is based on joint works with F. Kassel.

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© Toshiyuki Kobayashi