Mathematics Department Colloquium: Recent advances on `hearing the shape of a drum'

Colloquium | October 31 | 4:10-5 p.m. | 60 Evans Hall

 Steve Zelditch, Northwestern University

 Department of Mathematics

In 1966, M. Kac posed the problem, `Can one hear the shape of a drum', i.e. can one determine a bounded plane domain from its Dirichlet (or, Neumann) eigenvalues? He proved that indeed one can determine a disk from its spectrum. Disks remained the only domains known to be determined by their eigenvalues until recently. My talk concerns a recent result due to Hamid Hezari and myself, showing that ellipses of small eccentricity are also determined by their eigenvalues. The proof uses some recent advances in dynamical inverse theory due to Avila, de Simoi, Kaloshin, Sorrentino, Wei and others which partly resolve another famous inverse problem of G.D. Birkhoff: are ellipses the only billiard tables with integrable billiards? The solution of the inverse eigenvalue problem is based on their result that ellipses of small eccentricity are the only nearly circular billiard tables with rationally integrable billiards. My talk will review both the eigenvalue and dynamical inverse spectral results.