My research lies at the intersection of geometry and analysis. For instance, you might ask how much land you can enclose with 100 miles of fencing; this is the classical isoperimetric problem, which people have studied for over 2000 years. If you have a very large, flat field, the optimal shape is the first one you'd guess: a circle. However, the answer changes if, say, the land is a narrow strip between two forests, or if there's a river running through the field, or if the land isn't flat. (Is it better to surround the top of a hill or the pass between two peaks?) The best shape will satisfy a differential equation, and the solutions to this equation depend very much on the geometry of the land. Problems like that of finding isoperimetric regions form a rich area of study, with connections to other areas of mathematics such as function theory, probability, and topology, and even some applications.

Coauthors: Tom Carroll, UCC, Ireland; Andrejs Treibergs, U. Utah, US; Nick Korevaar, U. Utah, US; Rob Kusner, U. Mass, US; John Sullivan, TU Berlin, Germany; Karsten Grosse-Brauckmann, TU Darmstadt, Germany

Papers (preprint versions available on the the arXiv):

• Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of nonpositive curvature, (joint with T. Carroll) to appear in the Indiana U. Math. J.
• A reverse Holder inequality for extremal Sobolev functions. (joint with T. Carroll) Potential Analysis 42 (2015), 283-292.
• A numerical investigation of level sets of extremal Sobolev functions (joint with S. Juhnke), to appear in Involve.
• An isoperimetric inequality for extremal Sobolev functions. (joint with T. Carroll), RIMS Kokyuroku Bessatsu B43 (2013), 1-16.
• Two isoperimetric inequalities for the Sobolev constant. (joint with T. Carroll), Z. Angew. Math. Phys. 63 (2012), 855-863.
• Isoperimetric inequalities and variations on Schwarz's lemma. (joint with T. Carroll), preprint which has mostly been subsumed by later papers.
• Interpolating between torsional rigidity and principal frequency. (joint with T. Carroll) J. Math. Anal. Appl. 379 (2011), 818-826.
• Eigenvalues of Euclidean wedge domains in higher dimensions. Calc. Var. and PDE. 42 (2011), 93-106.
• Coplanar k-unduloids are nondegenerate (joint with K. Grosse-Brauckmann, N. Korevaar, R. Kusner, and J. Sullivan) Int. Math. Res. Not. 2009, 3391-3416.
• A Payne-Weinberger eigenvalue estimate for wedge domains on spheres (joint with A. Treibergs) Proc. Amer. Math. Soc. 137 (2009), 2299-2309.
• A caputre problem in Brownian motion and eigenvalues of spherical domains (joint with A. Treibergs) Trans. Amer. Math. Soc. 361 (2009) 391-405.
• On the nondegeneracy of constant mean curvature surfaces (joint with N. Korevaar and R. Kusner) Geom. Funct. Anal. 16 (2006), 891-923.
• An end-to-end gluing construction for metrics of constant positive scalar curvature. Indiana U. Math. J. 52 (2003), 703-726.
• An end-to-end gluing construction for surfaces of constant mean curvature PhD dissertation, U. Washington, 2001.

I've also written some miscellaneous notes related to my research, which you can read (if you're curious and somewhat masochistic) by clicking here.