Monge-Ampere Equations and Geometric Structures on Manifolds

Project Details



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.

Effective start/end date8/1/047/31/08


  • National Science Foundation: $107,973.00


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