Monge-Ampere Equations and Geometric Structures on Manifolds

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

Abstract Award: DMS-0405873 Principal Investigator: John Loftin A geometric structure on a manifold is a set of restricted coordinate charts and gluing functions. There are often partial differential equations which behave well with respect to these gluing functions, and in turn these PDEs can be used to study the geometric structure. Dr. Loftin will continue to apply the theory of Monge-Ampere equations to manifolds with geometric structure. In particular, the relationship between Goldman's Fenchel-Nielsen type coordinates on the deformation space of convex real projective surfaces and the holomorphic coordinates introduced by Loftin will be further studied. Together with Eric Zaslow and S.T. Yau, Dr. Loftin will study parabolic affine sphere metrics on certain affine manifolds with singularities. These metrics are degenerate real slices of the Calabi-Yau metrics. Yau, Zaslow, and Loftin will follow the conjecture of Strominger-Yau-Zaslow to relate these metrics to a larger program arising from string theory of understanding mirror symmetry of Calabi-Yau manifolds. Monge-Ampere equations have an illustrious history in differential geometry. Yau's celebrated theorem constructed important notions of distance on a large class of spaces (so-called Calabi-Yau manifolds) by solving a Monge-Ampere equation. Dr. Loftin will study Monge-Ampere equations on other spaces. Some of these spaces are useful in the study of string theory, a physical theory which proposes a unification of all the forces in nature. In particular, there are singular Calabi-Yau spaces which Dr. Loftin, together with Yau and Zaslow, will study in order to shed light on the physics of string theory. Also, Dr. Loftin will put also into practice curricular extensions in teaching precalculus. These extensions were developed in a program involving other faculty and high school teachers, in July, 2003, associated with Columbia University's VIGRE program. He will also continue to develop similar curricular extensions.