RESEARCH STATEMENT ( revised 10 / 01 / 08 )

My work focuses on the geometry and differential equations invariant under groups of affine and projective motions (in R and RP respectively). In particular, affine differential geometry, the study of properties of hypersurfaces in R which are invariant under affine volume-preserving motions, informs most of my work. Affine differential geometry is an old subfield of geometry, with Blaschke making much progress in the 1920s, but it is intimately related with many other topics of current interest: real Monge-Ampère equations, mirror symmetry of Calabi-Yau manifolds, and RP structures on surfaces. My research primarily consists of exploring these relationships. A slightly different point of view of my research is to consider manifolds with canonical coordinate charts—(X,G) structures in Thurston’s parlance, with coordinate charts in X = R or RP and gluing maps in G the affine or projective group, respectively. The rigidity of this construction allows for more interesting geometric and analytic structure on such a manifold. For example, the Hessian of a smooth function is a tensor on an affine manifold, and its determinant (the real Monge-Ampère operator) also makes sense. The model here is complex geometry: On a complex manifold (with holomorphic coordinate gluing maps), one defines Kähler metrics locally as complex Hessians of potential functions, and then natural complex Monge-Ampère equations lead to the rich geometry of Kähler-Einstein metrics. Much of my work involves similar PDEs on affine and projective manifolds, with the successes of complex geometry as a model.