Hearing the shape of a triangle

Abstract

In 1966 Mark Kac asked the famous question 'Can one hear the shape of a drum?'. While this was later shown to be false in general, it was proved by C. Durso that one can hear the shape of a triangle. After an introduction to the general inverse spectral problem we will give a new proof of this fact. The central point of the argument is to show that area, perimeter and the sum of the reciprocals of the angles determine a triangle uniquely. This is proved using convexity arguments and the partial fraction expansion of $\sin^{-2}x$.Comment: 13 pages, 4 figures (tikz), to appear in Notices of the AMS; compared to first version: added more references and pictures and related problems, added a discussion of wave invariants, slightly changed titl