Perimeter Institute for Theoretical Physics

**e-mail:** theojf@pitp.ca

My primary research is in an area I would call "homotopical physics": a mathematical physicist, I focus on quantum field theory, topological field theory, phases of condensed matter, "moonshine" phenoma, perturbative quantization, category theory, representation theory, and algebraic topology. For more details, see my publications and seminar presentations. For fewer details, see my Curriculum Vitae (PDF), which contains a strict subset of the information on this webpage.

I received my PhD from UC Berkeley in 2013, under the supervision of Nicolai Reshetikhin. I was then an NSF postdoc and Boas Assistant Professor at Northwestern University, and then at the Perimeter Institute. My postdoc mentors are Kevin Costello and Davide Gaiotto. Other collaborators include Alex Chirvasitu, Owen Gwilliam, Claudia Scheimbauer, and David Treumann.

- Research papers: publications, other preprints
- Other documents: textbook, liveTeXed notes, miscellaneous mathematics
- Conference and seminar presentations: 2018, 2017, 2016, 2015, 2014, 2013, 2012, 2015, 2015, 2010, 2009, 2008, 2007,
- Teaching: at Northwestern, at Berkeley

H

^{4}(Co_{0};**Z**)=**Z**/24. With David Treumann. 2018. To appear in*International Mathematics Research Notices*. (abstract, arXiv: 1707.07587.)How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism. With Owen Gwilliam.

*Topology and Quantum Theory in Interaction*, proceedings from the NSF-CBMS Regional Conference in the Mathematical Sciences that took place in Bozeman, Montana in 2017. (abstract, PDF, arXiv: 1202.1554.)The Moonshine Anomaly. 2018.

*Accepted pending revisions to Communications in Mathematical Physics.*(abstract, arXiv:1707.08388.)Spin, statistics, orientations, unitarity.

*Algebraic & Geometric Topology*17 (2017) 917–956. (abstract, arXiv: 1507.06297, DOI: 10.2140/agt.2017.17.917.)(Op)lax natural transformations, twisted field theories, and the "even higher" Morita categories. With Claudia Scheimbauer.

*Advances in Mathematics*, 307 (2017) 147–223. (abstract, arXiv: 1502.06526, DOI: 10.1016/j.aim.2016.11.014.)The quaternions and Bott periodicity are quantum Hamiltonian reductions.

*Symmetry, Integrability and Geometry: Methods and Applications*, 12 (2016), 116, 6 pages. (abstract, arXiv: 1603.06603, DOI: 10.3842/SIGMA.2016.116.)Tree- versus graph-level quasilocal Poincaré duality on

*S*^{1}.*Journal of homotopy and related structures*, June 2016, Volume 11, Issue 2, pp 333–374. (abstract, arXiv: 1412.4664, DOI: 10.1007/s40062-015-0110-2.)Homological perturbation theory for nonperturbative integrals.

*Letters in Mathematical Physics*, November 2015, Volume 105, Issue 11, pp 1605–1632. (abstract, arXiv: 1206.5319, DOI: 10.1007/s11005-015-0791-9.)Reflexivity and dualizability in categorified linear algebra. With Martin Brandenburg and Alexandru Chirvasitu.

*Theory and Applications of Categories*, Vol. 30, No. 23, 2015, pp. 808–835. (abstract, arXiv: 1409.5934, published version (open access).)Poisson AKSZ theories and their quantizations. In

*Proceedings of the conference String-Math 2013*, volume 88 of*Proceedings of Symposia in Pure Mathematics*, pages 291–306, Providence, RI, 2014. Amer. Math. Soc. (abstract, PDF (published version), arXiv: 1307.5812, DOI: 10.1090/pspum/088.)The fundamental pro-groupoid of an affine 2-scheme. With Alex Chirvasitu.

*Applied Categorical Structures*, Vol 21, Issue 5 (2013), pp. 469–522. (abstract, arXiv: 1105.3104, DOI: 10.1007/s10485-011-9275-y).The formal path integral and quantum mechanics.

*Journal of Mathematical Physics*, 51, 122103 (2010). (abstract, published PDF, DOI:10.1063/1.3503472, arXiv: 1004.4305, equation and theorem numbering differs between preprint and published versions).Feynman-diagrammatic description of the asymptotics of the time evolution operator in quantum mechanics.

*Letters in Mathematical Physics*, November 2010, Volume 94, Issue 2, pp 123–149. (abstract, arXiv: 1003.1156, available Open Access from Springer Link at DOI: 10.1007/s11005-010-0424-2).

Exact triangles, Koszul duality, and coisotopic boundary conditions. 2016. (abstract, arXiv: 1608.08598.)

Symmetry protected topological phases and generalized cohomology. With Davide Gaiotto. 2017. (abstract, arXiv: 1712.07950.)

Heisenberg-picture quantum field theory. 2015. (abstract, arXiv: 1508.05908.)

Chains(

**R**) does not admit a geometrically meaningful properadic homotopy Frobenius algebra structure. 2013. (abstract, arXiv: 1308.3423.)Peturbative techniques in path integration. Ph.D. Thesis. 2013. (abstract, PDF.)

On the coordinate (in)dependence of the formal path integral. 2010. (abstract, PDF, arXiv: 1003.5730).

The following draft textbook has been accepted for publication by World Scientific:

- R. Borcherds, M. Haiman, T. Johnson-Freyd, N. Reshetikhin, and V. Serganova.
*Berkeley Lectures on Lie Groups and Quantum Groups*. PDF.

I occasionally "Live-TeX" notes from classes and lectures. As with any notes, mine are replete with omissions and errors, undoubtedly; typing does allow me to catch questions from the audience and jokes from the professors, so these are included as well. Needless to say, anything good about the notes, and in particular presentation of the mathematical material, is due to the professor of the class. Anything bad about them, and in particular every inaccuracy, is mine. Use them with care. Also, please e-mail me with corrections: typos are trivial to fix, and mathematical errors should not be allowed to propagate. I was inspired to start typing lecture notes after watching Anton Geraschenko do it, and appreciate his advice.

- Philippe Di Francesco. Chern-Simons Research Lectures: Discrete Integrable Systems and Cluster Algebras, 10-12 April 2012. Prepared slides: Day 1 (8 MB), Day 2 (16 MB), Day 3 (13 MB). Unedited notes from the first two days.
- Denis Bernard. Chern-Simons Research Lectures: Stochastic Schramm-Loewner Evolution (SLE) from Statistical Conformal Field Theory (CFT): An Introduction for (and by) Amateurs, 19-23 March 2012. Prepared notes: arXiv:math-ph/0602049v1. Mildly edited notes.
- David Kazhdan. Chern-Simons Research Lectures: The classical master equation in the finite-dimensional case, 9-13 Jan 2012. Mildly edited notes.
- Jorg Teschner. Chern-Simons Research Lectures: Quantization of Hitchin's Moduli Spaces and Liouville Theory, 17-21 Oct 2011. Unedited notes.
- Fedor Smirnov. Chern-Simons Research Lectures: Correlation functions in integrable Quantum Field Theory, 28 Sept - 4 Oct 2011. Unedited notes. Slides.
- MSRI, Symplectic Geometry, Noncommutative Geometry and Physics, May 10, 2010 to May 14, 2010. (All slides and transcripts (tar.gz).)
- V. Serganova. Math 261B: Quantum Groups. Spring 2010. (Unedited: PDF, TAR.GZ.)
- N. Reshetikhin. Math 261B: Quantum Groups. Spring 2009. (Unedited: PDF (size exceeds 800 KB), TAR.GZ (size exceeds 9 MB).)
- M. Haiman. Math 261A: Lie Groups. Fall 2008. (Edited: PDF, TeX. Unedited: PDF (size exceeds 1 MB), TAR.GZ (size exceeds 6 MB).)
- M. Rieffel. Math 208: C* Algebras. Spring 2008. (Unedited: PDF (about 600 KB), TAR.GZ (about 4 MB).)

Please note that the TAR.GZ files include TeX sources and plenty of other detritus: auxiliary files, partly completed problem sets, etc. You are welcome to download them, but I make no promises that the files will load on your computer: that will depend on whether your TeX installation exactly matches mine.

Here are two ideosyncratic notes surveying various mathematics:

- Exact triangles as modules over an
*A*_{∞}-category. 2015. (PDF.) - Poisson Lie linear algebra in the graphical language. 2009. (PDF. TeX source.)

As many graduate students discover when taking their language exams, you don't need any training in French in order to read Mathematical French. (It helps that Mathematical English is more French-inflected than is Colloquial English.) As evidence, I have translated into English Deligne's article *Catégories Tensorielles* (French original).

One of my more satisfying activities is that I am a reviewer for MathReviews and zbMath. Of my reviews, two appeared in the September 2012 print addition: MR2742432 (2012i:55005): Stolz and Teichner, Supersymmetric field theories and generalized cohomology, 2011 and MR2752518 (2012i:81001): Baez and Lauda, A prehistory of n-categorical physics, 2011.

T-duality for finite groups. June 4. Representation Theory, Mathematical Physics and Integrable Systems, CIRM, Luminy, France. (abstract. notes.)

The fourth cohomology of some sporadic groups. April 17. Geometry, Symmetry and Physics, Yale. (abstract.)

Moonshine anomalies. April 9. Algebra seminar, University at Buffalo. (abstract.)

Infinitely-categorified commutative algebra. March 17--19. Recent developments in noncommutative algebra and related areas, University of Washington. (abstract.)

Moonshine anomalies. February 9. QMAP seminar, UC Davis. (abstract.)

Higher algebraic closures and phases of matter. January 22. Northeastern University. (abstract. handout.)

Moonshine anomalies. January 19. UT Austin. (abstract.)

Higher categories, generalized cohomology, and condensed matter. November 15. Representation Theory and Mathematical Physics, UC Berkeley. (abstract. notes.)

576 Fermions. October 24. Algebra Seminar, Emory. (abstract.)

Bott periodicity from Hamiltonian reduction. October 5. NT&AG, Boston College. (abstract.)

Exceptional structures, fermions, anomalies, and Hamiltonian reduction. October 3. Research Seminar in Mathematics, Northeastern. (abstract. notes.)

The Moonshine Anomaly. July 25. Higher Structures Lisbon, Instituto Superior Técnico, Lisbon, 24–27 July 2017. (abstract.)

The Moonshine Anomaly. July 19. Maximals Seminar, University of Edinburgh. (abstract.)

Orbifolds of conformal field theories and cohomology of sporadic groups. April 14. RTGC Seminar, UC Berkeley. (abstract. notes.)

Advanced integration by parts: the BV formalism. Feb 9. Geometric Structures Laboratory, Fields Institute. (abstract. notes.)

Fermionic hamiltonian reduction and periodicity. Feb 1. Geometry and Physics Seminar, Boston University. (abstract.)

Ideals in derived algebra and boundary conditions in AKSZ-type field theories. Jan 27. RTGC Seminar, UC Berkeley. (abstract. notes.)

576 fermions, the Conway group, and tmf. Sept 27. Institute for Theoretical Physics, Stanford.

Moonshine, topological modular forms, and 576 fermions. Sept 22, Mathematical Physics Seminar, Perimeter Institute. (abstract. video.)

Bott periodicity via quantum Hamiltonian reduction. Sept 2, Representation Theory, Geometry, and Combinatorics Seminar, UC Berkeley. (abstract)

Bott periodicity via quantum Hamiltonian reduction. May 12, Factorization Algebras and Functorial Field Theories meeting, Oberwolfach. (abstract)

"Spin-statistics" is a categorification of "Hermitian". February 22, RTGC Seminar, Berkeley. (abstract, handout)

Where does the Spin-Statistics Theorem come from?. November 23, Geometry, Topology and Dynamics Seminar, UIC. (abstract, handout)

Spin--Statistics and Categorified Galois Groups. October 23, Conference on Condensed Matter Physics and Topological Field Theory, Perimeter Institute. (abstract, handout)

A higher category theorist's take on the spin--statistics theorem. September 28, Topology Seminar, UIUC. (abstract, handout)

The Stokes groupoids of Gualtieri, Li, and Pym. May 4&6, Math 448: Topics in Geometry and Topology, Northwestern. (abstract, notes)

Some non-dualizable categories. April 17, Representation Theory, Geometry, and Combinatorics, UC Berkeley. (abstract, handout)

Local Poincare duality & deformation quantization. April 2, Center for Geometry and Physics, Institute for Basic Science, Pohang, South Korea. (abstract, handout, video (follow talk link))

Some comments on Heisenberg-picture qft. March 18, Max Planck Institute for Mathematics, Bonn, Germany. (abstract, handout)

Twisted field theories and higher-categorical (op)lax transfors. March 3, Topology Seminar, University of Notre Dame, South Bend, IN. (abstract, handout)

Functorial axioms for Heisenberg-picture quantum field theory. January 12, CRG Geometry and Physics Seminar, UBC, Vancouver, BC. (abstract, handout)

Heisenberg-picture quantum field theory. November 17, Quantum Mondays Seminar, Center for Geometry and Physics, Institute for Basic Science, Pohang, South Korea. (abstract, video (follow talk link))

The CS-WZW correspondence. October 30, CFT Seminar, Northwestern University, Evanston, IL. (abstract, notes)

Poisson AKSZ theories. October 3, Homological Methods in Quantum Field Theory, Simons Center for Geometry and Physics, Stony Brook, NY. (abstract, outline, video, live-TeX'ed notes by Gabriel C. Drummond-Cole)

Lie bialgebra quantization in 2- and 3-dimensional field theory. May 28, Associators, Formality and Invariants Seminar, Northwestern, Evanston, IL. (abstract, notes)

The Jones polynomial and the Temperley--Lieb TQFT. May 9, Graduate Student Seminar, Northwestern, Evanston, IL. (abstract, notes)

Heisenberg-picture TQFTs. May 2, Representation Theory, Geometry, and Combinatorics, UC Berkeley. (abstract, notes)

Poisson AKSZ theory and homotopy actions of properads. February 21, Modern trends in topological quantum field theory, Erwin Schrödinger International Institute for Mathematical Physics. (abstract, handout)

Up-to-homotopy Frobenius structures on manifolds, and how they relate to perturbative QFT. January 22, Topology Seminar, UC Berkeley. (abstract, handout)

Poisson AKSZ theories and quantization. October 24, Geometry Seminar, UT Austin. (abstract, handout)

Poisson AKSZ theories and quantization. October 12, Higher Structures in Algebra, Geometry and Physics, Fall Eastern Sectional Meeting of the AMS, Temple University, Philadelphia. (abstract, handout)

Poisson AKSZ theory, properads, and quantization. October 10, Geometry and Physics, Northwestern, Evanston. (abstract, handout)

A properad action on homology that fails to lift to the chain level. October 10, Geometry and Physics Pre-talk, Northwestern, Evanston. (abstract, handout)

A properadic approach to the deformation quantization of topological field theories. September 25, Algebra and Combinatorics, Loyola University, Chicago. (abstract, notes)

A salad of BV integrals and AKSZ field theories, served over a bed of properads; it comes spiced with chain-level Poincare duality and just a pinch of Poisson geometry. September 10, Research Seminar in Mathematics, Northeastern University, Boston. (abstract, notes)

Star quantization via lattice topological field theory. June 18, String-Math, Simons Center for Geometry and Physics, Stony Brook. (abstract, slides)

Lattice Poisson AKSZ Theory. February 4, Algebraic Geometry Seminar, University of British Columbia. (abstract, handout)

Feynman diagrams for quantum mechanics. October 10–12, Topics in Applied Mathematics, UC Berkeley. (abstract, notes)

Nonperturbative integrals, imaginary critical points, and homological perturbation theory. August 28, New Perspectives in Topological Field Theories, Center for Mathematical Physics, Hamburg. (abstract, notes)

Nonperturbative integrals, imaginary critical points, and homological perturbation theory. August 24, QGM Lunch Seminar, Aarhus. (abstract, notes)

Wick-type theorems beyond the Gaussian. March 2, Representation Theory (and related topics) seminar, Northeastern. (abstract, handout, notes)

Twisted N=1 and N=2 supersymmetry on R^4. February 10, GRASP seminar, UC Berkeley. (abstract, notes)

Notes on Floer / Gromov–Witten TQFT, based on conversations with Zack Sylvan. November 15, Witten in the 80s, UC Berkeley. (notes)

Gauge-fixed integrals for Lie algebroids. November 1, Talks in Mathematical Physics, Universität Zürich & Eidgenössische Technische Hochschule Zürich. (abstract, handout)

Introduction to BV Integrals. November 1, Selected topics in classical and quantum geometry, Universität Zürich. (abstract, handout)

Asymptotics of oscillating integrals via homological perturbation theory. October 19, GRASP seminar, UC Berkeley. (abstract, handwritten notes, too-long typed notes)

BRST Gauge Fixing: I. Introduction to Q-manifolds and BRST. II. Chern-Gauss-Bonnet, Morse Theory, and topological sigma models. October 11-20, Witten in the 80s, UC Berkeley. (abstract, notes I, notes II)

(Topological) duality of Hopf algebras. June 13, Cluster Algebras and Lusztig's Semicanonical Basis, University of Oregon. (abstract, notes)

Homological perturbation and factorization algebras. May 26, Geometry/Physics Seminar, Northwestern University. (abstract, notes)

On Atoms, Mountains, and Rain. April 20, NUMS Seminar, Northwestern University. (abstract)

Feynman Diagrams for Schrodinger's Equation. Feb 15, GADGET Seminar, UT Austin. (abstract, handout)

E

_{2}operad, Gerstenhaber and BV, and Formality. Feb 10, Student String Topology Seminar, UC Berkeley. (abstract, notes, handout)

Crash course in Tannaka-Krein theory. Dec 3, Student Subfactors Seminar, UC Berkeley. (abstract, notes)

Formal calculus, with applications to quantum mechanics. Sept 10, GRASP seminar, UC Berkeley. (abstract, notes).

Introduction to Vassiliev invariants. April 2, GRASP seminar, UC Berkeley. (abstract).

How to quantize infinitesimally-braided symmetric monoidal categories. March 19, Subfactors Seminar, UC Berkeley. (abstract, notes).

The Formal Path Integral in Quantum Mechanics. Feb 26, Subfactors Seminar, UC Berkeley. (abstract, slides).

What the Hell is a Feynman Diagram? Sept 29, Ph.d. seminar, Institut for Matematiske Fag, Aarhus Universitet. (abstract, notes).

On Atoms, Mountains, and Rain. March 31, Many Cheerful Facts, UC Berkeley. (abstract, notes).

Combinatorial Calculus: From Taylor Series to Feynman Diagrams. July 7-12, Canada/USA Mathcamp. (abstract, notes, exercises).

Enriching Yoneda. December 11, QFT Mini Conference, UC Berkeley. (abstract, notes).

Divergent Series. October 18, Many Cheerful Facts, UC Berkeley. (abstract, notes).

- Winter 2016: Math 448: Topology and Geometry seminar — Reshetikhin–Turaev Invariants
- Fall 2015: First-year Seminar: Theories of mind and mathematics
- Spring 2014: Freshman Seminar: Theories of mind and mathematics
- Fall 2013: Math 300: Foundations of Higher Mathematics

- Summer 2009: Math 1B: Calculus
- Spring 2009: Math 1A: Calculus
- Fall 2008: Math 32: Precalculus
- Summer 2008: Math 1B: Calculus
- Spring 2008: Math 53: Multivariable Calculus
- Fall 2007: Math 1B: Calculus