Theo Johnson-Freyd

Boas Assistant Professor and NSF Postdoctoral Fellow,
Department of Mathematics, Northwestern University

e-mail: theojf at math dot northwestern dot edu
Office: 312 Lunt Hall, Northwestern University, Evanston, IL, 60202
Office Hours: Tuesdays and Thursdays, 12–2pm.

Fall 2015 Teaching: First-year Seminar: Theories of mind and mathematics.

Brief bio

My primary research is in an area I would call "topological physics": a mathematical physicist, I focus on quantum field theory, topological field theory, perturbative quantization, category theory, representation theory, and algebraic topology. A more detailed description of my current research interests may be found by reading my recent research statement for the 2015-16 hiring season. For even more details, see my publications and seminar presentations. Of course, if you want fewer details, perhaps you want my Curriculum Vitae (PDF) or perhaps the shorter version; they contain a strict subset of the information on this webpage.

I received my PhD from UC Berkeley in 2013, under the supervision of Nicolai Reshetikhin. As an NSF postdoc, my sponsoring scientist at Northwestern is Kevin Costello. Other collaborators include Alex Chirvasitu, Owen Gwilliam, and Claudia Scheimbauer.

Contents

• Research papers: publications, submitted for publication, other preprints
• Other documents: draft textbook, liveTeXed notes, grant and job application materials, miscellaneous mathematics
• Talks and Seminars: 2015, 2014, 2013, 2012, 2015, 2015, 2010, 2009, 2008, 2007,
• Teaching: at Northwestern, at Berkeley
• Non-math

Research papers

Published

1. Homological perturbation theory for nonperturbative integrals. Letters in Mathematical Physics, November 2015, Volume 105, Issue 11, pp 1605-1632. (abstract, arXiv: 1206.5319, DOI: 10.1007/s11005-015-0791-9.)

   Abstract: In this paper we study integrals of the form ∫_γ f e^s, where f and s are complex polynomials of n variables and γ ⊆ Cⁿ is an n-real-dimensional contour along which e^s enjoys exponential decay. Suppose s is generic of degree d. Using homological algebra, we automate the method of ``integration by parts,'' and show how to express any such integral as a linear combination of integrals of monomials which are of degree < d-1 in each variable. We conjecture that for generic contour γ the values of these (d-1)ⁿ integrals are inaccessible to pure algebra.
   
   More generally, we explain how homological algebra allows to ``integrate out the high-energy modes'' to turn any such integral problem into an integral over the scheme-theoretic critical locus {d s = 0}. Thus concentration onto the critical locus is not only a perturbative phenomenon. Our primary tool in this paper is the Homological Perturbation Lemma, which when applied to perturbative integrals recovers the method of Feynman diagrams --- ``perturbation theory'' is another not-only-perturbative phenomenon. Our motivation for this paper is to better understand the ``path'' integrals that appear in quantum field theory, and we make a few brief comments about these at the end. (hide abstract)

2. Tree- versus graph-level quasilocal Poincaré duality on S¹. Journal of homotopy and related structures, published online 20 May 2015. (abstract, arXiv: 1412.4664, DOI: 10.1007/s40062-015-0110-2.)

   Abstract: Among its many corollaries, Poincaré duality implies that the de Rham cohomology of a compact oriented manifold is a shifted commutative Frobenius algebra — a commutative Frobenius algebra in which the comultiplication has cohomological degree equal to the dimension of the manifold. We study the question of whether this structure lifts to a "homotopy" shifted commutative Frobenius algebra structure at the cochain level. To make this question nontrivial, we impose a mild locality-type condition that we call "quasilocality": strict locality at the cochain level is unreasonable, but it is reasonable to ask for homotopically-constant families of operations that become local "in the limit." To make the question concrete, we take the manifold to be the one-dimensional circle.
   
   The answer to whether a quasilocal homotopy-Frobenius algebra structure exists turns out to depend on the choice of context in which to do homotopy algebra. There are two reasonable worlds in which to study structures (like Frobenius algebras) that involve many-to-many operations: one can work at "tree level," corresponding roughly to the world of operadic homotopy algebras and their homotopy modules; or one can work at "graph level," corresponding to the world of PROPs. For the tree-level version of our question, the answer is the unsurprising ``Yes, such a structure exists'' — indeed, it is unique up to a contractible space of choices. But for the graph-level version, the answer is the surprising "No, such a structure does not exist." Most of the paper consists of computing explicitly this nonexistence, which is controlled by the numerical value of a certain obstruction, and we compute this value explicitly via a sequence of integrals.
   
   Note: This paper retells a story that I tried to tell in "Chains(R) does not admit a geometrically meaningful properadic homotopy Frobenius algebra structure", and I have lifted some discussion of properads and their Koszul duality theory (notably, most of Section 3) from that older version. This new version has been expanded and improved, and essentially all conventions have been changed. (hide abstract)

3. Reflexivity and dualizability in categorified linear algebra. With Martin Brandenburg and Alexandru Chirvasitu. Theory and Applications of Categories, Vol. 30, No. 23, 2015, pp. 808–835. (abstract, arXiv: 1409.5934, published version (open access).)

   Abstract: The "linear dual" of a cocomplete linear category C is the category of all cocontinuous linear functors C→VECT. We study the questions of when a cocomplete linear category is reflexive (equivalent to its double dual) or dualizable (the pairing with its dual comes with a corresponding copairing). Our main results are that the category of comodules for a countable-dimensional coassociative coalgebra is always reflexive, but (without any dimension hypothesis) dualizable if and only if it has enough projectives, which rarely happens. Along the way, we prove that the category Qcho(X)$ of quasi-coherent sheaves on a stack X is not dualizable if X is the classifying stack of a semisimple algebraic group in positive characteristic or if X is a scheme containing a closed projective subscheme of positive dimension, but is dualizable if X is the quotient of an affine scheme by a virtually linearly reductive group. Finally we prove tensoriality (a type of Tannakian duality) for affine ind-schemes with countable indexing poset. (hide abstract)

4. Poisson AKSZ theories and their quantizations. In Proceedings of the conference String-Math 2013, volume 88 of Proceedings of Symposia in Pure Mathematics, pages 291--306, Providence, RI, 2014. Amer. Math. Soc. (abstract, PDF (published version), arXiv: 1307.5812, DOI: 10.1090/pspum/088.)

   Abstract: We generalize the AKSZ construction of topological field theories to allow the target manifolds to have possibly-degenerate (homotopy) Poisson structures. Classical AKSZ theories, which exist for all oriented spacetimes, are described in terms of dioperads. The quantization problem is posed in terms of extending from dioperads to properads. We conclude by relating the quantization problem for AKSZ theories on R^d to the formality of the E_d operad, and conjecture a properadic description of the space of E_d formality quasiisomorphisms. (hide abstract)

5. The fundamental pro-groupoid of an affine 2-scheme. With Alex Chirvasitu. Applied Categorical Structures, Vol 21, Issue 5 (2013), pp. 469–522. (abstract, arXiv: 1105.3104, DOI: 10.1007/s10485-011-9275-y).

   Abstract: A natural question in the theory of Tannakian categories is: What if you don't remember Forget? Working over an arbitrary commutative ring R, we prove that an answer to this question is given by the functor represented by the étale fundamental groupoid π₁(spec(R))$, i.e. the separable absolute Galois group of R when it is a field. This gives a new definition for étale π₁(spec(R))$ in terms of the category of R-modules rather than the category of étale covers. More generally, we introduce a new notion of "commutative 2-ring" that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of π₁ for the corresponding "affine 2-schemes." These results help to simplify and clarify some of the peculiarities of the étale fundamental group. For example, étale fundamental groups are not "true" groups but only profinite groups, and one cannot hope to recover more: the "Tannakian" functor represented by the étale fundamental group of a scheme preserves finite products but not all products. (hide abstract)

6. The formal path integral and quantum mechanics. Journal of Mathematical Physics, 51, 122103 (2010). (abstract, published PDF, DOI:10.1063/1.3503472, arXiv: 1004.4305, equation and theorem numbering differs between preprint and published versions).

   Abstract: Given an arbitrary Lagrangian function on ℝ^d and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by "Feynman diagrams," although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a "Fubini theorem" expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous-quadratic in velocity such that its homogeneous-quadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by "cutting and pasting" and choosing volume-compatible local coordinates, our construction defines a Feynman-diagrammatic "formal path integral" for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field. (hide abstract)

7. Feynman-diagrammatic description of the asymptotics of the time evolution operator in quantum mechanics. Letters in Mathematical Physics, November 2010, Volume 94, Issue 2, pp 123-149. (abstract, arXiv: 1003.1156, available Open Access from Springer Link at DOI: 10.1007/s11005-010-0424-2).

   Abstract: We describe the "Feynman diagram" approach to nonrelativistic quantum mechanics on ℝⁿ, with magnetic and potential terms. In particular, for each classical path γ connecting points q₀ and q₁ in time t, we define a formal power series V_γ(t, q₀, q₁) in Planck's constant h, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V_γ) satisfies Schrödinger's equation, and explain in what sense the t → 0 limit approaches the δ distribution. As such, our construction gives explicitly the full h → 0 asymptotics of the fundamental solution to Schrödinger's equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman's path integral in diagrams. (hide abstract)

Submitted for Publication

• Spin, statistics, orientations, unitarity. 2015. (abstract, arXiv: 1507.06297.)

  Abstract: The notions of "oriented" and "unitary" quantum field theory are the two possible real forms of the same complexified notion. Indeed, oriented and unitary quantum field theories are precisely the unoriented quantum field theories fibered over the two Z/2-torsors over Spec(R). By the same token, we show that spin and super field theories are two "bosonic forms" of the same "superified" notion. Undergirding this is a coincidence between the homotopy 0- and 1-types of the stable orthogonal group and the absolute Galois and "categorified absolute Galois" groups of R. (hide abstract)

• (Op)lax natural transformations, relative field theories, and the "even higher" Morita category of E_d-algebras. With Claudia Scheimbauer. 2015. (abstract, arXiv: 1502.06526.)

  Abstract: Motivated by the challenge of defining relative, also called twisted, quantum field theories in the context of higher categories, we develop a general framework for both lax and oplax transformations and their higher analogs between strong (∞,n)-functors. Namely, we construct a double (∞,n)-category built out of the target (∞,n)-category that we call its (op)lax square, which governs the desired diagrammatics. Both lax and oplax transformations are functors into parts thereof. We then define a lax or oplax relative field theory to be a symmetric monoidal lax or oplax natural transformation between field theories. We verify in particular that lax trivially-twisted relative field theories are the same as absolute field theories. Finally, we use the (op)lax square to extend the construction of the higher Morita category of E_d-algebras in a symmetric monoidal (∞,n)-category C to an even higher level using the higher morphisms in C. (hide abstract)

Other Preprints

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

  Abstract: This paper discusses what we should mean by "Heisenberg-picture quantum field theory." Atiyah–Segal-type axioms do a good job of capturing the "Schrödinger picture": these axioms define a "d-dimensional quantum field theory" to be a symmetric monoidal functor from an (∞,d)-category of "spacetimes" to an (∞,d)-category which at the second-from-top level consists of vector spaces, so at the top level consists of numbers. This paper argues that the appropriate parallel notion "Heisenberg picture" should also be defined in terms of symmetric monoidal functors from the category of spacetimes, but the target should be an (∞,d)-category that in top dimension consists of pointed vector spaces instead of numbers; the second-from-top level can be taken to consist of associative algebras or of pointed categories. The paper ends by outlining two sources of such Heisenberg-picture field theories: factorization algebras and skein theory. (hide abstract)

• Chains(R) does not admit a geometrically meaningful properadic homotopy Frobenius algebra structure. 2013. (abstract, arXiv: 1308.3423.)

  Abstract: The embedding of Chains(R) into Cochains(R) as the compactly supported cochains might lead one to expect Chains(R) to carry a nonunital commutative Frobenius algebra structure, up to a degree shift and some homotopic weakening of the axioms. We prove that under reasonable "locality" conditions, a cofibrant resolution of the dioperad controlling nonunital shifted-Frobenius algebras does act on Chains(R), and in a homotopically-unique way. But we prove that this action does not extend to a homotopy Frobenius action at the level of properads or props. This gives an example of a geometrically meaningful algebraic structure on homology that does not lift in a geometrically meaningful way to the chain level.
  
  Note: An expanded retelling of the story in this paper, with different conventions and clearer results, is available in "Tree- versus graph-level quasilocal Poincaré duality on S¹." (hide abstract)

• Peturbative techniques in path integration. Ph.D. Thesis. 2013. (abstract, PDF.)

  Abstract: This dissertation addresses a number of related questions concerning perturbative "path" integrals. Perturbative methods are one of the few successful ways physicists have worked with (or even defined) these infinite-dimensional integrals, and it is important as mathematicians to check that they are correct. Chapter 0 provides a detailed introduction.
  
  We take a classical approach to path integrals in Chapter 1. Following standard arguments, we posit a Feynman-diagrammatic description of the asymptotics of the time-evolution operator for the quantum mechanics of a charged particle moving nonrelativistically through a curved manifold under the influence of an external electromagnetic field. We check that our sum of Feynman diagrams has all desired properties: it is coordinate-independent and well-defined without ultraviolet divergences, it satisfies the correct composition law, and it satisfies Schrödinger's equation thought of as a boundary-value problem in PDE.
  
  Path integrals in quantum mechanics and elsewhere in quantum field theory are almost always of the shape ∫ f e^s for some functions f (the "observable") and s (the "action"). In Chapter 2 we step back to analyze integrals of this type more generally. Integration by parts provides algebraic relations between the values of ∫ (-) e^s for different inputs, which can be packaged into a Batalin–Vilkovisky-type chain complex. Using some simple homological perturbation theory, we study the version of this complex that arises when f and s are taken to be polynomial functions, and power series are banished. We find that in such cases, the entire scheme-theoretic critical locus (complex points included) of s plays an important role, and that one can uniformly (but noncanonically) integrate out in a purely algebraic way the contributions to the integral from all "higher modes," reducing ∫ f e^s to an integral over the critical locus. This may help explain the presence of analytic continuation in questions like the Volume Conjecture.
  
  We end with Chapter 3, in which the role of integration is somewhat obscured, but perturbation theory is prominent. The Batalin–Vilkovisky homological approach to integration illustrates that there are generalizations of the notion of "integral" analogous to the generalization from cotangent bundles to Poisson manifolds. The AKSZ construction of topological quantum field theories fits into this approach; in what is usually called "AKSZ theory," everything is still required to be symplectic. Using factorization algebras as a framework for (topological) quantum field theory, we construct a one-dimensional Poisson AKSZ field theory for any formal Poisson manifold M. Quantizations of our field theory correspond to formal star-products on M. By using a ``universal'' formal Poisson manifold and abandoning configuration-space integrals in favor of other homological-perturbation techniques, we construct a universal formal star-product all of whose coefficients are manifestly rational numbers. (hide abstract)

• How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism. With Owen Gwilliam. 2011. (abstract, PDF, arXiv: 1202.1554.)

  Abstract: The Batalin-Vilkovisky formalism in quantum field theory was originally invented to avoid the difficult problem of finding diagrammatic descriptions of oscillating integrals with degenerate critical points. But since then, BV algebras have become interesting objects of study in their own right, and mathematicians sometimes have good understanding of the homological aspects of the story without any access to the diagrammatics. In this note we reverse the usual direction of argument: we begin by asking for an explicit calculation of the homology of a BV algebra, and from it derive WickÕs Theorem and the other Feynman rules for finite-dimensional integrals. (hide abstract)

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

  Abstract: When path integrals are discussed in quantum field theory, it is almost always assumed that the fields take values in a vector bundle. When the fields are instead valued in a possibly-curved fiber bundle, the independence of the formal path integral on the coordinates becomes much less obvious. In this short note, aimed primarily at mathematicians, we first briefly recall the notions of Lagrangian classical and quantum field theory and the standard coordinate-full definition of the "formal" or "Feynman-diagrammatic" path integral construction. We then outline a proof of the following claim: the formal path integral does not depend on the choice of coordinates, but only on a choice of fiberwise volume form. Our outline is an honest proof when the formal path integral is defined without ultraviolet divergences. (hide abstract)

Other documents

Textbook

I have an unfinited draft of a textbook on Lie theory, based on lectures given at UC Berkeley in 2006–08:

• R. Borcherds, M. Haiman, N. Reshetikhin, V. Serganova. Berkeley Lectures on Lie Groups and Quantum Groups. Part I: Lie Groups, and start of Part II: Quantum Groups. Edited by T. Johnson-Freyd and A. Geraschenko. (DRAFT. PDF, TeX (tar.gz). Last updated Friday, 27-Sep-2013 13:35:14 CDT.)

Class Notes

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

• Philippe Di Francesco. Chern-Simons Research Lectures: Discrete Integrable Systems and Cluster Algebras, 10-12 April 2012. Prepared slides: Day 1 (8 MB), Day 2 (16 MB), Day 3 (13 MB). Unedited notes from the first two days.
• Denis Bernard. Chern-Simons Research Lectures: Stochastic Schramm-Loewner Evolution (SLE) from Statistical Conformal Field Theory (CFT): An Introduction for (and by) Amateurs, 19-23 March 2012. Prepared notes: arXiv:math-ph/0602049v1. Mildly edited notes.
• David Kazhdan. Chern-Simons Research Lectures: The classical master equation in the finite-dimensional case, 9-13 Jan 2012. Mildly edited notes.
• Jorg Teschner. Chern-Simons Research Lectures: Quantization of Hitchin's Moduli Spaces and Liouville Theory, 17-21 Oct 2011. Unedited notes.
• Fedor Smirnov. Chern-Simons Research Lectures: Correlation functions in integrable Quantum Field Theory, 28 Sept - 4 Oct 2011. Unedited notes. Slides.
• MSRI, Symplectic Geometry, Noncommutative Geometry and Physics, May 10, 2010 to May 14, 2010. (All slides and transcripts (tar.gz).)
• V. Serganova. Math 261B: Quantum Groups. Spring 2010. (Unedited: PDF, TAR.GZ. Last updated Friday, 27-Sep-2013 13:35:48 CDT.)
• N. Reshetikhin. Math 261B: Quantum Groups. Spring 2009. (Unedited: PDF (size exceeds 800 KB), TAR.GZ (size exceeds 9 MB). Last updated Friday, 27-Sep-2013 13:35:56 CDT.)
• M. Haiman. Math 261A: Lie Groups. Fall 2008. (Edited: PDF, TeX. Unedited: PDF (size exceeds 1 MB), TAR.GZ (size exceeds 6 MB). Last updated Friday, 27-Sep-2013 13:36:13 CDT.)
• M. Rieffel. Math 208: C* Algebras. Spring 2008. (Unedited: PDF (about 600 KB), TAR.GZ (about 4 MB). Last updated Friday, 27-Sep-2013 13:36:22 CDT.)

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

Grant and job application materials

Linked below are various application materials. I produced the following materials in Fall of 2015 for the tenure-track search:

• Research Statement: on the interface between topology and physics
• Teaching Statement: some experiences teaching at a research university

• NSF Summary
• NSF Project Proposal
• NSF References
• NSF Biographical Sketch
• NSF Data Plan
• MathJobs Cover Letter Template
• MathJobs Project Proposal
• MathJobs Publication List
• MathJobs Teaching Statement

Miscellaneous mathematics

Here are two ideosyncratic notes surveying various mathematics:

• Exact triangles as modules over an A_∞-category. 2015. (PDF.)
• Poisson Lie linear algebra in the graphical language. 2009. (PDF. TeX source.)

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

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

If you are curious, you can read the syllabus for my PhD Candidacy Qualifying Exam (UC Berkeley, 11 June 2009).

Seminar presentations

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)

2008

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

  Abstract: A Feynman diagram is many things (a picture, a process, an event, a morphism). For me, a Feynman diagram is a combinatorial integral. This class will explain some of the beautiful combinatorics that underlies calculus, beginning with derivatives and Taylor's theorem, and concluding with integrals and Feynman Diagrams. For example, the generalized Chain Rule (dⁿ[f(u(x))]/dxⁿ in terms of df/du and du/dx) also generalizes the number of partitions of n objects. Along the way, we will develop some multi-variable calculus — certainly not a whole course, but whatever is needed to get at the full combinatorial elegance. High school calculus is strongly recommended. (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)

Teaching

At Northwestern

• Fall 2015: First-year Seminar: Theories of mind and mathematics
• Spring 2014: Freshman Seminar: Theories of mind and mathematics
• Fall 2013: Math 300: Foundations of Higher Mathematics

At Berkeley

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

Non-math

My husband and I are avid cooks. For a while we collected recipes and discussions at Local Seasoning. We are on hiatus right now, but may start up again.

In a previous life, I was a kitchen manager at Columbae House at Stanford University. Around the same time, I was also a dancer, and I choreographed the Opening Committee performance for the 2007 Stanford Viennese Ball (video).

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