Research

Broadly speaking, I work on problems that lie between geometry and analysis. Most recently I've been studying isoperimetric problems for the first eigenvalue of the Laplace operator. In other words, of all drumheads with the same size, which one has the lowest fundamental tone? I've also worked with constant mean curvature surfaces (i.e. soap bubbles) and constant scalar curvature metrics. You can read more in my research statement.

Publications and Preprints

• Eigenvalues of Euclidean wedge domains in higher dimensions, submitted.
• A Payne-Weinberger eigenvalue estimate for wedge domains on spheres, joint with Andrejs Treibergs, Proc. Amer. Math. Soc. 137 (2009), 2299-2309.
• Coplanar k-unduloids are nondegenerate, joint with K. Grosse-Brauckmann, N. Korevaar, R. Kusner, and J. Sullivan, Int. Mat. Res. Not. 2009 (2009), 3391-3416.
• A Capture Problem in Brownian Motion and Eigenvalues of Spherical Domains, joint with Andrejs Treibergs, Trans. Amer. Math. Soc. 361 (2009), 391-405. Consider n Brownian cops chasing one Brownian thief with all motion restricted to a line. We find how many cops you need so that the expected capture time of the thief is finite.
• On the Nondegeneracy of Constant Mean Curvature Surfaces, joint with Nick Korevaar and Rob Kusner, Geom. Funct. Anal. 16 (2006), 891--923. This paper gives a general bound for the space of L^2 Jacobi fields on a coplanar, genus zero CMC surface. In particular, we show almost all triunduloids are nondegenerate (i.e. have no L^2 Jacobi fields).
• An End to End Gluing Construction for Metrics of Constant Positive Scalar Curvature Indiana Univ. Math. J. Vol. 52 (2003), 703--726.
• An End-to-End Gluing Construction for Constant Mean Curvature Surfaces PhD Thesis, 2001