Math Colloquium: Nodal Statistics of Graph Eigenfunctions

Wednesday, January 31, 2018

3:00 PM-4:00 PM

Gregory Berkolaiko, Texas A&M University

Abstract: Zeros of vibrational modes have been fascinating physicists for several centuries. Mathematical study of zeros of eigenfunctions goes back at least to Sturm, who showed that, in dimension d=1, the n-th eigenfunction has n-1 zeros. Courant showed that in higher dimensions only half of this is true, namely zero curves of the n-th eigenfunction of the Laplace operator on a compact domain partition the domain into at most n parts (which are called "nodal domains"). Graph eigenfunctions lie between the 1- and higher-dimensional pictures: the nodal count deviates from n-1, but only by a bounded amount, which we will call "nodal surplus".

It recently transpired that this nodal surplus can be interpreted as an index of instability of the eigenvalue with respect to a suitably chosen perturbation. This discovery allows us to study the distribution of the nodal surpluses and to show that for a special family of graphs it takes a universal form.

Based on joint work with Yves Colin de Verdiere, Tracy Weyand, Lior Alon and Ram Band.

Contact Information

Georgi Medvedev
gsm29@drexel.edu