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.
[ poster (pdf) ]
© Toshiyuki Kobayashi