Survey of Spectra of Laplacians on Finite Symmetric Spaces
Audrey Terras
Here we survey what is known about spectra of combinatorial Laplacians (or adjacency operators) of graphs on the simplest finite symmetric spaces. This work is joint with J. Angel, N. Celniker, A. Medrano, P. Myers, S. Poulos, H. Stark, C. Trimble, and E. Velasquez. For each finite field $F$ with $q$ elements ($q$ odd) we consider graphs associated to finite Euclidean and non-Euclidean symmetric spaces over $F$. We are mainly interested in three questions:
- Are the graphs Ramanujan in the sense of Lubotsky, Phillips and Sarnak?
- What can you say about the distribution of eigenvalues of $A$ as $q$ goes to infinity?
- What can you say about the "level curves" of the eigenfunctions of $A$?