Alexandre Girouard (Cardiff):

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"Shape optimisation for lower eigenvalues of the Laplace operator" 

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ABSTRACT: 

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The Polya conjecture (from 1954) states that the k-th Neumann eigenvalue
of a planar domain is bounded above by 4k Pi.

In this talk I will present a sharp isoperimetric inequality for the second
nonzero eigenvalue. This implies Polya conjecture for k=2. I will also
discuss similar results for the Steklov spectrum and for the spectrum of
the Laplace-Beltrami operator on a closed surface.

This is joint work with Nikolai Nadirashvili and Iosif Polterovich.