Toshiyuki Kobayashi

The University of Tokyo

"Universal sounds" of anti-de Sitter manifolds

Wednesday, May 3, 2017

7:00PM

Arvo J Oopik '78 Auditorium

Abstract: In musical instruments, shorter strings produce a higher pitch than longer strings. The question, "Can one hear the shape of a drum?" (M. Kac, 1966), shows a typical aspect of spectral geometry, which asks the relationship between analysis (spectrum of Laplacian) and the Riemannian geometry. What will happen about "music instrument" beyond Riemannian geometry? A basic case is Lorentz geometry familiar to us as the spacetime of relativity theory. Recently, a new phenomenon has been discovered in anti-de Sitter manifolds, analog of spheres in Lorentz geometry, asserting that "universal sounds exist", namely, some eigenvalues of the Laplacian do not vary under the deformation of geometric structure. I plan to explain this strange phenomenon and the methods.

NB: PDF version of this announcement (suitable for posting).

Local to global — geometry of symmetric spaces with indefinite-metric.

Thursday, May 4, 2017

4:30PM

Kemeny 007

Abstract: How local geometric structure affects the global nature of manifolds? The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry of general signature, surprising little has been known about global properties of the geometry until recently even if we impose a locally homogeneous structure. I plan to survey this young topic in geometry such as the existence problem of compact locally homogeneous manifolds and their deformation theory.

NB: PDF version of this announcement (suitable for posting).

Analysis on locally pseudo-Riemannian symmetric spaces

Friday, May 5, 2017

4:00PM

Kemeny 007

Abstract: Analysis on Riemann surface, or more generally, locally Riemannian symmetric spaces, has been developed extensively in connection with automorphic form theory and representation theory of reductive groups. In the more general setting where the metric is not positive definite, new difficulties arise from analysis, geometry, and representation theory. I will discuss some new developments and methods in my lecture.

NB: PDF version of this announcement (suitable for posting).