A.J. Tolland

I'm on the job market this year. If you're hiring, please have a look at my research statement, and my curriculum vitae.

I was once a physics PhD student (at UChicago), and I remain strongly interested in quantum field theory. My research these days focuses mainly on the relationships between QFT and topology, geometry, and representation theory. Mathematically, this means that I'm interested in things like the moduli stack of curves, elliptic cohomology, Gromov-Witten theory, and the geometric Langlands program. Somewhat more philosophically, I'm interested in developing technology which allows us mathematicians to give proofs that conform with the intuition derived from path integrals, rather than simply using this intuition to make conjectures which we then prove using more traditional methods.

My main project right now (which is joint with E. Frenkel & C. Teleman) is an example of this sort of thinking. We have constructed a two-dimensional topological QFT, a U(1)-gauge theory analogue of Gromov-Witten theory, by computing the (K-theoretic) intersection numbers of a moduli stack of algebraic curves and principal GL(1)-bundles. What's interesting about this work is that we aren't studying a compact finite-type moduli stack; instead we work on (a completion of) the infinite type Artin stack of all GL(1)-bundles on stable curves. This stack is huge, so it's far from obvious that integrating over it will produce well-defined invariants; we prove that it does, using an algebro-geometric form of Witten's "non-abelian localization" principle.

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