## EVENT DETAILS AND ABSTRACT

**Colloquium**
**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.

**Date:** Wednesday, April 16, 2008

**Time:** 4:10pm

**Where:** Lunt 105

**Contact Person:** Prof. Jeff Xia

**Contact email:** xia@math.northwestern.edu

**Contact Phone:** 847-491-5487

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