The geometry of non-arithmetic hyperbolic manifolds is mysterious in spite of how plentiful they are. McMullen and Reid independently conjectured that such manifolds have only finitely many totally geodesic hyperplanes and their conjecture was recently settled by Bader-Fisher-Miller-Stover in dimensions larger than 3. Their works rely on superrigidity theorems and are not constructive.

In this talk, we strengthen their result by proving a quantitative finiteness theorem for non-arithmetic hyperbolic manifolds that arise from a gluing construction of Gromov and Piatetski-Shapiro. Perhaps surprisingly, the proof relies on an effective density theorem for certain periodic orbits. The effective density theorem uses a number of ideas including Margulis functions, a restricted projection theorem, and an effective equidistribution result for measures that are nearly full dimensional. This is joint work with K. W. Ohm.

The extreme eigenvalues of a random matrix have been important objects of study since the inception of random matrix theory and also have a variety of applications.  The Laplacian matrix is the workhorse of spectral graph theory and is the key player in many practical algorithms for graph clustering, network control theory and combinatorial optimization.  In this talk, we discuss the fluctuations of the extreme eigenvalues of a random Laplacian matrix with gaussian entries.  The proof relies on a broad set of techniques from random matrix theory and free probability.  We will also describe some recent progress on a broader class of random Laplacian matrices.

This is joint work with Andrew Campbell and Sean O'Rourke.

https://sites.uci.edu/inverse/