Current Research
My research applies homotopy theory to study the topology of various moduli spaces, including Yang-Mills instanton moduli spaces and moduli spaces of holomorphic curves. The applications range from open problems in algebraic geometry to the mathematical development of quantum invariants with values in K-theory and elliptic cohomology, as currently studied by physicists.
Fundamental topological questions about moduli spaces are whether they are orientable (leading to a virtual fundamental class in ordinary homology, and to classical Donaldson invariants), admit a spin structure (so that one may use the Dirac operator for quantization, and define a virtual fundamental class in K-homology), or further refinements to elliptic homology. The higher differential topology of moduli spaces (spin, string, and fivebrane structures) is controlled by higher categorical analogues of the Quillen determinant line bundle, and one of my research objectives is to construct these.
The systematic study of these virtual fundamental classes leads to vertex algebra structures, as invented in conformal field theory. They encode sophisticated symmetries, for example, a conformal vector induces a Virasoro action. My research in this area is about the connections to generalized cohomology and bordism theory.
