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Annual Lecture Series in Topology and Geometry
in memory of
Prof. Alexander Zabrodsky

First talk:    Thursday, March 20, Colloquium, 14:30-15:30    Lecture Hall 2 Shapes and sizes of Laplace eigenfunctions    (slides) A gathering in memory of Prof. Alexander Zabrodsky will be held at 15:30 in the faculty lounge. Abstract : Eigenfunctions of the Laplacian on a Riemannian manifold (M, g) represent modes of vibrations of drums and membranes. In quantum mechanics they represent stationary states of atoms. Understanding shapes and sizes of eigenfunctions allows one to visualize these objects. An intriguing problem is to relate the shapes and sizes to the underlying classical mechanics, such as the geodesic flow of (M, g) or the dynamics of billiard trajectories on a billiard table. In this talk we will explain the role of eigenfunctions in quantum mechanics and discuss both classic and new results describing nodal (zero) sets of eigenfunctions. The new results relate nodal sets to classical dynamics. No prior knowledge of quantum mechanics is assumed.

Second talk:    Tuesday, March 25, 15:00-16:00. Geometry of Riemannian manifolds with the biggest possible eigenfunctions    (slides) Third talk:    Wednesday, March 26, 15:00-16:00. Eigenfunctions and Nodal sets    (slides) The lectures will be held in Lecture Hall 110 Abstract for the two more specialized talks: In these talks we discuss several recent results on shapes and sizes of eigenfunctions. By shape is meant the nodal geometry of eigenfunctions, i.e. their zero sets. Eigenfunctions of eigenvalue N2 are similar to polynomials of degree N, and one expects their zero sets to be similar to those of real algebraic varieties of degree N. I will discuss some classic and new results on the hypersurface volume of nodal sets which reflect this expectation and on the number of nodal domains (joint work in part with J. Toth and J. Jung as well as recent results of Ghosh-Reznikov-Sarnak). By the size of eigenfunctions is meant their Lp norms. I will discuss some joint work with C. Sogge relating the size of eigenfunctions to the geometry of geodesic loops. The relations between nodal sets and norms with classical mechanics are stronger if we complexify the manifold, analytically continue the eigenfunctions and study complex zeros. The complexification of M is the phase space of the mechanical system and the complex zeros reflect the ergodicity or integrability of the dynamics.

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