I am an associate professor in mathematics at the University of Massachusetts, Amherst. Before coming to UMass in the fall of 2018, I spent four years as a postdoctoral fellow at the Max Planck Institute for Mathematics in Bonn. Before that, I was an NSF postdoc at UC Berkeley following graduate school at Northwestern.
An NSF CAREER grant currently supports an annual summer school on physical mathematics that I co-organize with Chris Elliott, Matteo Lotito, and Samantha Kirk.
My research revolves around quantum field theory, focusing both on applications of homotopical ideas to QFT itself and applications of QFT to geometry and representation theory. Most of my papers can be found on the arXiv.
With Kevin Costello, I wrote a two-volume book about how the observables of a perturbative QFT is encoded in a factorization algebra on its spacetime. An overview of these ideas and how they fit into recent research can be found in our article for the Encyclopedia of Mathematical Physics.
Our results build upon Costello's earlier work developing a renormalization machine. In our books we prove a kind of deformation quantization theorem for field theory, as well as a factorization refinement of the Noether theorem. The first volume introduces factorization algebras and develops examples, primarily from free field theories. (This is not as boring as it might sound: we recover vertex algebras in complex dimension 1 and a quantum group for abelian Chern-Simons theory in dimension 3.) The second volume develops interacting classical and quantum field theory using the Batalin-Vilkovisky formalism and proves the deformation quantization and Noether theorems.
I'm currently thinking about issues in QFT that might count as ``nonperturbative'' and about building more machinery to manipulate factorization algebras. If you'd like to know more, feel free to contact me.
In my graduate courses, I often have students write expository papers on topics that combine the course subject and their own interests. You might enjoy reading the Journal of Math 79x where these articles are collected.
For the last several years I have supervised groups of students through the UMass REU. In summer 2023 the group of Nikos Orginos, Nathaniel Reid, and Sam Silver wrote a nice paper about twists (square-zero odd nilpotent elements) of the 2d superconformal algebras.
With Claudia Scheimbauer, I taught a course on derived deformation theory and Koszul duality at the University of Bonn in 2016. Here is our website.
I've also been lucky to be an organizer of several conferences (at MPIM, the Simons Center, Banff, IBS-CGP, and Oberwolfach).