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Abstract: (joint work with Eran Makover)
In the '70's, McKean announced the following result, which he called ``The Riemann Hypothesis for Riemann Surfaces'':
Theorem: Let S be a Riemann surface of genus g >1. Then the first eigenvalue of the Laplacian of S is at least 1/4.
Unfortunately, McKean's result is quite false, as simple examples show. Nonetheless, one can still hope that the first eigenvalue of a ``typical'' Riemann surface is large.
We show that there is a positive constant C such that, for a randomly picked Riemann surface, the probability that the first eigenvalue is at least C tends to 1 as the genus goes to infinity. Part of the problem is to define what one means by a randomly picked Riemann surface, in a way that gives one control over the geometry. We do this by modeling picking a random Riemann surface on the problem of picking a random 3-regular graph.
This talk will be accessible to graduate students.
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