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Mathematics Colloquium


Date:  Tuesday, February 08, 2011

Title:  Eigenfunctions of the Laplacian, ergodicity and complex geometry of nodal sets

Abstract:  Eigenfunctions of the Laplacian (or of a Schrodinger operator) represent vibrational modes of drums, Riemannian manifolds or atoms. Ever since Chladni in 1800 scientists have studied the nodal (zero) sets of eigenfunctions, which are the stationary points of the vibrating drum. As the eigenvalue tends to infinity the pattern of zeros becomes ever more complex, and a natural question is to find the length of the nodal line (or hypersurface area in dimensions > 2) and to find the spatial pattern. I will discuss some recent results on this problem. It turns out that for real analytic metrics, one can prove strong results on the complex zero set of analytic continuations of eigenfunctions to the complexification of the Riemannian manifold. As in algebraic geometry, the complex zeros are more accessible than the real zeros. When the geodesic flow is ergodic one can find the limit distribution of complex zeros and, in work in progress, the intersections of nodal lines with geodesics.

Speaker:  Steve Zelditch
Institution:  Northwestern Univ


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