Colloquium
3:00 p.m., Friday (April 7, 2006)
MATX 1100
Uzi Vishne
IAS and Bar Ilan University
Isospectral manifolds and Cayley graphs
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 nonisomorphic 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 noncommensurable manifolds in dimension d > 2. Time permitting, I will also explain how a positivecharacteristic analogue of these techniques provides isospectral noncommensurable finite complexes, and isospectral nonisomorphic Cayley graphs of finite simple groups.
This is a joint work with A. Lubotzky and B. Samuels.
Refreshments will be served at 2:45 p.m. (MATX 1115, Math Lounge).
