1. Research interests:
Topological and geometric aspects of quantum field theories. One goal of my research is to contribute to a mathematical definition of quantum field theory that is versatile enough to include many different flavors of field theories (e.g., topological field theories, conformal field theories, Euclidean field theories and supersymmetric versions of these field theories).
A conjecture that my collaborator Peter Teichner (UC Berkeley and Max-Planck-Institut Bonn, Germany) pursue is to show that families of supersymmetric Euclidean field theories parametrized by a manifold $X$ represent elements of $TMF^*(X)$, the generalized cohomology theory constructed by Hopkins-Miller known as the "topolological modular form theory". Another conjecture that motivated much of my interest in field theories is the following: Let $X$ be a closed Riemannian manifold of dimension $4k$ with string structure (i.e., its tangent bundle restricted to the 4-skeleton is trivial). Then if the Ricci curvature is positive, the Witten genus of $X$ vanishes. The Witten genus is a topologically defined invariant of $M$. Heuristically, it is the "partition function" of the "non-linear $\sigma$-model of $X$, a field theory associated to $X$ that has yet to be defined rigorously.
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