Abstract: In his famous paper from 1966, M. Kac asks "Can you hear the shape of a drum?"; namely -- can a compact manifold be determined from the spectrum of its Laplacian? The answer turned out to be negative, where various construction of isospectral non-isomorphic surfaces were discovered (by M.F. Vigneras, T. Sunada and others).
However, in all these construction, the pairs of manifolds are commensurable (namely they have a finite common cover). This raises a natural question: can you hear the shape of a drum, at least roughly (i.e. up to commensurability)?
I will present a construction of families of isospectral non-commensurable manifolds in dimension d>2. Time permitting, I will also explain how a positive-characteristic analogue of these techniques provides isospectral non-commensurable finite complexes, and isospectral non-isomorphic Cayley graphs of finite simple groups.
This is a joint work with A. Lubotzky and B. Samuels.