I've moved to Berkeley. See here for my current page.
Welcome to my bare-bones home page. I am a fifth year graduate student in mathematics at Northwestern University, and my advisor is Kevin Costello. I spend most of my time trying to understand what a quantum field theory is and how to use the QFT toolkit I've learned from Kevin to approach problems in geometry and topology. If you'd like to contact me, my email address is ''gwilliam at math dot northwestern dot edu''.
I'm graduating this year and welcome any opportunities to discuss my work or give a talk. Here is my curriculum vitae. Next year I will begin an NSF postdoc at Berkeley.
- My main research project so far is joint with Kevin. He recently finished a lovely book on perturbative QFT where he develops a renormalization machine, but he does not discuss observables in his book. Inspired by the work of Beilinson-Drinfeld and Francis-Gaitsgory-Lurie, we have developed a version of factorization algebras appropriate to Kevin's version of QFT and proved a kind of deformation quantization theorem for field theory.
We are writing a book that discusses factorization algebras, proves the theorem, and includes examples. Our current draft is available in conventional book format as a pdf here
Factorization algebras in perturbative quantum field theory
or as a wiki here
Factorization algebras in perturbative quantum field theory.
- Ryan Grady and I are using this formalism -- of perturbative QFT and factorization algebras -- to develop another proof of the Atiyah-Singer index theorem and the mathematics around it, like K-theory. Many aspects resemble already existing approaches, but it's fun to see how everything emerges from Feynman diagram computations. So far, we have constructed a very lovely 1-dimensional field theory that we call 1-dimensional perturbative Chern-Simons, as its fields consist of the derived space of g-connections on a 1-manifold, where g is an arbitrary curved L-infinity algebra. When we encode a manifold X as a curved L-infinity algebra over its de Rham space X_dR, we recover the A-hat class of X from the global observables of the field theory of maps from the circle into X. You can download the paper below. We also hope to finish the second paper during the upcoming year, where we construct the factorization algebra for this QFT and thus give another proof of the Nest-Tsygan theorem.
One-dimensional Chern-Simons Theory and the A-hat genus (also available here arXiv:1110.3533)
- Lately, I've been speaking a lot with Theo Johnson-Freyd about the Batalin-Vilkovisky formalism. We wrote a short expository note explaining how the homological algebra of a BV algebra naturally encodes Wick's lemma and Feynman diagram expansions.
How to derive Feynman diagrams for finite-dimensional integrals
directly from the BV formalism
- I've just finished writing my thesis, which focuses on free theories and their factorization algebras. In it, I show that BV quantization provides a determinant-type functor when restricted to perfect complexes, recover several well-known vertex algebras in a concrete way from some simple factorization algebras on Riemann surfaces, and prove an index theorem for families of elliptic complexes using the local nature of factorization algebras. In the next year, I intend to extend these results and integrate them into various writing projects. If you would like a copy of the thesis, however, just email me.
For the past few years, I have co-organized the Talbot workshops with John Francis, Sheel Ganatra, and Hiro Tanaka. This year, Saul Glasman has joined us as an organizer.
This summer I helped organize an informal workshop on Chiral Differential Operators with Kevin Costello and Ryan Grady. The schedule and notes from the workshop are available on that page.
A plug for my friend Wes Swing
If you have a hankering for some new music, check out his new album at his website