Owen Gwilliam
I am an assistant professor in mathematics at the University of Massachusetts, Amherst. In summer 2018 I finished 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.
Research
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. My books and papers are described below.
 Inspired by the work of BeilinsonDrinfeld and FrancisGaitsgoryLurie on factorization algebras, Kevin Costello and I have developed a version of factorization algebras appropriate to perturbative QFT, building upon Costello's earlier work developing a renormalization machine. We have proved a kind of deformation quantization theorem for field theory, as well as a factorization refinement of the Noether theorem. We develop these ideas in a twovolume book ``Factorization Algebras in Quantum Field Theory'' to be published by Cambridge University Press, with the first volume currently available. 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 ChernSimons theory in dimension 3.) The second volume develops interacting classical and quantum field theory using the BatalinVilkovisky formalism and proves the deformation quantization and Noether theorems. It was submitted for publication in 2020.
 The Higgs mechanism and related ideas play a crucial role in theoretical physics, so it's gratifying that the key mechanism admits a nice articulation in derived geometry. Chris Elliott and I wrote an exposition about how the BV formalism gives a clean approach to this topic in Spontaneous symmetry breaking: a view from derived geometry.
 Eugene Rabinovich is developing analogs for manifolds with boundary of the factorization/BV results in my books. Along with Brian Williams, we showed in Factorization algebras and abelian CS/WZWtype correspondences how factorization algebras allow for a nice formulation of bulkboundary correspondences of free theories (e.g., abelian gauge theories), including the Poisson σmodel into a Poisson vector space, the wellknown ChernSimons/WessZuminoWitten correspondence, and a higher dimensional generalization of CS/WZW using higher abelian CS theory for connections on gerbes and using a (derived) intermediate Jacobian to formulate a higher abelian WZW model.
 Following a suggestion of Costello, Brian Williams and I studied a beautiful gauge fixing for ChernSimonstype field theories, which makes sense whenever the 3manifold admits a nice foliation by Riemann surfaces. We show that perturbative ChernSimons theory and its cousins (e.g., RozanskyWitten theories) admit a oneloop exact quantization. Going further, we also examine the case of a chiral boundary condition and see that the levelshifting phenomenon appears cleanly with this gauge. With Eugene Rabinovich, we have extended these analytic results to arbitrary theories that are partially topologicalholomorphic, and we are currently writing that up.
 In Higher KacMoody algebras and symmetries of holomorphic field theories Brian Williams and I examine how currents (in the physicists' sense) are encoded in the language of factorization algebras. Our central examples are generalizations of the chiral current algebras familiar in 2d chiral CFT, whose mathematical avatars are KacMoody vertex algebras; our construction works on arbitrary complex manifolds, but recovers the KacMoody case when restricted to Riemann surfaces. We develop as well a higher dimensional version of free field realization.
 In Duals and adjoints in higher Morita categories, Claudia Scheimbauer and I explored higher analogs of the Morita (bi)category that are of interest from the perspective of functorial field theories as well as factorization algebras. In a companion paper we use this machinery to explore a version of relative TFTs, focusing on lowdimensional examples where the consequences for algebra and category theory are easy to appreciate.
 In Chiral differential operators via quantization of the holomorphic σmodel, written with Vassily Gorbounov and Brian Williams, we construct the curved βγ system using a combination of the BV formalism and GelfandKazhdan formal geometry, modified to work with factorization algebras. We then show that the associated vertex algebra is the chiral differential operators. Our methods use BV techniques to realize mathematically the physical arguments given by Witten and Nekrasov recovering CDOs from the βγ system.
 Expanding on from the CDO project, Brian Williams and I wrote an exposition about the BV quantization of the holomorphic string with linear target space. It has a version of the dimensional anomaly and recovers the chiral sector of the standard bosonic string. We still need to write up the case of a curved target.
 With Kasia Rejzner, I have begun to explore the relationship between perturbative AQFT, particularly in conjunction with the BV formalism, and the BV/factorization package that I've developed with Costello. To get our bearings, we examined the case of the free field theories in some detail. We are writing up a construction that yields a factorization algebra when EpsteinGlaser renormalization allows for a BV quantization of a perturbative Lorentzian theory.
 With Rune Haugseng, I carefully studied linear BV quantization and formulated it as a functor of infinitycategories and then as a map of derived stacks in the imaginativelynamed Linear BV quantization as a functor of infinitycategories.
 With Dmitri Pavlov, I reexamined filtered derived categories from the perspective of model categories and infinitycategories in Enhancing the filtered derived category.
 With Ryan Grady, I have pursued several projects touching on derived geometry and QFT. In Onedimensional ChernSimons Theory and the Ahat genus, we constructed a onedimensional TFT whose partition function recovers the Ahat genus of a manifold. In Linfinity spaces and derived loop spaces, we clarified and further developed Costello's approach to derived geometry via Linfinity spaces. Finally, in Lie algebroids as Linfinity spaces, we showed that Lie algebroids (and associated constructions like representations up to homotopy) fit naturally into this version of derived geometry.
 My thesis showed how the BV/factorization package works in several simple contexts. The thesis has a few distinct pieces. First, I provide an expository introduction to the BatalinVilkovisky formalism. Second, I prove that the BV formalism provides a determinantal functor on perfect complexes. Third, I show by example how our factorization algebra procedure recovers vertex algebras, notably free bosons, free fermions, and affine KacMoody vertex algebras. Finally, I prove an index theorem based on our techniques. Some of my recent work is about extending and enhancing these results.
There are several other projects at various stages of development. If you'd like to know more, feel free to contact me.
Notes and videos
 With Theo JohnsonFreyd, I wrote an expository introduction to the BV formalism, emphasizing how one would rediscover Feynman diagrams for purely homological reasons.
 Based on a lecture by Costello, I wrote up an explanation for how the βγ system arises from the usual twodimensional sigma model. To be more precise, after rewriting the sigma model in the firstorder formalism, one can look at scaling the metric on the target to infinity and then take the chiral sector.
 For an MPIM seminar on topological insulators, I made slides introducing some basic, relevant ideas for a mathematical audience. It might be helpful to others wanting to see a quick overview, but beware that in this seminar I was the blind leading the blind.

I made slides for some talks in my last year at MPIM:
 I gave a lecture series in November 2017 at the Hausdorff Institute on my joint work with John Francis and Kevin Costello about how perturbative ChernSimons theory recovers quantum groups. If you're interested in seeing video, please contact me. For a onehour lecture about this research (that's a bit smoother, honestly), watch the video of a talk at MSRI from March 2020.
 At MSRI in March 2020 I also gave a survey of factorization algebras in TFT at a great workshop.
Activities
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, IBSCGP, and Oberwolfach).
The summer school that Chris Elliott, Matteo Lotito, Brian Williams, and I were organizing for August 2020 was postponed due to the pandemic. We hope to run it in summer 2021, as we had an awesome lineup of speakers. In the meanwhile, we're trying our hand at making masterclasses.