EVENT DETAILS AND ABSTRACT

Title: Nodal hypersurfaces, ergodicity and complex analysis
Speaker: Professor Steve Zelditch
Speaker Info: Johns Hopkins University
Brief Description:
Special Note:
Abstract:

Nodal hypersurfaces are zero sets of eigenfunctions of the Laplacian on a Riemannian manifold (M, g). Since the time of Chladni, mathematicians and physicists have used nodal hypersurfaces to `visualize' eigenfunctions in their role as modes of vibrations or states of atoms and molecules. It is diffiuclt however to determine how the nodal hypersurfaces are distributed, or even how large their hypersurface volume is. But when (M, g) is real analytic, one can sometimes determine the distribution of complex zeros of analytic continuations of eigenfunctions to the complexification of M. When the geodesic flow is ergodic, one can precisely determine the limit distribution of complex zeros. When it is integrable, one can often determine the limit(s) as well (work in progress). Complexification also illuminates the pattern of nodal domains on analytic domains. We will discuss a number of such applications of complex analysis and dynamics to patterns of complex nodal sets.