Theo Johnson-Freyd

Talks that I have given (or will give)

For conferences that I organized rather than spoke at, click here.

2024

Musings on topological modular forms and moonshine. December 16, Vertex algebras and related topics, Academia Sinica, Taipei. (abstract.)

Abstract: The character of a (rational, unitary) VOA is expected to refine to a vector-valued Topological Modular Form. This expected refinement implies, among other things, nontrivial divisibility statements on the character as well as torsion refinements of the character. Many mathematical details remain open, including about the very definition of ``vector-valued Topological Modular Form.'' Particularly interesting is the question of how exactly to control cuspidal behaviour; guidance comes from attempting to conjecture topological refinements of moonshine-type statements. I will describe some ongoing conversations with Lennart Meier and Mayuko Yamashita on this subject. (hide abstract)
The unitary cobordism hypothesis. November 12, SCGCS Annual Meeting Satellite Workshop, New York University. (abstract.)

Abstract: A dagger category is a category equipped with extra "unitarity" data: among its isomorphisms, some are marked as "unitary"; to each 1-morphism, there is an "adjoint" 1-morphism, and this assignment sends unitary (but not general!) isomorphisms to their inverses. If a monoidal higher category has duals, it is interesting to ask for a choice of duality functor for which the units and counits are adjoints; whereas duality functors are unique up to contractible choice, so-defined "unitary duality" functors are not. I will explain a higher-categorical generalization of these ideas, and explain how, for any stable tangential structure H, a construction of Freed and Hopkins makes the extended bordism category Bord_n^H(n) into a dagger symmetric monoidal n-category with unitary duality. This category satisfies a unitary cobordism hypothesis: whereas nonunitary functors Bord_n^H(n) → C are classified by H(n)-fixed points in C, unitary functors are classified by fixed points for the stablized group H(∞). This talk is based on joint work in preparation with Cameron Krulewski, Lukas Mueller, and Luuk Stehouwer. (hide abstract)
The unitary cobordism hypothesis. November 5, Fields Geometry and Physics Seminar, University of Toronto. (abstract. video.)

Abstract: A strict dagger category is a strict 1-category C with an antiinvolutive functor from C to C^op that is the identity on objects. I will explain a natural coherent version of dagger higher category. In the case of higher categories with lots of duals and adjoints, it is natural to ask for "unitary adjoints", and I will explain how such notion is also natural, and also selects a good definition of higher pivotality. A special case is the extended cobordism category: building on a construction of Freed and Hopkins, the cobordism category (extended or unextended) is naturally dagger (with unitary duals, if extended), but only when the tangential structure is stable. The stably-framed cobordism (∞,n)-category satisfies the unitary cobordism hypothesis: it is freely generated by the point among symmetric monoidal dagger (∞,n)-categories with unitary duals. In particular, whereas the non-dagger cobordism category with tangential structure H is a noninvertible refinement of MTH(n), the dagger structure makes it act more like a noninvertible refinement of MTH. This talk is based on arXiv:2403.01651 with many authors and on joint work in progress with Cameron Krulewski, Lukas Mueller, and Luuk Stehouwer. (hide abstract)
Exact sequences of Hopf algebras. October 30, Geometry and Physics Seminar, Boston University. (abstract.)

Abstract: The notion of "Hopf algebra" makes sense in any braided monoidal category. I will explain an equational version of "exact sequence of finite-dimensional Hopf algebras" that I call "BC-exactness" because it involves a funny variation of Beck-Chevalley condition. Being equational, BC-exactness is preserved by all functors; BC-exactness recovers other versions of exactness when the Hopf algebras in the sequence are separable and coseparable. There is a functor that inputs a retract in a 3-category with duals and outputs a finite dimensional Hopf algebra. A different Beck-Chevalley-type condition supplies a notion of "BC-exact sequence" of retracts, and the retract-to-Hopf-algebra functor takes BC-exact sequences of retracts to BC-exact sequences of Hopf algebras. The framed bordism category with boundary contains a BC-exact sequence of retracts and hence a BC-exact sequence of Hopf algebras that I call the "quantum Puppe sequence" because when evaluated via a sigma model with target X and boundary condition Y → X with fibre F, it recovers the Puppe sequence of homotopy groups for the fibration F → Y → X. This talk is based on joint work in preparation with David Reutter. (hide abstract)
The homotopy groups of a TQFT. October 22, Representation theory and tensor categories seminar, University of California, Berkeley. (abstract.)

Abstract: To any open-closed TQFT (possibly framed, possibly not fully extended, possibly not compact) I associate a sequence of Hopf algebras that I interpret as "homotopy groups" of the "quantum target space" of the TQFT — the interpretation is justified because when the TQFT does come from a sigma model into some target space, then these Hopf algebras encode the homotopy groups of the target space. To any relative open-closed TQFT, I associate a long exact sequence of Hopf algebras — I interpret this as the "Puppe sequence" for a "quantum fibre bundle". Every tensor higher category is expected to determine an open-closed TQFT; my Hopf algebras generalize the canonical Hopf algebra object in any braided fusion category. Exactness of the Puppe sequence in this case gives a version of S-matrix theory for tensor higher categories, relating (non)degeneracy of Hopf links with Morita (non)invertibility. This talk is based on joint work in preparation with David Reutter. (hide abstract)
Wormholes and an exact sequence. September 10, Applications of Generalized Symmetries and Topological Defects to Quantum Matter, Simons Center for Geometry and Physics, Stony Brook. (video.)
Quantum homotopy groups. June 17, Categorical Symmetries in Quantum Field Theory, International Centre for Mathematical Sciences, Edinburgh. (abstract. video.)

Abstract: An open-closed tqft is a tqft with a choice of boundary condition. Example: the sigma model for a sufficiently finite space, with its Neumann boundary. Slogan: every open-closed tqft is (sigma model, Neumann boundary) for some “quantum space”. In this talk, I will construct homotopy groups for every such “quantum space” (and recover usual homotopy groups). More precisely, these “groups” are Hopf in some category. Given a “quantum fibre bundle” (a relative open-closed tqft), I will construct a Puppe long exact sequence. Retracts in 3-categories and a higher Beck-Chevalley condition will cameo appearances. This project is joint work in progress with David Reutter. (hide abstract)
Quantum homotopy groups. June 12, Thematic Program in Field Theory and Topology, University of Notre Dame. (abstract. video.)

Abstract: An open-closed tqft is a tqft with a choice of boundary condition. Example: the sigma model for a sufficiently finite space, with its Neumann boundary. Slogan: every open-closed tqft is (sigma model, Neumann boundary) for some “quantum space”. In this talk, I will construct homotopy groups for every such “quantum space” (and recover usual homotopy groups). More precisely, these “groups” are Hopf in some category. Given a “quantum fibre bundle” (a relative open-closed tqft), I will construct a Puppe long exact sequence. Retracts in 3-categories and a higher Beck-Chevalley condition will cameo appearances. This project is joint work in progress with David Reutter. (hide abstract)
The universal target category. May 2, Workshop on Global Categorical Symmetries, Center of Mathematical Sciences and Applications, Harvard University. (abstract. slides.)

Abstract: Hilbert's Nullstellensatz says that the complex numbers C satisfy a universal property among all R-algebras: every not-too-large nonzero commutative R-algebra maps to C. Deligne proved a similar statement in categorical dimension 1: every not-too-large symmetric monoidal category over R maps to the category sVec_C of complex super vector spaces. In other words, sVec_C (and not Vec_C!) is "algebraically closed". These statements help explain why quantum field theory requires imaginary numbers and fermions. I will describe the universal symmetric monoidal higher category that extends the sequence C, sVec_C, .... This is joint work in progress with David Reutter, and builds on closely-related work by GCS collaborators Freed, Scheimbauer, and Teleman and Schlank et al. (hide abstract)
The universal target category. April 11, Topology Seminar, Johns Hopkins University. (abstract. notes.)

Abstract: Hilbert's Nullstellensatz says that the complex numbers C satisfy a universal property among all R-algebras: every not-too-large nonzero commutative R-algebra maps to C. Deligne proved a similar statement in categorical dimension 1: every not-too-large symmetric monoidal category over R maps to the category sVec of super vector spaces. These statements help explain why quantum field theory involves imaginary numbers and fermions. I will describe the universal symmetric monoidal higher category that extends the sequence C, sVec, .... In particular, I will tell you what you have to add at each categorical dimension --- in other words, I will tell you about the ``∞-categorical absolute Galois group'' of R. This computation involves a version of surgery theory for topological quantum field theories instead of manifolds, and shows that this absolute Galois group is enticingly similar to the infinite piecewise-linear group PL. This talk is based on joint work in progress with David Reutter. (hide abstract)
Quantum homotopy groups. March 21, Higher Categorical Tools for Quantum Phases of Matter, Perimeter Institute for Theoretical Physics. (abstract. slides. video.)

Abstract: An open-closed tqft is a tqft with a choice of boundary condition. Example: the sigma model for a sufficiently finite space, with its Neumann boundary. Slogan: every open-closed tqft is (sigma model, Neumann boundary) for some “quantum space”. In this talk, I will construct homotopy groups for every such “quantum space” (and recover usual homotopy groups). More precisely, these “groups” are Hopf in some category. Given a “quantum fibre bundle” (a relative open-closed tqft), I will construct a Puppe long exact sequence. Retracts in 3-categories and a higher Beck-Chevalley condition will make appearances. This project is joint work in progress with David Reutter. (hide abstract)
Recent progress on the classification of fusion higher categories. SCGCS internal meeting, Zoom. (slides.)

2023

Dagger Categories. December 14, SCGCS Collaboration Assembly, Zoom. (video.)
Higher Dagger Categories. November 8, Geometry, Topology & Physics Seminar, New York University Abu Dhabi. (abstract. slides.)

Abstract: Hilbert spaces form more than a category: their morphisms maps can be composed, but also every morphism $f : X \to Y$ has a distinguished "adjoint" $f^\dagger : Y \to X$, making it into a "dagger category". This extra data is important for axiomatizing functional analysis, quantum mechanics, quantum information theory.... However, the assignment $f \mapsto f^\dagger$ is unsatisfying from a higher category theorist's perspective because it is "evil", i.e. it violates the principle of equivalence: a category equivalent to a dagger category may not admit a dagger structure. This in particular interferes with generalizing the notion of dagger category to the (non-strict) higher categories necessary for axiomatizing fully-local quantum field theory. In this talk I will propose a manifestly non-evil definition of "dagger $(\infty,n)$-category". The same machinery also produce a non-evil definitions of "pivotal $(\infty,n)$-category" and helps to clarify the relationship between reflection positivity and spin-statistics. This is based on joint work with B. Bartlett, G. Ferrar, B. Hungar, C. Krulewski, L. Müller, N. Nivedita, D. Penneys, D. Reutter, C. Scheimbauer, L. Stehouwer, and C. Vuppulury. (hide abstract)
Deeper Kummer theory. September 21, Mathematical Physics Seminar, Perimeter Institute. (abstract. video.)

Abstract: A tower is an infinite sequence of deloopings of symmetric monoidal ever-higher categories. Towers are places where extended functorial field theories take values. Towers are a "deeper" version of commutative rings (as opposed to "higher rings" aka E_∞-spectra). Notably, towers have their own opinions about Galois theory, and think that usual Galois groups are merely shallow approximations of deeper homotopical objects. In this talk, I will describe some steps in the construction and calculation of the deeper Galois group of a characteristic-zero field. In particular, I'll explain a homotopical version of the Kummer description of abelian extensions. This is joint work in progress with David Reutter. (hide abstract)
Quantum Homotopy Types. September 11, Researcher Presentations for New PSI Students, Perimeter Institute. (slides.)
SVOAs and some exceptional groups. August 22, Universität Hamburg. (abstract.)

Abstract: The goal of this talk is to present the answer to a fun classification problem: What are all N=1 SVOAs with no continuous symmetries and with bosonic part a simply connected WZW model, and what are their automorphism groups? It turns out that there are two infinite families, both related to alternating groups, and eleven exceptions, all of which are related to the "Suzuki chain" of exceptional subgroups of Conway's largest sporadic group. The Suzuki chain is "dual" to a certain chain of alternating subgroups of Conway's group, and my construction of the corresponding SVOAs implements this to a sort of "level-rank duality" inside the Conway Moonshine SCFT. Time permitting, I will also speculate that there might be an interesting family of SVOAs related to the Mathieu groups, and that this family is "dual" to a family of actions of the fusion categories SU(2)_k on the Conway Moonshine SCFT. (hide abstract)
Super Duper Vector Spaces II: The higher-categorical Galois group of R. August 18, Higher Structures in Functorial Field Theory, University of Regensburg. (abstract. notes.)

Abstract: A theorem of Deligne suggests that the complex numbers are not algebraically closed in a 1-categorical sense but that their 1-categorical algebraic closure is the category sVec of complex super vector spaces. In fact, this property uniquely (up to non-unique isomorphism) characterizes sVec amongst complex-linear symmetric monoidal categories. In these talks, we will outline work in progress on constructing complex-linear symmetric n-categories which are higher categorical analogues of sVec in that they are uniquely characterized by being the n-categorical separable closure of the complex numbers. We will explore the resulting higher-categorical absolute Galois group of the complex numbers, and outline a construction of that group very much akin to the surgery-theoretic description of the stable piecewise linear group PL. This is the second half of a 2-part lecture. The first part is given by David Reutter, with whom this work is joint in progress. The slides from Part I are available here. (hide abstract)
Topological Umbral Moonshine. July 17-21, Topological Moonshine, UIUC. (notes.)
Higher algebraic closure. May 19, OWSM Seminar, Mathematics Institute, Oxford. (abstract.)

Abstract: I will describe my construction, joint with David Reutter, of the universal target category for semisimple TQFTs. (hide abstract)
Higher algebraic closure. April 27, Colloquium, The Ohio State University. (abstract. slides. video.)

Abstract: The fundamental theorem of algebra, as Hilbert explained, asserts that every consistent system of polynomial equations over R has a solution over C. Together with David Reutter, we have established a "fundamental theorem of higher algebra": we have constructed and analyzed the n-category in which every consistent (and semisimple) system of "n-categorical polynomial equations" has a solution. In this talk, I will explain a bit about our construction, and why a quantum physicist might care. (hide abstract)
Higher algebraic closure. April 11, Higher Structures Seminar , Feza Gürsey Institute. (abstract. slides.)

Abstract: Deligne's work on Tannakian duality identifies the category sVec of super vector spaces as the "algebraic closure" of the category Vec of vector spaces (over C). I will describe my construction, joint with David Reutter, of the higher-categorical analog of sVec: the algebraic closure of the n-category of "n-vector spaces". The construction mixes ideas from Galois theory, quantum physics, homotopy theory, and fusion category theory. Time permitting, I will describe the higher-categorical Galois group, which turns out to have a surgery-theoretic description through which it is almost, but not quite, the group PL. (hide abstract)

2022

Homotopy Quantum Groups. November 18, 2022 Simons Collaboration on Global Categorical Symmetries Annual Meeting, Simons Foundation. (abstract. slides. video.)

Abstract: Systems of global categorical symmetry can be thought of as quantum higher groups; I will define and describe their homotopy quantum groups in which two operators represent the same class if they are related by a quantum (noninvertible) homotopy. When the categorical symmetry is a usual higher group, these homotopy quantum groups recover its usual homotopy groups. For fusion higher categories, two operators are in the same “quantum homotopy class” if and only if they are related by a condensation of operators of higher homotopical degree. Although in general the homotopy quantum groups can reflect the noninvertible nature of categorical symmetries, it happens remarkably often that they are honest groups — that all operators are “invertible up to quantum homotopy,” but that the homotopy itself is noninvertible. This provides an interesting middle ground between fully invertible and fully noninvertible symmetries. This talk is based on joint work with David Reutter. (hide abstract)
SVOAs and some exceptional groups. October 25, Quantum Symmetries, Centre de Recherches Mathématics. (abstract. video. notes.)

Abstract: This talk will be in two parts. In the first part of the talk, I'll present the answer to a fun classification problem: What are all N=1 SVOAs with some natural properties, and what are their automorphism groups? It turns out that there are two infinite families, both related to alternating groups, and eleven exceptions, all of which are related to the "Suzuki chain" of exceptional subgroups of Conway's largest sporadic group. The Suzuki chain is "dual" to a certain chain of alternating subgroups of Conway's group, and my construction of the corresponding SVOAs implements this to a sort of "level-rank duality" inside the Conway Moonshine SCFT. In the second half of my talk, I will say a bit more about various constructions about duality, gauging, and coset models. In particular, I will speculate that there might be an interesting family of SVOAs related to the Mathieu groups, and that this family is "dual" to a family of actions of the fusion categories $SU(2)_\ell$ on the Conway Moonshine SCFT. (hide abstract)
Hypergroups and fusion higher categories. October 6, Mathematical Physics Group Meeting, Perimeter Institute. (abstract.)

Abstract: Hypergroups are a piece of "lower" algebra in which rather than a well-defined group law, group elements multiply to each other in a probabilistic way. I will explain that all fusion higher categories, and all (possibly-relative, possibly-unextended) TQFTs — both very much objects of "higher" algebra — have "homotopy hypergroups". They encode the "fusion rules", they have "S-matrices", a "Verlinde formula", and all that jazz. This is based on joint work in progress with David Reutter. (hide abstract)
A 4D TQFT which is not (quite) a gauge theory. October 4, Symmetry Seminar, Oxford. (abstract. slides. video.)

Abstract: Some people have accused me of proving that every (nice enough) 4D TQFT is equivalent to a gauge theory. A version of this statement is true, but only if your notion of "gauge theory" allows dynamical spin structures among the "gauge fields", and some rather complicated terms in the Lagrangian that couple the spin structures to the other gauge fields. That said, many of these 4D TQFTs also admit "dual" descriptions as true gauge theories for higher-form groups. In this talk, I will explain exactly which 4D TQFTs are equivalent to true (higher-group) gauge theories theories, and present a minimal counterexample to the belief that all 4D TQFTs are true gauge theories. (hide abstract)
Hypergroups and fusion higher categories. September 29, Higher categories and topological order, AIM. (notes.)
Global categorical symmetry and higher fusion categories. September 19, PSI faculty presentations, Perimeter Institute. (slides.)
What is a fusion higher category? September 9, Higher Symmetry and Quantum Field Theory, Aspen Center for Physics.
Classification of (semisimple) TQFTs. March 29, Math QFT seminar , MPIM. (abstract. slides.)

Abstract: If you ask a mathematician for a classification of (fully extended, framed) TQFTs, she will probably tell you that they are classified by fully-dualizable objects of the target n-category, and in particular the classification depends on your choice of target n-category. If you ask a physicist, on the other hand, she will tell you that "(T)QFT" is a primitive notion, and that the classification question has a well-defined answer. These two perspectives combine into the following challenge: define, construct, and analyze the "universal target category" in which all TQFTs take values. In this talk, I will describe the solution to this challenge in the case of semisimple TQFTs. This is joint work in progress with David Reutter. (hide abstract)
Why the spaces of N=(0,1) susy QFTs form a spectrum? March 7, Stolz–Teichner Seminar, Oxford. (slides)
Categorified algebraic closure. February 15, ATCAT, Dalhousie. (abstract. notes. video.)

Abstract: A famous theorem of Deligne's says that any (abelian, C-linear) symmetric monoidal category satisfying certain mild size constraints admits a symmetric monoidal functor to the category sVec of super vector spaces. Deligne used this result to classify such symmetric monoidal categories in terms of representation theories of algebraic groups — this is the "Tannakian Duality". A few years ago, I pointed out that Deligne's theorem has a neat interpretation: it says that the symmetric monoidal category sVec is the "algebraic closure" of the symmetric monoidal category Vec of vector spaces; that the extension of Vec into sVec is "Galois"; and that the Tannakian Duality is a categorified version of the Galois correspondence. In this talk, I will explain the statement of Deligne's theorem and my interpretation, and mention some aspects of Deligne's proof. The version of the story I will present here was developed in conversation with D. Reutter. (hide abstract)

2021

Algebraically closed higher categories. December 5, Geometry and Topology Seminar, Haifa. (abstract. slides. video.)

Abstract: Super Tannakian duality can be interpreted as saying that the symmetric monoidal category Vec is not algebraically closed, but rather its algebraic closure is the category sVec of super vector spaces. In this talk, I will explain how to construct the algebraic closure of the symmetric monoidal n-category nVec. The Galois group is almost, but not quite, the stable orthogonal group. The cokernel of the J homomorphism appears in an interesting way. This is based on joint work in progress with David Reutter. (hide abstract)
Algebraically closed higher categories. December 3, Mathematical Physics Seminar, Perimeter Institute. (abstract. video.)

Abstract: I will report on my progress, joint with David Reutter, to construct and analyze the algebraic closure of nVec — in other words, the universal n-category of framed nD TQFTs. The invertibles are Pontryagin dual to the stable homotopy groups of spheres. The Galois group is almost, but not quite, the stable PL group. An invertible TQFT can be condensed from the vacuum if and only if it trivializes on (possibly-exceptional) spheres. (hide abstract)
Classification of topological quantum field theories. November 18, CTP Seminar, QMUL. (abstract. slides.)

Abstract: Modulo some vitally important ansätze, subtleties, provisos, and work in progress, all topological quantum field theories are gauge theories for higher finite groups. (hide abstract)
TMF and SQFT: questions and conjectures. November 4, Generalized Cohomology and Physics, ICTP. (abstract. slides. video.)

Abstract: The Monster is a particularly magical finite group, with spooky relations to both number theory and quantum physics. I will explain its significance, and hint at some of the mysteries that still remain. (hide abstract)
The Monster. October 27, Honours Seminar, Dalhousie. (abstract.)

Abstract: The Monster is a particularly magical finite group, with spooky relations to both number theory and quantum physics. I will explain its significance, and hint at some of the mysteries that still remain. (hide abstract)
A menagerie of N=1 SVOAs. October 6, NAAP, Kavli IPMU. (abstract. slides.)

Abstract: The Conway Moonshine module V^f♮ is specific "N=1" supersymmetric vertex operator algebra; its name reflects that its automorphism group is the Conway sporadic group Co₁. It is a supersymmetric analogue of the Monstrous Moonshine module, and a quantum analogue of the Leech lattice. I will tell you about some interesting subalgebras of V^f♮, which seem to correspond to some interesting subgroups of Co₁. Some of these subalgebras fit within a theorem about WZW algebras, and others fit within a conjecture about umbral moonshine. Along the way, I will highlight some of the techniques for building and analyzing SVOAs and superconformal field theories. (hide abstract)
Semisimple higher categories. July 26, Western Hemisphere Colloquium on Geometry and Physics. (abstract. slides. video.)

Abstract: Semisimple higher categories are a quantum version of topological spaces (behaving sometimes like homotopy types and sometimes like manifolds) in which cells are attached along superpositions of other cells. Many operations from topology make sense for semisimple higher categories: they have homotopy sets (not groups), loop spaces, etc. For example, the extended operators in a topological sigma model form a semisimple higher category that can be thought of as a type of "cotangent bundle" of the target space. The "symplectic pairing" on this "cotangent bundle" is measured an S-matrix pairing aka Whitehead bracket defined on the homotopy sets of any (pointed connected) semisimple higher category, and the nondegeneracy of this pairing is a type of Poincare or Atiyah duality. This is joint work in progress with David Reutter. (hide abstract)
Operators and (higher) categories in quantum field theory. July 19-23, Seminar on Arithmetic Geometry and Quantum Field Theory, Korea Institute for Advanced Study. (abstract. digital chalk boards: I, II, III, IV, V. videos: I, II [first five minutes] and II, III, IV, V.)

Abstract:
I. A complete mathematical definition of quantum field theory does not yet exist. Following the example of quantum mechanics, I will indicate what a good definition in terms could look like. In this good definition, QFTs are defined in terms of their operator content (including extended operators), and the collection of all operators is required to satisfy some natural properties.
II. After reviewing some classic examples, I will describe the construction of Noether currents and the corresponding extended symmetry operators.
III. One way to build topological extended operators is by "condensing" lower-dimensional operators. The existence of this condensation procedure makes the collection of all topological operators into a semisimple higher category.
IV. Topological operators provide "noninvertible higher-form symmetries". These symmetries assign charges to operators of complementary dimension. This assignment is a version of what fusion category theorists call an "S-matrix".
V. The Tannakian formalism suggests a way to recognize higher gauge theories. It also suggests the existence of interesting higher versions of super vector spaces with more exotic tangential structures. (hide abstract)
Higher S-matrices. June 18, TQFT Club, Instituto Superior Técnico. (abstract. slides. video.)

Abstract: Each fusion higher category has a "framed S-matrix" which encodes the commutator of operators of complementary dimension. I will explain how to construct and interpret this pairing, and I will emphasize that it may fail to exist if you drop semisimplicity requirements. I will then outline a proof that the framed S-matrix detects (non)degeneracy of the fusion higher category. This is joint work in progress with David Reutter. (hide abstract)
Minimal nondegenerate extensions. June 11, Fusion Friday, AIM. (notes. video.)
Minimal nondegenerate extensions and an anomaly indicator. June 10, Quantum Matter in Mathematics and Physics, CMSA, Harvard. (abstract. slides. video.)

Abstract: Braided fusion categories arise as the G-invariant (extended) observables in a 2+1D topological order, for some (generalized) symmetry group G. A minimal nondegenerate extension exists when the G-symmetry can be gauged. I will explain what this has to do with the classification of 3+1D topological orders. I will also explain a resolution to a 20-year-old question in mathematics, which required inventing an indicator for a specific particularly problematic anomaly, and a clever calculation of its value. Based on arXiv:2105.15167, joint with David Reutter. (hide abstract)
Classification of topological orders. June 9, Quantum Mathematics: Quantum Matter and Quantum Information, 75+1 CMS Summer Meeting. (abstract. slides. video.)

Abstract: Topological orders have a mathematical axiomatization in terms of their higher fusion categories of extended operators; the characterizing property of these higher fusion categories is that they are satisfy a nondegeneracy condition. After overviewing some of the higher category theory that goes into this axiomatization, I will describe what we do and don't know about the classification of topological orders in various dimensions. (hide abstract)
Higher S-matrices. May 20, Higher Structures & Field Theory Seminar, UniVie/Erlangen/Würzburg/TUM. (abstract. slides.)

Abstract: Each fusion higher category has a "framed S-matrix" which encodes the commutator of operators of complementary dimension. I will explain how to construct and interpret this pairing, and I will emphasize that it may fail to exist if you drop semisimplicity requirements. I will then outline a proof that the framed S-matrix detects (non)degeneracy of the fusion higher category. This is joint work in progress with David Reutter. (hide abstract)
The classification of topological orders. April 1, Mathematics Department Colloquium, OSU. (abstract. slides. pretalk slides.)

Abstract: The Landau Paradigm classifies phases of matter by how "ordered" they are, i.e. by their symmetry groups (and symmetry breaking). The difference between liquids and solids fits into this paradigm, as does the Higgs mechanism that gives particles masses in high-energy physics. However, starting around the turn of the (21st) century, it has become clear that there are patterns of "order" or "symmetry" in quantum matter systems which cannot be described by groups. In particular, there are topological phases of matter, characterized by having no local observables whatsoever, which Landau would have thought were completely trivial but which in fact have subtle long-range topological order. To describe these topological orders requires the mathematics of fusion higher categories. In this talk, I will describe the classification of these topological orders in various dimensions and the extent to which the Landau Paradigm does or does not hold. (hide abstract)
Fusion n-category Q&A. March 12, Fusion categories and tensor networks, AIM.
Orbifolds. March 4, Moonshine Learning Seminar, IAS. (prepared typed notes. live hand-written slides.)
Higher Galois closures. March 3, AGQFT, University of Warwick. (abstract. slides. video.)

Abstract: I will describe a mostly-conjectural picture of the higher-categorical separable closure of R. In particular, I will speculate about unitary topological field theory, higher analogues of spin-statistics, homotopy groups of spheres, and the j-homomorphism. (hide abstract)
Strongly-fusion 2-categories are grouplike. March 1, Representation Theory Seminar, UMass Amherst. (abstract. slides. video.)

Abstract: A fusion category is a finite semisimple monoidal category in which the unit object is indecomposable, equivalently has trivial endomorphism algebra. There are two natural categorifications of this notion: a fusion 2-category is a finite semisimple monoidal 2-category in which the unit object is indecomposable, and a strongly fusion 2-category is one in which the unit object has trivial endomorphism algebra. As I will explain in this talk, fusion 2-categories are extremely rich, with a seemingly-wild classification, whereas strongly-fusion 2-category are very simple: they are essentially just finite groups. Based on joint work with Matthew Yu. (hide abstract)
Condensations and components. February 18, University Quantum Symmetries Lectures, NCSU. (abstract. slides.)

Abstract: The 1-categorical Schur's lemma, which says that a nonzero morphism between simple objects is an isomorphism, fails for semisimple n-categories when n≥2. Rather, when two simple objects are related by a nonzero morphism, they each arise as a condensation descendant of the other. Because of this, for many purposes the natural n-categorical version of "set of simple objects" is the set of components: the set of simples modulo condensation descent. I will explain this phenomenon and describe some conjectures, including conjectures about "higher categorical S-matrices" and, time permitting, about the image of the j-homomorphism in the homotopy groups of spheres. (hide abstract)

2020

A topological umbral moonshine conjecture. November 17, Modularity in Quantum Systems, Kavli Institute for Theoretical Physics. (abstract. slides. video.)

Abstract: I will propose a "topological" description of M₂₄ umbral moonshine. Specifically, I will describe a specific M₂₄-equivariant SCFT, and explain that if it is M₂₄-equivariantly nullhomotopic in the space of SQFTs — if it can be continuously deformed to an SQFT with spontaneous supersymmetry breaking — then that nullhomotopy would produce the mock modular forms of generalized M₂₄-moonshine. I will not construct such a nullhomotopy, but I will provide some evidence of its existence: it is expected that the obstruction for and SQFT to be nullhomotopic is valued in a space of "topological modular forms", and I have calculated that the obstruction vanishes "perturbatively at odd primes". Time permitting, I will suggest that the "optimal growth condition" of umbral moonshine corresponds to working with "topological cusp forms", and I will outline a version of the construction for the umbral groups 2M₁₂ and 2AGL₃(2). (hide abstract)
Holomorphic SCFTs of small index. November 13, Topology, Algebraic Geometry, and Dynamics Seminar, George Mason University. (abstract. slides.)

Abstract: I will explain how some questions in theoretical physics and algebraic topology led to a curious result about error-correcting ternary codes. No knowledge of the terms in the title or abstract will be assumed. Based on joint work with Davide Gaiotto. (hide abstract)
Pseudounitary slightly degenerate braided fusion categories admit minimal modular extensions. November 10, Wales Mathematical Physics and Physical Mathematics, Cardiff University. (abstract. slides.)

Abstract: A braided fusion category is "slightly degenerate" if its Muger centre is a copy of SVec: they arise as the line operators of 3d spin-TFTs. It is a longstanding conjecture that any such braided fusion category admits a "minimal modular extension", i.e. an index-2 extension to a nondegenerate braided fusion category. I will outline a proof, which is joint work in progress with David Reutter, of this conjecture in the pseudounitary case. The proof involves traveling into four, and briefly five, dimensions. (hide abstract)
Some examples in fusion 2-categories and 3+1D topological order. November 6, Global Categorical Symmetries Seminar. (abstract. slides. video.)

Abstract: I have spent the last couple months computing everything I can about the (extended) operator content of a trio of closely related 3+1D bosonic topological orders (≈TFTs): Z₂ gauge theory, spin-Z₂ gauge theory, and an anomalous version of spin-Z₂ gauge theory. [Nontrivial theorem: these are precisely the three 3+1D topological orders with a unique nontrivial particle.] I will tell you some of the results of my computations. In other words, I will tell you the "global categorical symmetries" of these three topological orders. My hope is to illustrate some phenomena that can occur in fusion 2-categories. (hide abstract)
3+1d topological orders with (only) an emergent fermion. October 26, Heidelberg-Munich-Vienna Seminar on Mathematical Physics. (abstract. slides.)

Abstract: There are exactly two bosonic 3+1d topological orders whose only nontrivial quasiparticle is an emergent fermion (and exactly one whose only nontrivial quasiparticle is an emergent boson). I will explain the meaning of this sentence: I will explain what a "3+1d topological order" is, and how I know that these are the complete list. Time permitting, I will tell you some details about these specific topological orders, and say what this classification has to do with "minimal modular extensions". (hide abstract)
3+1d topological orders with (only) an emergent fermion. October 20, Representation Theory and Mathematical Physics, UC Berkeley. (abstract.)

Abstract: There are exactly two bosonic 3+1d topological orders whose only nontrivial quasiparticle is an emergent fermion (and exactly one whose only nontrivial quasiparticle is an emergent boson). I will explain the meaning of this sentence: I will explain what a "3+1d topological order" is, and how I know that these are the complete list. Time permitting, I will tell you some details about these specific topological orders, and say what this classification has to do with "minimal modular extensions". (hide abstract)
Separable and central simple (higher) algebras. October 13, ATCAT, Dalhousie. (abstract. slides.)

Abstract: Most of my talk will be a review of the famous characterization of separable algebras in terms of dualizability/adjunctibility conditions, and of central simple algebras in terms of invertibility conditions, in the bicategory of algebras and bimodules. Time permitting, I will describe my recent generalization of these results in which algebras are replaced by monoidal (higher) categories. Prerequisites: some familiarity with the bicategory of algebras and bimodules, as explained for instance in last week's talk by Robert Paré. (hide abstract)
Algebraic definition of topological order. August 7, Topological Orders and Higher Structures, ESI Vienna. (abstract. slides. video.)

Abstract: I will explain a fully mathematically rigorous definition of (n+1)-dimensional "topological order" in terms of its multifusion n-category of extended operators. This involves understanding "categorical condensation", which is a universal way to build extended operators as networks of lower-dimensional extended operators. It also allows a complete proof of the Lan--Kong--Wen classification of (3+1)-dimensional topological orders. (hide abstract)
Mock modularity and a secondary invariant. July 31, String Math 2020, Stellenbosch University, South Africa. (abstract. slides. video.)

Abstract: (1+1)d supersymmetric field theories admit a famous deformation invariant called, variously, the Witten Index or the Elliptic Genus, which is valued in integral (weak) modular forms. I will present a "secondary" variation of this invariant, which is measures the obstruction to being a shadow of an integral (weak, generalized) mock modular form. It sees beyond the Witten/Elliptic index/genus: in particular, it sees some torsion in the space of SQFTs. Based on joint work with Davide Gaiotto. (hide abstract)
SPT phases and generalized cohomology. June 23, Algebraic structures in quantum computation IV, UBC. (abstract. slides. video.)

Abstract: A priori, classifying n-dimensional SPT phases requires understanding the homotopy type of the topological space I^n of n-dimensional invertible phases of matter. As I will explain, the spaces I^n, for different values of n, compile into a structure called an "Omega-spectrum". This provides a huge advantage: whereas topological spaces are flimsy, Omega-spectra are rigid, and algebraic topologists have developed many powerful techniques for computing with them. In particular, for each n, there is a finite set of groups G such that knowledge of the classification of n-dimensional G-SPTs for those groups determines the classification for all groups. Based on joint work with D. Gaiotto. (hide abstract)
Phases of SQFTs. June 8, Mathematics Colloquium, Dalhousie. (abstract. slides.)

Abstract: A "phase" of quantum systems (of some type) is a connected component in the space of all quantum systems (of that type). For example, phases of minimally-supersymmetric quantum mechanics models (1D QFTs) are perfectly classified by K-theory, leading to myriad applications in mathematics and physics. I will report on what we do (and don't) know about phases of minimally-supersymmetric 2D QFTs. These are expected to be classified by the generalized cohomology theory of Topological Modular Forms (TMF). In particular, I will describe a new invariant of SQFTs called the "secondary Witten genus" which, on the one hand, sees torsion in TMF, and, on the other hand, connects directly to mock modular forms, and thereby to the modern "umbral" moonshine of Niemeier lattices and K3 surfaces. (hide abstract)
On the classification of topological phases. June 1, Colloquium, Perimeter Institute. (abstract. slides. video.)

Abstract: There is a rich interplay between higher algebra (category theory, algebraic topology) and condensed matter. I will describe recent mathematical results in the classification of gapped topological phases of matter. These results allow powerful techniques from stable homotopy theory and higher categories to be employed in the classification. In one direction, these techniques allow for complete a priori classifications in spacetime dimensions ≤6. In the other direction, they suggest fascinating and surprising statements in mathematics. (hide abstract)
Gapped condensation in higher categories. March 17, Tensor categories and topological quantum field theories, MSRI. (abstract. video.)

Abstract: Idempotent (aka Karoubi) completion is used throughout mathematics: for instance, it is a common step when building a Fukaya category. I will explain the n-category generalization of idempotent completion. We call it "condensation completion" because it answers the question of classifying the gapped phases of matter that can be reached from a given one by condensing some of the chemicals in the matter system. From the TFT side, condensation preserves full dualizability. In fact, if one starts with the n-category consisting purely of ℂ in degree n, its condensation completion is equivalent both to the n-category of n-dualizable ℂ-linear (n-1)-categories and to an n-category of lattice condensed matter systems with commuting projector Hamiltonians. This establishes an equivalence between large families of TFTs and of gapped topological phases. Based on joint work with D. Gaiotto. (hide abstract)
A deformation invariant of 2D SQFTs. February 25, New High Energy Theory Center Seminar, Rutgers. (abstract. video.)

Abstract: The elliptic genus is a powerful deformation invariant of 2D SQFTs: if it is nonzero, then it protects the SQFT from admitting a deformation to one with spontaneous supersymmetry breaking. I will describe a "secondary" invariant, defined in terms of mock modularity, that goes beyond the elliptic genus, protecting SQFTs with vanishing elliptic genus. The existence of this invariant supports the hypothesis that the space of minimally supersymmetric 2D SQFTs provides a geometric model for universal elliptic cohomology. (hide abstract)
A deformation invariant of 2D SQFTs. January 24, Geometry, Topology, and Physics, UC Santa Barbara. (abstract. audio. notes by Dave Morrison.)

Abstract: The elliptic genus is a powerful deformation invariant of 2D SQFTs: if it is nonzero, then it protects the SQFT from admitting a deformation to one with spontaneous supersymmetry breaking. I will describe a "secondary" invariant, defined in terms of mock modularity, that goes beyond the elliptic genus, protecting SQFTs with vanishing elliptic genus. The existence of this invariant supports the hypothesis that the space of minimally supersymmetric 2D SQFTs provides a geometric model for universal elliptic cohomology. Based on joint works with D. Gaiotto and E. Witten. (hide abstract)
A deformation invariant of 1+1D SQFTs. January 14, Quantum Fields and Strings, Perimeter Institute. (abstract. video.)

Abstract: The elliptic genus is a powerful deformation invariant of 1+1D SQFTs: if it is nonzero, then it protects the SQFT from admitting a deformation to one with spontaneous supersymmetry breaking. I will describe a "secondary" invariant, defined in terms of mock modularity, that goes beyond the elliptic genus, protecting SQFTs with vanishing elliptic genus. The existence of this invariant supports the hypothesis that the space of minimally supersymmetric 1+1D SQFTs provides a geometric model for universal elliptic cohomology. Based on joint works with D. Gaiotto and E. Witten. (hide abstract)

2019

A deformation invariant of 2D SQFTs. December 11, Diff. Geom, Math. Phys., PDE Seminar, UBC. (abstract. notes.)

Abstract: The elliptic genus is a powerful deformation invariant of 2D SQFTs: if it is nonzero, then it protects the SQFT from admitting a deformation to one with spontaneous supersymmetry breaking. I will describe a "secondary" invariant, defined in terms of mock modularity, that goes beyond the elliptic genus, protecting SQFTs with vanishing elliptic genus. The existence of this invariant supports the hypothesis that the space of minimally supersymmetric 2D SQFTs provides a geometric model for universal elliptic cohomology. (hide abstract)
Spaces of quantum systems. December 10, Department Colloquium, UBC. (abstract. slides.)

Abstract: Physicists have long been interested in answering homotopical questions about (appropriately topologized) spaces of quantum systems --- for example, the connected components of such spaces classify phases of matter (in the solid-liquid-gas sense). Recent evidence suggests that such spaces may also be of interest to pure mathematicians, because in many cases they have the same homotopy types as objects of fundamental interest in topology. I will describe two examples of this phenomenon. First, an example from condensed matter: the classification of topological phases of matter leads to rich category theory, and, conjecturally, to a relationship between cobordism groups and a higher-categorical version of Galois theory. Second, an example from high energy physics: the space of minimally supersymmetric 2D quantum field theories provides, conjecturally, an analytic model for universal elliptic cohomology. (hide abstract)
The Monstrous Moonshine Anomaly. November 27, Cohomology of Groups Seminar, Perimeter Institute. (abstract. video. notes.)

Abstract: The action of the Monster sporadic group on the Moonshine CFT enjoys an 't Hooft anomaly. I will describe my (complete) calculation of its value, and my (almost complete) calculation of its home; I will also mention my joint work with Treumann on other sporadic groups. The talk will begin with elementary methods in the cohomology of finite groups, and proceed to more and more sophisticated techniques. In particular, I will interpret the Serre Spectral Sequence as providing a "finite-group T-duality" relationship between symmetries of 2d QFTs related by cyclic group orbifold, and use a series of such relationship between the Monster CFT and the Leech lattice to control the anomaly. (hide abstract)
Condensation in higher categories. November 20, Higher Structures in Geometry and Physics, Fields Institute. (abstract. video.)

Abstract: Idempotent (aka Karoubi) completion is used throughout mathematics: for instance, it is a common step when building a Fukaya category. I will explain the n-category generalization of idempotent completion. We call it "condensation completion" because it answers the question of classifying the gapped phases of matter that can be reached from a given one by condensing some of the chemicals in the matter system. From the TFT side, condensation preserves full dualizability. In fact, if one starts with the n-category consisting purely of ℂ in degree n, its condensation completion is equivalent both to the n-category of n-dualizable ℂ-linear (n-1)-categories and to an n-category of lattice condensed matter systems with commuting projector Hamiltonians. This establishes an equivalence between large families of TFTs and of gapped topological phases. Based on joint work with D. Gaiotto. (hide abstract)
TMF and SQFT. November 18. High Energy Theory Seminar, IAS, Princeton. (abstract. video.)

Abstract: I will describe my work, all joint with D. Gaiotto and some also joint with E. Witten, to understand the homotopy type of the space of (1+1)d N=(0,1) SQFTs — what a condensed matter theorist would call "phases" of SQFTs. Our motivating hypothesis (due in large part to Stolz and Teichner) is that this space models the spectrum called "topological modular forms". Our work includes many nontrivial checks of this hypothesis. First, the hypothesis implies constraints on the possible values of elliptic genera, and suggests (but does not imply) the existence of holomorphic SCFTs saturating these constraints; we have succeeded in constructing such SCFTs in low central charge. Second, the hypothesis implies the existence of torsion-valued "secondary invariants" beyond the elliptic genus that protect SQFTs from admitting deformations that spontaneously break supersymmetry. I will explain such an invariant in terms of holomorphic anomalies and mock modularity. (hide abstract)
Bott periodicity from quantum Hamiltonian reduction. October 24. Colloquium, Dalhousie University, Halifax. (abstract.)

Abstract: The "quantization dictionary" posits that constructions in noncommutative algebra often parallel constructions in symplectic geometry. I will explain an example of this dictionary: I will produce the 8-fold periodicity of Clifford algebras as an example of quantum Hamiltonian reduction of a free fermion quantum mechanical system. The exceptional Lie group G_2 will make a cameo appearance. (hide abstract)
Condensation and idempotent completion. October 22. ATCAT Seminar, Dalhousie University, Halifax. (abstract.)

Abstract: Idempotent (aka Karoubi, aka Cauchy) completion appears throughout mathematics: for instance, it converts the category of free modules to the category of projective modules. I will explain the higher-categorical generalization of idempotent completion. I call it "condensation", because, as I will explain, if you start with a category of gapped phases of matter, then its idempotent completion consists of those phases that can be condensed from the phases you already have. In particular, if you start just with the vacuum phase, and idempotent complete, you recover a very large class of gapped phases, including the Turaev--Viro--Barrett--Westbury models. Moreover, every condensable-from-the-vacuum phase of matter is fully dualizable (i.e. determines a fully-extended TQFT), and conversely every condensable-from-the-vacuum TQFT has a commuting projector Hamiltonian model, and so one finds an equivalence between large classes of TQFTs and condensed phases. Based on joint work with Davide Gaiotto. (hide abstract)
Bott periodicity from quantum Hamiltonian reduction. September 16. McGill. (abstract.)

Abstract: The "quantization dictionary" posits that constructions in noncommutative algebra often parallel constructions in symplectic geometry. I will explain an example of this dictionary: I will produce the 8-fold periodicity of Clifford algebras as an example of quantum Hamiltonian reduction of a free fermion quantum mechanical system. The exceptional Lie group G₂ will make a cameo appearance. (hide abstract)
Phases of 2d SQFTs. August 9. Generalized Symmetries, Anomalies and Observables, Aspen Center for Physics. (notes. abstract.)

Abstract: I will describe my work, all joint with D. Gaiotto and some also joint with E. Witten, to understand the homotopy type of the space of (1+1)d N=(0,1) SQFTs --- what a condensed matter theorist would call "phases" of SQFTs. Our motivating hypothesis (due in large part to Stolz and Teichner) is that this space models the spectrum called "topological modular forms"; our work includes many nontrivial checks of this hypothesis. I will warm up with a brief discussion of N=1 quantum mechanics, fermionic anomalies, and K-theory. Then I will mention constraints that the Stolz--Teichner hypothesis places on the indexes of holomorphic SCFTs, and our checks of those constrains. Time permitting, I will end by explaining our realization of the Bunke--Naumann "secondary" invariants in terms of holomorphic anomalies and mock modularity. (hide abstract)
Exceptional Mathematics: from Egyptian fractions to heterotic strings. July 23. Colloquium, Canada/USA Mathcamp. (slides without animation (8MB), slides with embedded video (best viewed in Acrobat Reader, 200MB), slides in handout format. abstract.)

Abstract: Most of the mathematical universe is regular and repeating, but every once in a while there is an exception, and it leads to all sort of interesting and irregular phenomena. I will explain how the exceptional solutions to a simple problem from antiquity — find all integer solutions to (1/a) + (1/b) + (1/c) > 1 — lead to 20th and 21st century highlights: topological phases of matter, heterotic string theory, and E₈, the most exceptional Lie group. (hide abstract)
0D QFT and Feynman diagrams. June 17. QFT for Mathematicians, Perimeter Institute. (notes, video.)
Secondary invariants and mock modularity. May 27. Topology Seminar, Oxford. (abstract.)

Abstract: A two-dimensional, minimally Supersymmetric Quantum Field Theory is "nullhomotopic" if it can be deformed to one with spontaneous supersymmetry breaking, including along deformations that are allowed to "flow up" along RG flow lines. SQFTs modulo nullhomotopic SQFTs form a graded abelian group SQFT_•. There are many SQFTs with nonzero index; these are definitely not nullhomotopic, and indeed represent nontorision classes in SQFT_•. But relations to topological modular forms suggests that SQFT_• also has rich torsion. Based on an analysis of mock modularity and holomorphic anomalies, I will describe explicitly a "secondary invariant" of SQFTs and use it to show that a certain element of SQFT₃ has exact order 24. This work is joint with D. Gaiotto and E. Witten. (hide abstract)
The Galois action on VOA anomalies. March 18. Higher Symmetries Conference 2019, Aspen Center for Physics. (abstract.)

Abstract: An important role for higher symmetries arises when an ordinary group action suffers an 't Hooft anomaly. I will focus on the case of (anomalous) actions on (two-dimensional) holomorphic CFTs. After reviewing the construction and comparing to the 1d case, I will discuss how the anomaly transforms under Galois conjugation of the VOA — it does not transform in the most obvious way, because the relation between VOAs and MTCs is not Galois-equivariant. The actual transformation law explains many of the "24"s that arise in moonshine, and suggests connections between VOAs, algebraic K-theory, and "higher" Brauer groups. (hide abstract)
Galois actions on VOA gauge anomalies. February 27. Conference on Number Theory, Geometry, Moonshine & Strings III. Simons Foundation, NYC. (abstract. video available from conference website)

Abstract: Assuming a widely believed conjecture, any action of a finite group G on a holomorphic VOA V over C determines a gauge anomaly α living in H³(G;U(1)). I will explain that this anomaly does not transform in the most obvious way under Galois conjugation. Rather, if V is conjugated by an element γ ∈ Aut(C), then α transforms to γ²(α). This explains many of the appearances of the number 24 in moonshine. (hide abstract)
Bott periodicity from quantum Hamiltonian reduction. February 25. Analysis & PDE Seminar, Stanford. (abstract.)

Abstract: The "quantization dictionary" posits that constructions in noncommutative algebra often parallel constructions in symplectic geometry. I will explain an example of this dictionary: I will produce the 8-fold periodicity of Clifford algebras as an example of quantum Hamiltonian reduction of a free fermion quantum mechanical system. No knowledge of the words "quantization", "Clifford algebra", "free fermion", or "Hamiltonian reduction" will be assumed. The exceptional Lie group G₂ will make a cameo appearance. (hide abstract)
Galois actions on VOA gauge anomalies. February 22. Alg No Th Seminar, UCSC. (abstract.)

Abstract: Symmetries of a physical system can be "anomalous"; when they are, there is a "gauge anomaly" living in the cohomology of the group of symmetries. I will explain the definition of this anomaly in the case of quantum mechanics (also called Azumaya algebra) and holomorphic conformal field theory (also called holomorphic vertex operator algebra). I will then explain an anomaly of the VOA case: if a finite group G acts on a holomorphic VOA V over C, then the anomaly lives in H³(G; C^×), but a Galois automorphism γ does not act simply on the coefficients by α ↦ γ(α), but rather by α ↦ γ²(α). This explains many appearances of the number 24 in moonshine and suggests many questions relating VOAs to K-theory. (hide abstract)
Bott periodicity from quantum Hamiltonian reduction. February 19. Colloquium, UCSC. (abstract.)

Abstract: The "quantization dictionary" posits that constructions in noncommutative algebra often parallel constructions in symplectic geometry. I will explain an example of this dictionary: I will produce the 8-fold periodicity of Clifford algebras as an example of quantum Hamiltonian reduction of a free fermion quantum mechanical system. No knowledge of the words "quantization", "Clifford algebra", "free fermion", or "Hamiltonian reduction" will be assumed. The exceptional Lie group G₂ will make a cameo appearance. (hide abstract)

2018

Poisson and coisotropic. November 21 and December 12 (three-hour lecture). Koszul Duality Seminar, PITP. (abstract. video of first hour. video of third hour (self-contained).)

Abstract: I will explain some subset of the following, probably not in this order: a Poisson version of the AKSZ construction; derived-algebraic versions of the words "ideal" and "coisotropic"; Koszul duality between "Frobenius" and "Lie Bi"; the difference between trees and graphs; a nontrivial fact about Poincare duality in DeRham(S¹). (hide abstract)
Holomorphic SCFTs of small index. November 27. Quantum Algebra and Quantum Topology, OSU. (abstract.)

Abstract: Stolz and Teichner have conjectured that the moduli space of D=1+1, N=(0,1) QFTs provides a geometric model for Topological Modular Forms. Some important building blocks in this moduli space are the holomorphic superconformal field theories, and the conjecture leads to predictions about the possible values the supersymmetric index of such SCFTs can take. Specifically, the conjecture leads one to predict the existence of SCFTs of small nonzero index, and that the minimal possible index depends in an interesting way on the central charge of the SCFT. I will explain a construction of some SCFTs of indexes equal to the predicted minimal values. The construction leads to a new divisibility result in the seemingly unrelated field of algebraic coding theory. Based on joint work with Davide Gaiotto. (hide abstract)
Holomorphic SCFTs of small index. November 8. Mathematical Physics, UIUC. (abstract.)

Abstract: Stolz and Teichner have conjectured that the moduli space of D=1+1, N=(0,1) QFTs provides a geometric model for Topological Modular Forms. Some important building blocks in this moduli space are the holomorphic superconformal field theories, and the conjecture leads to predictions about the possible values the supersymmetric index of such SCFTs can take. Specifically, the conjecture leads one to predict the existence of SCFTs of small nonzero index, and that the minimal possible index depends in an interesting way on the central charge of the SCFT. I will explain a construction of some SCFTs of indexes equal to the predicted minimal values. The construction leads to a new divisibility result in the seemingly unrelated field of algebraic coding theory. Based on joint work with Davide Gaiotto. (hide abstract)
Galois action on gauge anomalies. October 18. Fusion Categories and Subfactors, BIRS, Banff, Canada. (abstract. video.)

Abstract: Assuming a widely-believed conjecture, any action of a finite group G on a holomorphic vertex algebra A determines a "gauge anomaly" ω ∈ H^3(G; U(1)). The construction is fusion category theoretic: any conformal inclusion W ⊂ V, with V holomorphic, determines a fusion category; W = V^G gives a pointed fusion category. I will explain a subtlety in the construction coming from Galois actions. Specifically, if γ is a Galois automorphism, then the anomaly for γ(V) is not, as might be assumed naively, γ(ω), but is rather γ²(ω). The proof relies on a recent construction by Evans and Gannon of vertex algebras with a given gauge anomaly. (hide abstract)
T-duality for finite groups. June 4. Representation Theory, Mathematical Physics and Integrable Systems, CIRM, Luminy, France. (abstract. notes.)

Abstract: I will describe a version of "T-duality" in which circles are replaced by finite cyclic groups. This T-duality appears naturally in fusion category theory and in the construction of "twisted orbifolds" of conformal field theories. As an application, I will compute the 't Hooft anomaly of monstrous moonshine. (hide abstract)
The fourth cohomology of some sporadic groups. April 17. Geometry, Symmetry and Physics, Yale. (abstract.)

Abstract: Whenever a finite group G acts on a 2d quantum field theory, it determines a "gauge anomaly" living in the fourth integral group cohomology of G. I will describe what I know about H^4 for certain sporadic groups, focusing on the most charismatic ones: the Monster and Conway's largest group. In particular, I will suggest that often H^4(G) is cyclic and generated by the gauge anomaly of a distinguished "moonshine" representation of G. (hide abstract)
Moonshine anomalies. April 9. Algebra seminar, University at Buffalo. (abstract.)

Abstract: Surprisingly many finite simple groups G have cyclic fourth integral group cohomology. Of particular interest are sporadic groups, at least some of which arise, via "moonshine", as automorphism groups of conformal field theories. Any action of a group G on a conformal field theory produces an "anomaly" living in the fourth cohomology of G, and I will speculate that most sporadic groups have distinguished "moonshine" actions on conformal field theories and that the corresponding anomalies generate the cohomology in question. This speculation is supported by various examples, including O'Nan's group O'N and Conway's largest group Co0; I will explain the techniques we used to calculate their fourth cohomologies. I will also tell you what I know about the Monster: although I cannot prove the full speculation, I can calculate the anomaly of the "monster moonshine" theory; if the speculation holds, then all Monster representations have vanishing second Chern class. Time permitting, I will explain a finite-group version of T-duality that I used to compute the Monster's moonshine anomaly. This talk is based in part on joint work with D. Treumann. (hide abstract)
Infinitely-categorified commutative algebra. March 17--19. Recent developments in noncommutative algebra and related areas, University of Washington. (abstract.)

Abstract: I will introduce an "infinite categorification" of commutative rings that I refer to as "towers" and that are a higher-categorical version of coconnective Ω-spectra. I will describe some basic constructions, including a "suspension" type construction that turns any commutative ring R or any symmetric monoidal linear category C into a tower Σ^•R or Σ^•-1C. I will emphasize the role that finitely generated projective modules and that separable associative algebras play in these constructions. Through these, suspension towers are closely related to constructions in condensed matter and in topological field theory. I will end by suggesting an infinitely-categorified Galois theory, and in particular I will predict that the infinitely-categorified absolute Galois group of the real numbers if the stable orthogonal group. This talk is based in part on joint work with Davide Gaiotto. (hide abstract)
Moonshine anomalies. February 9. QMAP seminar, UC Davis. (abstract.)

Abstract: Many sporadic groups G admit distinguished "moonshine" representations on conformal field theories --- the original and most famous example being the "defining" representation of the Monster group. Any such action produces an accompanying gauge anomaly living in H^4(G;Z). I will discuss the values of these anomalies, and suggest that often the anomaly of the distinguished "moonshine" action generates the corresponding cohomology group. In particular, I will report on my calculation, joint with David Treumann, of the fourth cohomology of the largest Conway group and of the O'Nan group, and on my calculation of the order of the Monster's moonshine anomaly. The latter calculation uses a construction I call "finite group T-duality" which may be of independent interest. (hide abstract)
Higher algebraic closures and phases of matter. January 22. Northeastern University. (abstract. handout.)

Abstract: Algebraic closures and Galois groups have been of central importance for hundreds of years. I will present a setting for "higher" commutative algebra in which these notions can be extended. In this setting C is not algebraically closed: its algebraic closure knows about super vector spaces and Deligne's "existence of fiber functors". I will conjecture that the "higher" absolute Galois group of R is the infinite orthogonal group O(∞), and suggest that the stable homotopy groups of spheres arise naturally in "higher" algebraic closures. My setting and conjectures are based on questions coming from the classification of condensed matter systems. In particular, the "spin-statistics theorem" and the experimentally-observed role for cobordism groups in the classification of condensed matter systems both arise naturally from my conjectures. Parts of this talk are based on joint work in progress with Davide Gaiotto and with Mike Hopkins. (hide abstract)
Moonshine anomalies. January 19. UT Austin. (abstract.)

Abstract: Many sporadic groups G admit distinguished "moonshine" representations on conformal field theories --- the original and most famous example being the "defining" representation of the Monster group. Any such action produces an accompanying gauge anomaly living in H^4(G;Z). I will discuss the values of these anomalies, and suggest that often the anomaly of the distinguished "moonshine" action generates the corresponding cohomology group. In particular, I will report on my calculation, joint with David Treumann, of the fourth cohomology of the largest Conway group and of the O'Nan group, and on my calculation of the order of the Monster's moonshine anomaly. The latter calculation uses a construction I call "finite group T-duality" which may be of independent interest. (hide abstract)

2017

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

Abstract: I will report on joint work in progress with Davide Gaiotto on the classification of gapped phases of matter. I will explain what symmetry protected phases are and why they are classified by reduced generalized group cohomology. I will also introduce the notion of "condensable n-algebra," and the higher category thereof, as an axiomatization of the algebraic structure enjoyed by gapped phases that can be condensed from the vacuum. Finally, I will interpret the Cobordism Hypothesis as the equivalence between (condensable) topological field theories and (condensable) gapped phases. (hide abstract)
576 Fermions. October 24. Algebra Seminar, Emory. (abstract.)

Abstract: The Stolz--Teichner conjectures predict that the generalized cohomology theory called Topological Modular Forms has a geometric model in terms of the space of 2-dimensional supersymmetric quantum field theories, and that holomorphic vertex operator superalgebras provide the geometric model for nontrivial degrees of TMF. Since TMF is periodic with period 576, these conjectures in particular predict an equivalence between holomorphic VOSAs of different central charge that had not been discovered by physicists. I will report on progress constructing this "periodicity" equivalence geometrically. Specifically, I will explain the solution to the warm-up problem of constructing geometrically the 8-fold periodicity of real K-theory: my solution realizes this periodicity as an example of super symplectic reduction. I will then explain why I believe the Conway group Co0 will play a role in the 576-fold periodicity problem, and why my recent computation of H^4(Co0) provides evidence for this belief. (hide abstract)
Bott periodicity from Hamiltonian reduction. October 5. NT&AG, Boston College. (abstract.)

Abstract: I will explain that the 8-fold periodicity of KO arises as a quantum Hamiltonian reduction of a free fermion system. The talk will be elementary: I will explain the words "8-fold periodicity", "quantum Hamiltonian reduction", and "free fermion". The exceptional Lie group G_2 will make an appearance. (hide abstract)
Exceptional structures, fermions, anomalies, and Hamiltonian reduction. October 3. Research Seminar in Mathematics, Northeastern. (abstract. notes.)

Abstract: My story will begin with Hamiltonian reduction and a super, quantum version thereof. It will end with the cohomology of the Monster group. Along the way, I will talk about Bott periodicity, topological modular forms, the Leech lattice, free chiral fermions, symmetry protected topological phases of matter, and a finite-group version of T-duality. I will, of course, assume no background in any of these areas: my goal will be to explain who all the characters are and why they are all characters in the same story. Parts of the story are still fantasy, and parts are joint with David Treumann. (hide abstract)
The Moonshine Anomaly. July 25. Higher Structures Lisbon, Instituto Superior Técnico, Lisbon, 24–27 July 2017. (abstract.)

Abstract: Whenever a finite group G acts on a holomorphic conformal field theory, there is a corresponding «anomaly» in H³(G,U(1)) — a sort of «characteristic class» of the action — measuring the obstruction to gauging the action. After a brief review of the general story, I will describe a construction that I call «finite group T-duality», which allows for information about anomalies to be compared between different field theories. The most famous example of a finite group acting on a conformal field theory is surely the Monster group acting on its natural «moonshine» representation. I will explain how T-duality can be used to calculate the anomaly. Along the way I will also discuss the Conway group and the anomaly for its natural action, the fermionic version of anomalies, the relation to String structures, and how I hope to construct physically the 576-fold periodicity of TMF. (hide abstract)
The Moonshine Anomaly. July 19. Maximals Seminar, University of Edinburgh. (abstract.)

Abstract: Conformal field theories, and the fusion categories derived from them, provide classes in group cohomology that generalize characteristic classes. These classes are called "anomalies", and obstruct the existence of constructing orbifold models. I will discuss two of the most charismatic groups — the Conway group Co₀ and the Fischer–Griess Monster group M — and explain my calculation that in both cases the anomaly has order exactly 24. The Monster calculation relies on a version of "T-duality" for finite groups which in turn relies on fundamental results about fusion categories. I will try to explain everything from the beginning, and assume no knowledge of the Monster or its cousins. (hide abstract)
Orbifolds of conformal field theories and cohomology of sporadic groups. April 14. RTGC Seminar, UC Berkeley. (abstract. notes.)

Abstract: I will report on work in progress to understand the fourth integral cohomology of the two most famous sporadic finite groups: the Fischer–Griess Monster and Conway's group Co₀. These cohomology classes arise when studying orbifolds of conformal field theories; in that world, they are called "anomalies". I will explain the connection between fermionic anomalies and "string structures" on representations. I will also describe how to move information about anomalies through orbifolds. (hide abstract)
Advanced integration by parts: the BV formalism. Feb 9. Geometric Structures Laboratory, Fields Institute. (abstract. notes.)

Abstract: The phrase "BV formalism" means many things. The first half of my talk will focus on its most basic meaning: a systematic way to organize the "integration by parts" method from undergraduate calculus into a packaging amenable to more general homological algebra (namely, into a twisted de Rham complex). Particularly useful is the Homological Perturbation Lemma. It assures that the algorithms we teach to undergraduates terminate; it produces Feynman diagrams, the ur-example of "perturbative" physics; and, as I will explain, it also applies to nonperturbative integrals, providing a nonperturbative version of "stationary phase approximation".

The second half of my talk will generalize the earlier discussion. The "BV formalism" suggests that any system with algebraic properties similar to a twisted de Rham complex should be thought of as an "oscillating integral problem". I will explain one origin of such systems, called the "AKSZ construction". BV-type systems are amenable to homological perturbation lemma techniques. Time permitting, I will explain how I had hoped to prove Kontsevich formality, why my proof failed, and what that failure says about Poincare duality. (hide abstract)
Fermionic hamiltonian reduction and periodicity. Feb 1. Geometry and Physics Seminar, Boston University. (abstract.)

Abstract: I will describe a "geometric" origin for the famous "Bott periodicity" Morita equivalence between Cliff(8) and R. Specifically, I will explain that that equivalence arises from quantizing the symplectic reduction of fermionic 8-dimensional space by an action by Spin(7). The quaternions and the Lie group G_2 will also make an appearance. Time permitting, I will speculate about a similar "periodicity" equivalence of conformal field theories predicted by conjectures in homotopy theory. In the CFT version, sporadic finite simple groups play a starring role. (hide abstract)
Ideals in derived algebra and boundary conditions in AKSZ-type field theories. Jan 27. RTGC Seminar, UC Berkeley. (abstract. notes.)

Abstract: For each dg operad P, I will present a homotopically-coherent version of "P-ideal". This presentation extends without change to a many-to-many generalization of operads with tree-level compositions called "dioperads". Whereas operads describe algebras, dioperads describe bialgebras, and "P-ideals" for a dioperad P are simultaneously ideals and coideals. In the case where P describes Frobenius algberas, P-ideals show up in relative Poincare duality. In the case where P describes Lie bialgebras, P-ideals are related to coisotropic submanifolds of derived Poisson manifolds. Koszul duality and exact triangles will also appear in my talk. (hide abstract)

2016

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.)

Abstract: I will report on progress understanding the 576-fold periodicity in TMF in terms of conformal field theoretic constructions. Sporadic finite groups and their cohomology will play a role. (hide abstract)
Bott periodicity via quantum Hamiltonian reduction. Sept 2, Representation Theory, Geometry, and Combinatorics Seminar, UC Berkeley. (abstract)

Abstract: I will describe the Morita equivalence between Cliff(8) and R in terms of the quantum Hamiltonian reduction of the spin module of Spin(7). Along the way I will mention fermions, the exceptional group G_2, the E_8 lattice, and the quaternions. Time permitting I will also speculate about Conway's group Co_0 and topological modular forms. I will aim to be elementary, at least in the non-speculative portion of the talk, and I always invite arbitrary questions and interruptions. (hide abstract)
Bott periodicity via quantum Hamiltonian reduction. May 12, Factorization Algebras and Functorial Field Theories meeting, Oberwolfach. (abstract, notes)

Abstract: I will describe a symplectic origin for the famous Morita equivalence between Cliff(8) and R. Specifically, I will explain that this Morita equivalence arises from quantizing the Hamiltonian reduction R^0|8//Spin(7), where Spin(7) acts on R⁸ via the spin representation. I will also quantize the reductions R^0|4//Spin(3) and R^0|7//G₂. (hide abstract)
"Spin-statistics" is a categorification of "Hermitian". February 22, RTGC Seminar, Berkeley. (abstract, handout)

Abstract: I will describe a "cobordism" language in which to pose requirements on a quantum field theory like being Hermitian (when complex conjugation = orientation reversal) or satisfying Spin-Statistics (when fermions = spinors). This language also a homotopy-theoretic origin for those two requirements: nature distinguished them among all possible similar requirements. Namely, Hermitian field theories arise because π₀(O(∞)) has a unique nontrivial torsor over Spec(R). Spin-statistics field theories arise for the same reason, except that one must replace π₀(O(∞)) with the fundamental groupoid of O(∞) and one must replace commutative algebras with symmetric monoidal categories. (hide abstract)

2015

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

Abstract: The "spin-statistics theorem" is a physical phenomenon in which spinors --- (-1)-eigenstates of rotation by 360° --- are the same as fermions --- (-1)-eigenstates of switching two identical particles. Physicists usually understand this phenomenon as a fact about certain representations of the Lorentz group. In this talk I will give a very different mathematical "origin" of the spin-statistics theorem. I will explain that spin-statistics arises in precisely the same was as does the physical phenomenon of "unitarity", which in turn depends on a fundamental but nontrivial coincidence: the absolute Galois group of R happens to equal the group of connected components of the orthogonal group. This talk will assume no knowledge of physics. (hide abstract)
Spin--Statistics and Categorified Galois Groups. October 23, Conference on Condensed Matter Physics and Topological Field Theory, Perimeter Institute. (abstract, handout)

Abstract: I will describe two coincidences in homotopy theory, the second a categorification of the first. The first coincidence is the origin of "unitarity" in field theory. The second is a topological origin for the so-called "spin--statistics theorem". (hide abstract)
A higher category theorist's take on the spin--statistics theorem. September 28, Topology Seminar, UIUC. (abstract, handout)

Abstract: This talk is about a pair of coincidences in homotopy theory which related quantum field theory with commutative algebra. The first coincidence is the fact that the etale homotopy type of Spec(R) matches the homotopy 1-type of BO(∞), the classifying space of the stable orthogonal group. This coincidence, I will argue, is the reason for "unitary" phenomena in physics. The second coincidence is a categorification of this: I will describe a setting in which Spec(R) has an "etale" homotopy type that matches the homotopy 2-type of BO(∞), and explain how this provides the "spin--statistics theorem" relating spinors to fermions. (hide abstract)
The Stokes groupoids of Gualtieri, Li, and Pym. May 4&6, Math 448: Topics in Geometry and Topology, Northwestern. (abstract, notes)

Abstract: An overview of the paper Gualtieri, Li, and Pym, "The Stokes Groupoids", 2013, arXiv:1305.7288. No results in this talk are due to the speaker. (hide abstract)
Some non-dualizable categories. April 17, Representation Theory, Geometry, and Combinatorics, UC Berkeley. (abstract, handout)

Abstract: Linear cocomplete categories provide a categorification of linear algebra. In this talk, I will describe recent work with M. Brandenburg and A. Chirvasitu in which we investigate which linear cocomplete categories are "dualizable" --- if this were the module theory for a commutative ring, these would be the finitely generated projective modules. In fact, I will explain that dualizability *of the category* is closely related to whether the category has enough finitely generated projectives. Examples will come from representation theory and from algebraic geometry: in particular, non-reductive groups and projective varieties provide non-dualizable examples. Applications come from quantum field theory: dualizable linear cocomplete categories arise both in "relative" and in "extended" quantum field theories, and so our results mean that "topological gauge theory for non-reductive groups" and "topological sigma models for projective varieties" cannot be described in this framework. (hide abstract)
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))

Abstract: Poincare duality implies, among other things, that the de Rham cohomology of a compact oriented manifold is a commutative Frobenius algebra. Then a version of "local Poincare duality" would be a "homotopy commutative Frobenius algebra" structure on the de Rham complex satisfying some locality conditions. It turns out that there are at least two inequivalent notions of "homotopy commutative Frobenius algebra", depending on whether you work at "tree level" or at "all loop order" in a certain "Feynman" diagrammatics. This choice affects whether local Poincare duality is or is not canonical. The "all loop order" version of local Poincare duality is closely related to Kontsevich-type problems in deformation quantization. In particular, "all loop order" local Poincare duality on S^1 is obstructed; the obstruction answers the question of which Poisson structures admit universal deformation quantizations that do not require taking traces. (hide abstract)
Some comments on Heisenberg-picture qft. March 18, Max Planck Institute for Mathematics, Bonn, Germany. (abstract, handout)

Abstract: The usual Atiyah–Segal "functorial" description of quantum field theory corresponds to the "Schrodinger picture" in quantum mechanics. I will describe a slight modification that corresponds to the "Heisenberg picture", which I will argue is better physically motivated. The example I am most interested in is a version of quantum Chern–Simons theory that does not require the level to be quantized; it provides a neat packaging of pretty much all objects of skein theory. (hide abstract)
Twisted field theories and higher-categorical (op)lax transfors. March 3, Topology Seminar, University of Notre Dame, South Bend, IN. (abstract, handout)

Abstract: A "Schrodinger picture" (extended) quantum field theory is a functor from some (higher) category of "spacetimes" to some (higher) category of "Hilbert spaces". This framework is powerful and well-studied. Unfortunately, it does not capture many important examples. Instead, most interesting quantum field theories are best described as "morphisms" of some sort between functors from the category of spacetimes --- these are called "twisted" or "relative" or "Heisenberg picture" quantum field theories. The most natural notion of "morphisms of functors" is "natural transformation." Unfortunately, plain (i.e. "strong") natural transformations still fail to accommodate most examples. Instead, what is needed are "lax" or "oplax" natural transformations. In this talk, based on joint work with Claudia Scheimbauer, I will describe the definition of "(op)lax natural transformation" between functors of higher categories, and discuss qualitative differences between "lax" and "oplax" twisted quantum field theories. (hide abstract)
Functorial axioms for Heisenberg-picture quantum field theory. January 12, CRG Geometry and Physics Seminar, UBC, Vancouver, BC. (abstract, handout)

Abstract: The usual Atiyah–Segal "functorial" description of quantum field theory corresponds to the "Schrodinger picture" in quantum mechanics. I will describe a slight modification that corresponds to the "Heisenberg picture", which I will argue is better physically motivated. The example I am most interested in is a version of quantum Chern–Simons theory that does not require the level to be quantized; it provides a neat packaging of pretty much all objects of skein theory. (hide abstract)

2014

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))

Abstract: The usual Atiyah–Segal axioms describe quantum field theory in terms of a "Schrodinger picture" of physics. I will argue that instead a "Heisenberg picture" is needed, and describe a small modification of those axioms that accommodates this. As an example, I will describe a skein-theoretic version of quantum Chern-Simons theory as a "fully extended oriented Heisenberg-picture tqft". It has the feature that it does not require the "level" to be quantized. It provides in particular a tqft packaging of skein theory, and my hope is that it will shed light on open conjectures in quantum topology. Bits of my talk will be based on joint work with M. Brandenburg, A. Chirvasitu, and C. Scheimbauer. (hide abstract)
The CS-WZW correspondence. October 30, CFT Seminar, Northwestern University, Evanston, IL. (abstract, notes)

Abstract: My overall goal is to at least explain an assertion that I have tied to Freed–Teleman that chiral WZW is not an absolute theory, but instead a relative theory valued in Chern–Simons theory. To get there, I will ramble for a while about "Heisenberg-picture field theory", and at least give a definition of Chern–Simons theory. I don't really have a complete definition of (quantum) WZW, but I do know what classical WZW theory is, and I'll end by giving the classical story of the correspondence, in which chiral WZW fields are a Lagrangian inside Chern–Simons fields. I will not have time to say anything important, like that this relates the space of conformal blocks for chiral WZW to the Hilbert space for Chern–Simons, because I will instead spend too much time being a bit polemical about how to set up the category theory necessary for qft. (hide abstract)
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)

Abstract: I will describe a version of the AKSZ construction that applies to possibly-open source manifolds and to possibly-infinite-dimensional Poisson (as opposed to symplectic) target manifolds (the cost being that the target must be infinitesimal). Quantization of such theories has to do with the relationship between dioperads and properads, and to the fact (due to Merkulov and Vallette) that formality in one world does not imply formality in the other. In particular, universal quantization of AKSZ theories on R^d is equivalent to the formality of a certain properad which is formal as a dioperad. I will conjecture that it is also equivalent to formality of the E_d operad. (hide abstract)
Lie bialgebra quantization in 2- and 3-dimensional field theory. May 28, Associators, Formality and Invariants Seminar, Northwestern, Evanston, IL. (abstract, notes)

Abstract: My goal for this talk is to describe "in pictures" a connection between the Etingof--Kazhdan and Tamarkin proofs of the existence of functorial quantization of Lie bialgebras. As I will explain, a Lie bialgebra provides the data for a 2- and 3-dimensional perturbative topological field theory --- a 3-dimensional field theory "of AKSZ type", a 2-dimensional field theory "of Poisson AKSZ type", and a way for the 2-dimensional theory to live as a "boundary field theory" for the 3-dimensional one. I will argue that Tamarkin's proof involves directly quantizing (the factorization algebra associated to) the 2-d theory (with certain boundary conditions), whereas the Etingof--Kazhdan proof involves quantizing (the Wilson lines in) the 3-d theory (again with appropriate boundary conditions). Joint with Owen Gwilliam. (hide abstract)
The Jones polynomial and the Temperley--Lieb TQFT. May 9, Graduate Student Seminar, Northwestern, Evanston, IL. (abstract, notes)

Abstract: The Jones polynomial was the first of by now many connections between low-dimensional topology and quantum groups. This talk will only barely touch the latter of these; instead, I will focus on the Jones polynomial and its close cousin, the Temperley--Lieb category. Indeed, Jones originally discovered his polynomial by investigating the algebras that Temperley and Lieb had written down, and I will give an ahistorical version of that discovery. My goal for the talk will be to put these objects, as well as many related objects from low-dimensional topology (going by names like "skein algebra" and "space of SL(2) local systems" and "quantum A polynomial"), into their natural packaging: a structure I call the "Temperley--Lieb TQFT". (hide abstract)
Heisenberg-picture TQFTs. May 2, Representation Theory, Geometry, and Combinatorics, UC Berkeley. (abstract, notes)

Abstract: The Atiyah–Segal axioms for quantum field theory generalize the "Schrödinger picture" of quantum mechanics (Hilbert spaces of states, partition functions, etc.). I will describe a small modification that corresponds instead to the "Heisenberg picture" (algebras of observables). As examples, I will describe some versions of "fully-extended" quantum Chern–Simons Theory: one built from the category of comodules of a quantized function algebra, and another built from the Temperley–Lieb category. The latter is defined over ℤ and fully extended, and (perhaps most importantly) "at generic level." (hide abstract)
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)

Abstract: I describe a generalization of the AKSZ construction of topological field theories to allow targets with possibly-degenerate up-to-homotopy Poisson structure. The construction requires investigating in what sense the chains on an oriented manifold carry a chain-level homotopy Frobenius structure. There are two versions of the construction: a "classical field theory" tree-level version, and a "quantum field theory" graph-level version. The tree-level version is well-behaved for all possible spacetimes and targets. The graph-level version is much more subtle, and intimately connected to the "formality" or "quantization" problem for the operad of little n-dimensional disks. (hide abstract)
Up-to-homotopy Frobenius structures on manifolds, and how they relate to perturbative QFT. January 22, Topology Seminar, UC Berkeley. (abstract, handout)

Abstract: The de Rham homology of an oriented manifold carries a well-known graded-commutative Frobenius algebra structure. Does this structure lift in a geometrically meaningful up-to-homotopy way to de Rham chains? The answer depends on the meanings of "geometrically meaningful" and "up-to-homotopy". I will describe two potential choices for the meanings of these words. Using the first choice, the answer to the question is always Yes. Using the second gives a more subtle situation, in which the answer is No in dimension 1, and related to the formality of the E_n operad in dimension n>1. To explain this relationship (and my interest in the problem) requires a short sojourn in the world of perturbative topological quantum field theory. (hide abstract)

2013

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

Abstract: In the late 1990s, Alexandrov, Kontsevich, Schwartz, and Zaboronsky introduced a very general construction of classical field theories of "topological sigma model" or "Chern--Simons" type, which is well-adapted to quantization in the Batalin--Vilkovisky formalism. I will describe a generalization, which is to the usual AKSZ construction as "Poisson" is to "symplectic". The perturbative quantization problem for such field theories includes the problem of wheel-free universal deformation quantization and the Etingof--Kazhdan quantization of Lie bialgebras; more generally, it has to do with the formality problem for the E_n operads. The terms "properad" and "Koszul duality" will also make appearances in my talk. (hide abstract)
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)

Abstract: I will describe a Poisson generalization of the AKSZ construction of topological field theories. This version of "classical" AKSZ theory exists for all oriented spacetimes, and resides in the world of dioperads and "quasilocal" factorization algebras. The quantization problem is generically obstructed; as I will discuss, "quantum" AKSZ theories are from the world of properads. The quantization problem is closely related to the formality problem for the En operad. It is also closely related to the question of finding a geometrically-meaningful properadic homotopy-Frobenius structure at the chain level, lifting the Frobenius-algebra structure on the homology of spacetime. (hide abstract)
Poisson AKSZ theory, properads, and quantization. October 10, Geometry and Physics, Northwestern, Evanston. (abstract, handout)

Abstract: In the late 1990s, Alexandrov, Kontsevich, Schwartz, and Zaboronsky introduced a very general construction of classical field theories of "topological sigma model" or "Chern--Simons" type, which is well-adapted to quantization in the Batalin--Vilkovisky formalism. I will describe a generalization, which is to the usual AKSZ construction as "Poisson" is to "symplectic". The perturbative quantization problem for such field theories includes the problem of wheel-free universal deformation quantization and the Etingof--Kazhdan quantization of Lie bialgebras; more generally, it has to do with the formality problem for the E_n operads. The technical tool needed to pose the quantum construction is the theory of properads (the classical construction corresponds to their genus-zero part, namely dioperads). This leads to a conjectured properadic description of the space of formality quasiisomorphisms for E_n. (hide abstract)
A properad action on homology that fails to lift to the chain level. October 10, Geometry and Physics Pre-talk, Northwestern, Evanston. (abstract, handout)

Abstract: A tenet of algebraic topology is that algebraic structures on the homology of a space should correspond to structures at the chain level, such that the axioms that hold on homology are weakened to coherent homotopies. For example, the homology of an oriented manifold is a Frobenius algebra --- what about the chains? In this talk, I will explain that for one-dimensional manifolds, the answer is No. I will save comenting on higher-dimensional manifolds for the 4pm talk.

To make this precise, I will spend some time discussing the notion of "properad", which generalizes the notion of "operad" to allow many-to-many operations. I will recall the Koszul duality for properads, and how to compute cofibrant replacements. I will not assume that the words "Koszul duality" or "cofibrant replacement" are particularly familiar. (hide abstract)
A properadic approach to the deformation quantization of topological field theories. September 25, Algebra and Combinatorics, Loyola University, Chicago. (abstract, notes)

Abstract: I will describe how Koszul duality and the bar construction for properads is related to the path integral quantization of topological field theories. As an application, I will give a class of Poisson structures that admit a canonical wheel-free deformation quantization. (hide abstract)
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)

Abstract: The "Batalin–Vilkovisky formalism" is a collection of algebraic structures that largely subsume the theory of oscillating integrals. As an appetizer, my talk will begin by motivating this formalism through an investigation of finite-dimensional integrals and integration by parts, in the "semiclassical" (or "rapidly oscillating") limit. Along the way, we will be led to invent Feynman diagrams, and we will find ourselves with a totally algebraic understanding of the (in)dependence of Feynman diagrams on the choice of coordinates, the choice of "gauge fixing", etc.

From here, my story moves into two branches, and how much of each branch I explain will depend on audience appetite. Of course, time permitting I will explain both, as they combine to give new results into deformation quantization problems. The antipasto has to do with "dioperads" and "properads", which are algebraic structures for which the multiplication is controlled by certain graphs (just like multiplication in associative algebras is controlled by putting beads on a string). In particular, Koszul duality for properads provides a host of examples of BV / Feynman diagrmamatic "integrals."

The main entree has to do with topological field theory. Factorization algebras are a "quantum" version of sheaves --- what makes them "quantum" is that a version of the Heisenberg uncertainty principle disallowing simultaneous measurements is build into the axioms. They provide a framework for understanding a deep construction of topological field theories due to Alexandrov, Kontsevich, Schwarz, and Zaboronsky. (A quantum field theory is "topological" if the classical equations of motion are "the fields are constant.") The AKSZ construction realizes important models, including topological quantum mechanics and Chern--Simons Theory, within the BV formalism. The properadic story from the second part provides new examples, and relates the classical and quantum AKSZ constructions to questions from algebraic topology about lifting Poincare duality to the chain level.

We will surely be too full for dessert, but time permitting I would love to describe an example using all of the above machinery: topological quantum mechanics valued in a Poisson manifold. The quantization problem for this theory is generically obstructed --- it is essentially equivalent to the problem of finding a "wheel free universal deformation quantization," and these do not exist. The properadic / BV / AKSZ story identifies (modulo combinatorial calculations that are too hard to do by hand, but should be trivial on a correctly-programmed computer) exactly which Poisson structures do admit a wheel-free deformation quantization. (hide abstract)
Star quantization via lattice topological field theory. June 18, String-Math, Simons Center for Geometry and Physics, Stony Brook. (abstract, slides)

Abstract: The deformation quantization problem for Poisson manifolds is well-known, and famously answered by Kontsevich more than a decade ago. I will describe a new, purely combinatorial, construction of deformation quantizations of infinitesimal Poisson manifolds. It is closely related to the "factorization algebra" perspective on effective quantum field theory recently introduced by Costello and Gwilliam, and also to a new "lattice" version of topological field theories of AKSZ type — time permitting, I will try to describe these connections. (hide abstract)
Lattice Poisson AKSZ Theory. February 4, Algebraic Geometry Seminar, University of British Columbia. (abstract, handout)

Abstract: AKSZ Theory is a topological version of the Sigma Model in quantum field theory, and includes many of the most important topological field theories. I will present two generalizations of the usual AKSZ construction. The first is closely related to the generalization from symplectic to Poisson geometry. (AKSZ theory has already incorporated an analogous step from the geometry of cotangent bundles to the geometry of symplectic manifolds.) The second generalization is to phrase the construction in an algebrotopological language (rather than the usual language of infinite-dimensional smooth manifolds), which allows in particular for lattice versions of the theory to be proposed. From this new point of view, renormalization theory is easily recognized as the way one constructs strongly homotopy algebraic objects when their strict versions are unavailable. Time permitting, I will end by discussing an application of lattice Poisson AKSZ theory to the deformation quantization problem for Poisson manifolds: a _one_-dimensional version of the theory leads to a universal star-product in which all coefficients are rational numbers. (hide abstract)

2012

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

Abstract: In this two-day guest-lecture in a semester-long class on quantum field theory, I describe some of my work on the Feynman-diagram expansion of the path integral in quantum mechanics. I begin by motivating the path integral, and then spend most of the time recalling the diagrammatic description of the asymptotics of finite-dimensional oscillating integrals. (hide abstract)
Nonperturbative integrals, imaginary critical points, and homological perturbation theory. August 28, New Perspectives in Topological Field Theories, Center for Mathematical Physics, Hamburg. (abstract, notes)

Abstract: The method of Feynman diagrams is a well-known example of algebraization of integration. Specifically, Feynman diagrams algebraize the asymptotics of integrals of the form ∫ f exp(s/h) in the limit as h→0 along the pure imaginary axis, supposing that s has only nondegenerate critical points. (In quantum field theory, s is the "action," and f is an "observable.") In this talk, I will describe an analogous algebraization when h=1 --- no formal power series will appear --- and s is allowed degenerate critical points. Nevertheless, some features from Feynman diagrams remain: I will explain how to algebraically "integrate out the higher modes" and reduce any such integral to the critical locus of s; the primary tool will be a homological form of perturbation theory (itself almost as old as Feynman's diagrams). One of the main new features in nonperturbative integration is that the critical locus of s must be interpreted in the scheme-theoretic sense, and in particular imaginary critical points do contribute. Perhaps this will shed light on questions like the Volume Conjecture, in which an integral over SU(2) connections is dominated by a critical point in SL(2,ℝ). (hide abstract)
Nonperturbative integrals, imaginary critical points, and homological perturbation theory. August 24, QGM Lunch Seminar, Aarhus. (abstract, notes)

Abstract: The method of Feynman diagrams is a well-known example of algebraization of integration. Specifically, Feynman diagrams algebraize the asymptotics of integrals of the form ∫ f exp(s/h) in the limit as h→0 along the pure imaginary axis, supposing that s has only nondegenerate critical points. (In quantum field theory, s is the "action," and f is an "observable.") In this talk, I will describe an analogous algebraization when h=1 --- no formal power series will appear --- and s is allowed degenerate critical points. Nevertheless, some features from Feynman diagrams remain: I will explain how to algebraically "integrate out the higher modes" and reduce any such integral to the critical locus of s; the primary tool will be a homological form of perturbation theory (itself almost as old as Feynman's diagrams). One of the main new features in nonperturbative integration is that the critical locus of s must be interpreted in the scheme-theoretic sense, and in particular imaginary critical points do contribute. Perhaps this will shed light on questions like the Volume Conjecture, in which an integral over SU(2) connections is dominated by a critical point in SL(2,ℝ). (hide abstract)
Wick-type theorems beyond the Gaussian. March 2, Representation Theory (and related topics) seminar, Northeastern. (abstract, handout, notes)

Abstract: Wick's theorem, proven by Isserlis in 1918, provides simple algebraic relations describing the moments (i.e. correlation functions, expectation values) of a Gaussian probability measure in terms of the quadratic moments. One can ask for similar explicit relations for probability measures of the form exp(cubic)dx or even higher-degree homogeneous polynomials in the exponent. In this talk I will present a homological-algebraic approach to finding such relations, based ultimately on a derived-geometry interpretation of Batalin--Vilkovisky integration. This is joint work with Owen Gwilliam and joint work in progress with Shamil Shakirov. (hide abstract)
Twisted N=1 and N=2 supersymmetry on R^4. February 10, GRASP seminar, UC Berkeley. (abstract, notes)

Abstract: The goal of this talk is to explain the title. In a little more detail, I will define the N=N super-translation and super-Poincare groups for R^4, including what is an "R-symmetry". I will then define what is "twisting data" for a supersymmetric theory, and why "twisting" a theory makes it simpler. Generic "twists" for N=2 supersymmetric theories on R^4 make it topological, but the most interesting twists make it holomorphic. This talk is an attempt to understand some talks by Kevin Costello, and contains no material due to me. (hide abstract)

2011

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)

Abstract: We describe the "BRST / Faddeev–Popov gauge-fixing" definition of integrals on (the quotient stack of) a Lie algebroid. As a central example, we compute the volume of the de Rham stack of a compact manifold. In the process, we find a new proof of the Chern–Gauss–Bonnet theorem. This is joint work with Dan Berwick-Evans. (hide abstract)
Introduction to BV Integrals. November 1, Selected topics in classical and quantum geometry, Universität Zürich. (abstract, handout)

Abstract: The BV method describes integrals (and in particular asymptotics of expectation values against rapidly oscillating measures) purely in terms of (homological) algebra, with the goal being to use this algebraic description as a definition of "integral" for generalized manifolds (stacks, infinite-dimensional spaces, etc.). In the first part of this talk, I will describe the translation of expectation values into homological algebra, and (somewhat telegraphically) mention the connections with super (Gerstenhaber and derived) geometry. In the second part of the talk, I will discuss some combinatorial and algebraic methods for carrying out the actual computations: one can directly derive the usual Feynman diagrams, or one can apply more general homological perturbation theory. The material in this talk is essentially "well-known" (the first part of the talk is based a paper of Witten's from 1990), and the Feynman diagrammatics I learned in joint work with Owen Gwilliam. (hide abstract)
Asymptotics of oscillating integrals via homological perturbation theory. October 19, GRASP seminar, UC Berkeley. (abstract, handwritten notes, too-long typed notes)

Abstract: The Batalin-Vilkovisky approach to integration converts the question of computing expectation values into a question in homological algebra, and reinterprets the asymptotics of oscillating integrals in terms of (quantum) deformations of (derived) intersections. The move to homological algebra makes these computations tractable by combinatorial means — a special case includes the Feynman-diagrammatic description of Gaussian integration. In this talk, I will try to explain both the derived geometry and the homological perturbation theory. Most of this story is known to experts, and a little of it is joint work with Owen Gwilliam. (hide abstract)
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)

Abstract:
Oct 11: Let X be a manifold equipped with a Lie algebra (or Lie algebroid) action. The derived quotient of X can be realized as a Q-manifold over X; Q-manifolds are (Z-graded) supermanifolds equipped with "cohomological" vector fields, and are a piece of derived geometry. I will recall the motation and definition. This talk is essentially contained within R. Mehta's thesis.
Oct 13: I will discuss what it would mean to "integrate" over a Q-manifold. The BRST argument explains how to improve a priori ill-defined integrals. The talk will conclude with a discussion of the "Faddeev-Popov construction" for Q-manifolds that arise from Lie algebroids. The description of the Faddeev-Popov construction is joint work with Dan Berwick-Evans (in prep).
Oct 18: Let X be a manifold. Denote the derived quotient of X modulo its tangent bundle by X_dR, as it is "spec" of the ring of de Rham forms on X. This derived manifold is formally zero-dimensional (if X is contractible, then X_dR is equivalent to a point), and so ought to be equipped with a canonical "counting measure". We will compute this measure by BRST gauge fixing. Along the way we will come up with a slick proof of the Chern–Gauss–Bonnet formula. A version of this argument will appear in the thesis by Dan Berwick-Evans; the version I will present is our joint work.
Oct 20: BRST gauge fixing ideas can be applied to topological field theories (with degenerate actions). In one dimension, BRST gauge fixing gives a heuristic proof of the Morse–de Rham equivalence. In two dimensions, BRST gauge fixing should give Witten's topological sigma model and Gromov-Witten theory. This talk will mostly follow papers by Rogers and Baulieu and Singer. Note: because of a schedule conflict, I did not end up giving this talk, and do not have completed notes.
(hide abstract)
(Topological) duality of Hopf algebras. June 13, Cluster Algebras and Lusztig's Semicanonical Basis, University of Oregon. (abstract, notes)

Abstract: I will begin by telling you what a "group" is, in a language that makes it easy to think about Lie groups, algebraic groups, universal enveloping algebras, etc., all at the same time. I will then tell you that the universal enveloping algebra of the Lie algebra T_eG of a Lie group G "is" the subgroup of G consisting of "the points infinitely close to the identity e∈G. To make this inclusion precise, I will describe the corresponding pairing between the universal enveloping algebra and the algebra of smooth functions. Replacing "smooth" with "polynomial" or "analytic" and forcing G to be commutative, we get a perfect pairing.

A perfect pairing isn't quite as good as you really want, because the structures involved are infinite-dimensional. In the case when G is the group of upper-triangular matrices (with 1s on the diagonal) you can do better: there are natural gradings on the universal enveloping algebra and on the algebra of polynomial functions, and each graded piece is finite-dimensional, and then the two Hopf algebras are precise graded duals of each other. (hide abstract)
Homological perturbation and factorization algebras. May 26, Geometry/Physics Seminar, Northwestern University. (abstract, notes)

Abstract: Much of quantum field theory concerns questions with the following flavor: you have some "classical" data, and you make a "small perturbation" to some part of it; how can you compatibly perturb the rest of the data to preserve some structure? One version of this question was solved in the 60s: given a homotopy equivalence of chain complexes and a small perturbation to the differential on the large complex, the homotopy perturbation lemma provides formulae that compute compatible perturbations to the differential on the small complex and to all the maps in the homotopy equivalence. In this talk I will recall this lemma, and then illustrate it with some examples from low-dimensional "topological" factorization algebras, where the homological perturbation lemma can be used to: compute asymptotics of oscillating integrals ("Feynman diagrams"); construct Weyl, Clifford, and Universal Enveloping algebras; explain how a topological quantum field theory on the bulk of a manifold can induce a tqft on the boundary. (hide abstract)
On Atoms, Mountains, and Rain. April 20, NUMS Seminar, Northwestern University. (abstract)

Abstract: This talk consists almost entirely of lies. A few lies we will tell: rocks are made of rock atoms, liquid water is a perfect cubic crystal lattice, and 1 = 2. Using these lies, we will derive from first principles the radius of an atom, the height of a mountain, and the volume of a raindrop. Doing so honestly, even if we knew all the fundamental equations of the universe, would be impossible; lying makes everything work out nicely. The talk is based on P. Goldreich, S. Mahajan, and S. Phinney, Order-of-Magnitude Physics: Understanding the World with Dimensional Analysis, Educated Guesswork, and White Lies, 1999. (hide abstract)
Feynman Diagrams for Schrodinger's Equation. Feb 15, GADGET Seminar, UT Austin. (abstract, handout)

Abstract: Feynman's path integral, an important formalism for quantum mechanics, lacks a completely satisfactory analytic definition. One possible definition is as a formal power series whose coefficients are given by sums of finite-dimensional integrals indexed by Feynman diagrams. This ``formal'' path integral is used extensively in every-day physics, but is not usually compared against (mathematically rigorous, nonperturbative) quantum mechanics. In this talk, I will explain the definition of the quantum-mechanical formal path integral, and point out many of its features --- it has ultraviolet divergences unless certain compatibility conditions are met, it is coordinate-independent, it solves Schrodinger's equation --- none of which are obvious from the definitions, but rather require the combinatorics of Feynman diagrams. These results provide justification for the formal path integrals in quantum field theory. (hide abstract)
E₂ operad, Gerstenhaber and BV, and Formality. Feb 10, Student String Topology Seminar, UC Berkeley. (abstract, notes, handout)

Abstract: I will briefly recall the notion of an operad, and then focus on the E₂ or "little 2-disks" operad (in spaces), and its framed cousin. Calculating its homology recovers the Gerstenhaber operad (in graded vector spaces), with the correct signs — most descriptions of Gerstenhaber have unfortunate sign conventions — or with framing the BV operad. I will then prove the Formality Theorem for (framed) E₂: as operads of dg vector spaces, the operad of simplicial chains in (framed) E₂ is quasiisomorphic to its homology. I will follow the Tamarkin/Severa proof, which requires developing some of the very rich theory of Drinfel'd associators. (hide abstract)

2010

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

Abstract: Tannaka-Krein theory asks two main questions: (Reconstruction) What about an algebraic object can you determine based on knowledge about its representation theory? (Recognition) Which alleged "representation theories" actually arise as the representation theories of algebraic objects? In this talk I'll mention some answers to the second question, but I'll focus more on the first. The punchline: essentially everything, provided you remember the underlying spaces of your representations --- there is an almost perfect dictionary between algebraic structures and categorical structures. My goal is to explain the results in as elementary and pared-down a way as possible, so the talk will be more or less reverse-chronological. The only prerequisite is some brief acquaintance with the following two-categories: (Category, Functor, Natural Transformation) and (Algebra, Bimodule, Intertwiner). The main Tannaka-Krein story that I will present is ``twentieth century'' and by now well known, but time permitting I will also mention some joint work in progress with Alex Chirvasitu. (hide abstract)
Formal calculus, with applications to quantum mechanics. Sept 10, GRASP seminar, UC Berkeley. (abstract, notes).

Abstract: "Formal" or "Feynman diagrammatic" calculus is nothing more nor less than the differential and integral calculus of formal power series. The latter name is because Feynman's diagrams provide a convenient notation for manipulating formal power series and for understanding their combinatorics. In this talk, I will outline the formal calculus, and then use it to write out the "path integral" description of the asymptotics of the time-evolution operator in quantum mechanics. The diagrammatics make it much easier to prove that the "path integral" is well-defined and satisfies the necessary requirements. (hide abstract)
Introduction to Vassiliev invariants. April 2, GRASP seminar, UC Berkeley. (abstract).

Abstract: "Vassiliev" or "finite-type" knot invariants include (up to a change of coordinates) most of the popular knot invariants (HOMFLYPT, ...). But they are also closely related to Lie algebraic questions. I will give an introduction to this story. (hide abstract)
How to quantize infinitesimally-braided symmetric monoidal categories. March 19, Subfactors Seminar, UC Berkeley. (abstract, notes).

Abstract: An infinitesimal braiding on a symmetric monoidal category is analogous to a Poisson structure on a commutative algebra: both tell you a "direction" in which to "quantize". In this expository talk, I will tell a story that was completed by the end of the 1990s, concerning the quantization problem for infinitesimally-braided symmetric monoidal categories. Along the way, other main characters will include: a Lie algebra, a quadratic Casimir, and a classical R-matrix; braided monoidal categories, associators, and pentagons and hexagons; Tannakian reconstructions theorems and Hopf and quasiHopf algebras; and everyone's favorite knot invariants. I'll explain all these words, and try to explain how they're all part of a single story. (hide abstract)
The Formal Path Integral in Quantum Mechanics. Feb 26, Subfactors Seminar, UC Berkeley. (abstract, slides).

Abstract: In his thesis (first published in 1948), Richard Feynman suggested a new formalism for quantum mechanics, now called the "Feynman Path Integral." Feynman knew that defining his path integral analytically would be difficult: modern analytic definitions generally start with a Wiener measure and place restrictions on the corresponding classical mechanical system. But within a few years Feynman and Freeman Dyson had defined a "perturbative" path integral: they declared the value of the integral to be a formal power series whose coefficients were given by sums of finite-dimensional integrals indexed by "Feynman diagrams." These days, this "formal" path integral is used extensively in every-day physics, and provided some of the first "quantum" knot invariants. However, it has not been compared carefully against (mathematically rigorous, nonperturbative) quantum mechanics.

In this talk, I will explain the definition of the quantum-mechanical formal path integral, and point out many of its features — it has ultraviolet divergences unless certain compatibility conditions are met, it is coordinate-independent, it solves Schrödinger's equation — none of which are obvious from the definitions, but rather require the combinatorics of Feynman diagrams. These results provide justification for the formal path integral. (hide abstract)

2009

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

Abstract: The goal of the talk is to introduce the notion of "Feynman Diagram" in a reasonably rigorous way, and to state some theorems proving that it is a good notion. I will organize the talk more-or-less via a "mathematician's history of mathematics," which is to say a false history, one that gives the impression that all ideas inevitably lead up to what we now know is the true and complete story. Thus, I will begin by describing why you might invent Feynman Diagrams. I'll then tell you about what the mathematicians have said about them. Time permitting, I'll finish with some speculation of my own. (hide abstract)
On Atoms, Mountains, and Rain. March 31, Many Cheerful Facts, UC Berkeley. (abstract, notes).

Abstract: This talk won't include very many facts, but it will include many almost facts, aka "lies". A few lies we will tell: rocks are made of rock atoms, liquid water is a perfect cubic crystal lattice, and 1 = 2. Using these and similar "facts", we will derive from first principles the radius of an atom, the height of a mountain, and the volume of a raindrop. Doing so honestly, even if we knew all the fundamental equations of the universe, would be impossible; lying makes everything work out nicely.

The material is almost entirely from to P. Goldreich, S. Mahajan, and S. Phinney, Order-of-Magnitude Physics: Understanding the World with Dimensional Analysis, Educated Guesswork, and White Lies, 1999. Available at http://www.inference.phy.cam.ac.uk/sanjoy/oom/. (hide abstract)

2007

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

Abstract: The goal of this expository talk is to formulate and prove the Yoneda embedding theorem for categories enriched over a closed monoidal category. The material for this talk is almost entirely from G.M. Kelly, Basic Concepts of Enriched Category Theory, Cambridge University Press, 2005. (hide abstract)
Divergent Series. October 18, Many Cheerful Facts, UC Berkeley. (abstract, notes).

Abstract: Mathematicians through the ages have varied from terrified of divergent sums to only mildly scared of them: Euler, most famously, made great use of divergent series, whereas Abel called them "the invention of the devil". In this talk, I will survey the most important methods of summing divergent series, and make general vague remarks about them. I will quote many results, but will studiously avoid proving anything. The material is almost entirely from G.H. Hardy, Divergent Series, 1949. (hide abstract)