My primary research interests are in mathematical physics, and specifically 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.
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.
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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.
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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 ℝd 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.
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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.
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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 Vγ(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(Vγ) 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.
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Poisson Lie linear algebra in the graphical language. 2009. (PDF, TeX)
I have also given many
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.
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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.
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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.
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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 XdR, 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 XdR 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.
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(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 TeG 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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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/.
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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.
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.
