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in memory of

Prof. Alexander Zabrodsky

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Shapes and Sizes of eigenfunctions
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Prof. *Steve Zelditch* (Northwestern University)

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, 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 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 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 |

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