Singularities in Kahler geometry and Lagrangian mean curvature flow

Singularities arise naturally in many problems in geometry and physics, and it is important to develop new techniques that can be applied in singular settings. The project deals with two related settings, singular Kahler-Einstein metrics, and the Lagrangian mean curvature flow. In both settings the goal is to understand the connection between singularities from the geometric and algebraic points of view. The project also incorporates the training of graduate students, and organizing activities such as seminars and conferences as well as outreach to the local elementary schools. One of the two main directions of the project is to further our geometric understanding of singular Kahler-Einstein spaces. In particular, a goal is to show that such spaces always satisfy the RCD property, which leads to many geometric consequences. Eventually such results will lead to new geometric approaches to questions about singular varieties in algebraic geometry. The other main direction is the study of singularities along the Lagrangian mean curvature flow. This high codimension mean curvature flow is so far little understood, but conjecturally has important applications to the study of special Lagrangian submanifolds and mirror symmetry. A major goal of the project is to show that generically only very simple singularities can form along the flow, in analogy with recent progress on the mean curvature flow of hypersurfaces. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.