I am broadly interested in geometric evolution equations. My primary interest is the mean curvature flow. I am also interested in other flow techniques such as the Ricci flow and Hermitian curvature flow.
All of these techniques are based on the fundamental idea of finding the minimisers of various geometric quantities by flowing down Lyapunov directions for the functional to be minimised. In the case of the mean curvature flow, the functional in question is the area of a submanifold, and the mean curvature flow is in fact the downward gradient of the area functional. Since none of these flows is a linear PDE, we expect to encounter singularities before reaching minimisers; these singularities need to be resolved in some coherent scheme in order to continue the flow and proceed toward the minimiser.
In order to do this, one must have a very clear picture of what exactly happens to cause the flow to become singular. Developing such a picture for the Lagrangian mean curvature flow, in particular for Lagrangian surfaces in complex 2-space, is the focus of my current research.
- ``A Characterization of the Singular Time of the Mean Curvature Flow", 2011, Proceedings of the AMS
- ``Mean Curvature Flow in Higher Codimension", 2011, Ph.D. Thesis, Michigan State University. Adviser: Jon G. Wolfson