I am interested in mirror symmetry, both homological and physical. My work has thus far concerned two main topics; quantum sheaf cohomology and the Calabi-Yau/Landau-Ginzburg correspondence.
Quantum sheaf cohomology is a physics-inspired branch of study that proposes new invariants of manifolds generalizing Gromov-Witten invariants; see my talk (aimed at mathematicians) for a synopsis. Currently there is no mathematical definition of these invariants — only a collection of physics-inspired algorithms exist to compute them. As part of my thesis, I computed the quantum sheaf cohomology of an arbitrary deformation of the tangent bundle of the product of two rational curves. See, for instance, my talk (aimed at physicists, where QSC was called (0,2) quantum cohomology) illustrating the algorithm. Physicists have additionally proposed a generalization of mirror symmetry that exchanges bundle data in addition to the symplectic and complex-structure data exchanged by ordinary mirror symmetry. This is another area of active research.
The Landau-Ginzburg/Calabi-Yau correspondence correspondence relates the data of certain Calabi-Yau manifolds to the singularity theory of their defining polynomials. My work approaches the topic from the point of view of symplectic quotients, and relies on combinatorial algebra associated to the ambient toric varieties of the Calabi-Yau hypersurfaces. This program is expected to completely classify compact subvarieties of non-complete toric varieties. It is hoped that the techniques will produce many examples (with defining polynomials) of non-complete-intersection Calabi-Yau's.
This year I am organizing the Math/Physics seminar, whose calendar is available in both iCal and XML feeds. As the name implies, the seminar covers a range of topics of interest to both mathematicians and physicists. Students from both departments are encouraged to attend.