# Research

*This is the intersection of a coplanar constant mean curvature surface with its plane of symmetry; see papers by N. Korevaar, R. Kusner, and myself*.

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.