This project lies at the interface between Differential Geometry, General Relativity, and Partial Differential Equations. One theme will be the study of manifolds of positive curvature. Specifically the proposer plans to further his study of manifolds of positive isotropic curvature (PIC) using Ricci flow and minimal surface techniques. He also hopes to prove sphere theorems under pointwise pinching conditions with pinching slightly below 1/4. In relativity he plans to study the geometry of static matter solutions hoping to show that matter bodies cannot be separated by convex sets in such solutions. He also plans to study the general Penrose inequality; that is, the conjectured inequality for black hole initial sets with nonzero second fundamental form. He will also pursue a range of questions concerning minimal submanifolds satisfying free boundary conditions and connections to eigenvalue problems. Finally he plans to continue his study of minimal lagrangian and special lagrangian submanifolds of Kahler-Einstein manifolds. He will attempt to prove a conjecture concerning the invariance of the subgroup of the integral homology of a Calabi-Yau manifold which is generated by minimal lagrangian cycles when one deforms the ambient Calabi-Yau structure.
Understanding spaces in terms of their curvature properties is a fundamental idea in mathematics and science. The laws of nature often provide information about curvature properties, and from those we must deduce information about the spaces. A prime example of this is General Relativity where the equations of motion are described by curvature, and from those we must determine concrete properties of the universe. This project deals with a range of questions of this type such as the way in which black holes form from smooth initial data. Furthermore, geometry is important in application areas such as imaging and computer graphics. The research of this project will advance the core geometric ideas which form the basis of such applications.