A heat trace anomaly on polygons, with R. Mazzeo.
In the 1960s, McKean, Singer, Ray, Kac, Fedosov (and others) observed that the t^0 coefficient in the small time heat trace expansion for polygonal domains includes a mysterious contribution from the angles at the corners.
which was beautifully computed in an unpublished work by Ray. The heat trace "anomaly" is that this term does not arise as a limit in the corresponding t^0 coefficient of the heat trace for smoothly bounded domains which approximate the polygonal domain. In other words, there is a defect.
In our work we uncover the cause of this anomaly...
Dynamics of asymptotically hyperbolic manifolds,
This work is a first step (step zero) in generalizing and extending results concerning the quantum and classical mechanics of infinite
volume hyperbolic manifolds to similar results for conformally compact manifolds with sectional curvatures asymptotic to -1. The main result is a prime orbit theorem for the geodesic flow, and my favorite result is the final corollary which demonstrates a relationship between the dynamics of the geodesic flow (classical mechanics) and the existence of pure point spectrum (quantum mechanics).
A note on the spectral theory and dynamics of asymptotically hyperbolic manifolds, is somewhere between a survey and a continuation of my work on asymptotically hyperbolic and conformally compact manifolds and was written
for the conference proceedings of the internation conference on spectral theory and geometry at the Institut Fourier. Some excellent authors on this subject include David Borthwick who is one of the talented physicists able to prove rigorous mathematical theorems, and Collin Guillarmou
whose work is elegant and thorough in the tradition of French mathematics.
The Fundamental Gap Conjecture for Polygonal Domains, with Z. Lu.
This paper is a preliminary investigation of the gap between the first two eigenvalues of the Dirichlet Laplacian on polygonal domains. The main result is a compactness theorem when the domain is a triangle.
The fundamental gap conjecture, with Z. Lu and T. Betcke is a continuation of Z. Lu and my work on the fundamental gap.
Spectral Geometry and Aysmptotically Conic Convergence, was my doctoral dissertation published by Stanford University in 2006. It is 115 pages and has color pictures, although my pictures are not nearly
as beautiful as those in Emily Dryden's thesis. My thesis introduces a parametrized parabolic operator calculus which is developed as a technical tool with which to study the spectral geometry of manifolds with isolated
and iterated conic singularities. The simpler techniques are present in my work with Mazzeo. These technical tools are waiting patiently to be exploited...
The theory of erosion and connections to optimal transport, with B. Birnir is in prepration. We prove existence an uniqueness of (weak) solutions to the nonlinear PDEs
demonstrated by Birnir, Hernandez, Merchant, Smith (and others?) to model the deterministic aspect of erosion (large time scale). We then investigate the natural question, "Is erosion transported optimally in some sense
according to the model PDEs?" The answer may (or may not) suprise you...
Conformal deformations of singular metrics to constant negative scalar curvature, with T. Jeffres is currently in preparation. We determine necessary and sufficient conditions to
construct a conformal deformation to constant negative scalar cuvature for incomplete Riemannian manifolds with generalized conic metrics such that the resulting metric (with cnsc)
is also an incomplete generalized conic metric.
A new look at an old spectral anomaly is a talk based on my work with Mazzeo which reveals the cause of the heat trace anomaly (defect) on polygonal domains.
Can one hear the shape of a drum? is a talk based on the famous question posed by Mark Kac (in his paper of the same name) and its resolution. Note the date this talk was given and peut-etre vous trouveriez un poisson d'avril...