Singularities of Minimal Hypersurfaces and Lagrangian Mean Curvature Flow

Overview The PI proposes to investigate singularities in two related settings in geometric analysis: minimal hypersurfaces and the Lagrangian mean curvature flow. The unifying theme in the proposal is the study of situations where singularities are not isolated. The major questions that will be addressed are uniqueness of tangent cones and tangent flows, and the behavior of minimal surfaces and the Lagrangian mean curvature flow near singularities. In the setting of minimal hypersurfaces important consequences will be a partial classification of entire minimal graphs, an improved understanding of the fine structure of the singular set, and progress towards understanding the smoothness of generic minimal hypersurfaces in higher dimensions. With regards to the Lagrangian mean curvature flow the proposed research will make progress on the question of how the flow can be continued through certain well behaved singularities, and under what conditions other types of singularities can occur. Intellectual merit A basic problem in many areas of geometry and PDE is to understand the formation of singularities under different processes. The proposed research will lead to a deeper under- standing of this in the settings of minimal hypersurfaces and the Lagrangian mean curvature flow. The focus of the proposal is on settings where the singular set does not consist of isolated points, and there are many different problems in geometric analysis where this situation arises. As such, progress on the proposed research problems will not only have important geometric consequences for minimal surfaces and the Lagrangian mean curvature flow, but will lead to the development of techniques that can be used in a wide variety of other problems in geometric analysis. Broader impacts The PI is currently advising four Ph.D. students. The PI has served, and is continuing to serve, as a co-organizer of several educational programs. One is a yearly week-long under- graduate summer workshop in geometry and topology, partly aimed at students with little background beyond Calculus and Linear Algebra, so that they can gain a broader perspective of different mathematical areas. The other is a graduate bridge program for incoming Ph. D. students, aimed at getting them up to speed and help them succeed in the graduate program. The PI also co-organizes the Geometric Analysis seminar geared towards graduate students, and has co-organized multiple conferences and workshops in the past 5 years. The PI will organize a week long summer Math Circle camp aimed at students in grades three to five. This will fill an important gap, as most such activities are currently aimed at high school students.