Theo Johnson-Freyd

PhD, Mathematics, UC Berkeley

Beginning Fall 2013, I will be an NSF Postdoctoral Fellow and Boas Assistant Professor at Northwestern University.

e-mail: theojf at math dot northwestern dot edu
Office: 1065 Evans Hall, Berkeley, CA, 94720

Contents

• Research
• Talks and Seminars
• Class Notes
• Teaching
• Non-math

Research

My primary research interests are in mathematical physics, focusing on quantum field theory, and specifically perturbative quantization and path integrals and so on. I am also very interested in questions in category theory and representation theory. My adviser is Nicolai Reshetikhin.

My papers and preprints include:

• A combinatorial universal star product. 2013. (abstract, PDF).

  Abstract: We construct combinatorially a canonical universal star quantization of infinitesimal Poisson manifolds equipped with affine coordinates. Our construction improves the existing celebrated result by Kontsevich in two ways. First, we avoid all period integrals and other transcendental techniques, so that the coefficients of our star quantization are rational numbers. Second, we use no "wheels," so that our quantization is defined in infinite dimensions and in other settings where traces are undefined. Our construction relies on the theory of Koszul duality for properads, and also uses a "quasilocal" version of factorization algebras and effective quantum field theory. (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).

• Homological perturbation theory for nonperturbative integrals. 2012. (abstract, PDF, arXiv: 1206.5319).

  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).

• 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).

• The fundamental pro-groupoid of an affine 2-scheme. With Alex Chirvasitu. 2011. Applied Categorical Structures. (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).

• The formal path integral and quantum mechanics. 2010. Journal of Mathematical Physics, 51, 122103. (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).

• 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).

• Feynman-diagrammatic description of the asymptotics of the time evolution operator in quantum mechanics. 2010. Letters in Mathematical Physics, 94(2):123–149. (abstract, preprint PDF, 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).

• Poisson Lie linear algebra in the graphical language. 2009. (PDF, TeX)

One of my more satisfying activities is that I am a reviewer for MathReviews. 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 Qualifying Exam (11 June 2009).

Talks and Seminars

I co-organize with Harold Williams the Geometry, Representations, And Some Physics student seminar, which meets Fridays, 2—3, in 939 Evans.

Some talks I have given are listed below. You can open all abstracts or close all abstracts if you like.

2013

• 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).

Class Notes

I occasionally "Live-TeX" notes from my 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.

• 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 Thursday, 22-Sep-2011 23:54:17 PDT.)

  Chapters 1–6 of the full volume are almost exactly the same as the edited notes from M. Haiman (2008), but there have been some formatting changes. Chapter 7 is based mostly on lectures 20–31 of Anton's notes from the 2006 Tag Team Lie Groups course (PDF). Chapter 8 is drawn from V. Serganova (2010). The first chapter of Part II: Quantum Groups has now been incorporated. Ultimately Part II will consist of between five and six chapters, based on the lectures by N. Reshetikhin (2009) and V. Serganova (2010).

• 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 Sunday, 02-May-2010 12:26:20 PDT.)
• N. Reshetikhin. Math 261B: Quantum Groups. Spring 2009. (Unedited: PDF (size exceeds 800 KB), TAR.GZ (size exceeds 9 MB). Last updated Friday, 08-May-2009 13:35:25 PDT.)
• 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, 23-Jan-2009 17:43:44 PST.)
• M. Rieffel. Math 208: C* Algebras. Spring 2008. (Unedited: PDF (about 600 KB), TAR.GZ (about 4 MB). Last updated Monday, 12-May-2008 11:15:05 PDT.)

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.

Teaching

I have taught the following classes at UC Berkeley:

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

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. I recommend that you use an RSS reader, so that if we do have new posts, you'll see them, but don't have to regularly check.

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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