Mailing Address: Department of Mathematics Columbia University Room 509, MC 4406 2990 Broadway New York, NY 10027, USA Email: lastname@math.columbia.edu |
I think about problems in symplectic topology. Symplectic manifolds have no local invariants (they all look the same at sufficiently small scales), but they have interesting global ones which, almost without exception, come from studying maps from Riemann surfaces satisfying an appropriate PDE which generalises the notion of a holomorphic curve in complex geometry. The data produced by studying such maps can be packaged into different algebraic structures; the one I tend to use is called the Fukaya category, which should be thought of as a "categorification" of the intersection number of half-dimensional (Lagrangian) submanifolds. I've been studying these categories, and developing applications to mirror symmetry and to the study of Lagrangian embeddings.