Fall 2017 GRASP Seminar

This is the website for the GRASP (Geometry, Representation Theory, And Some Physics) student seminar.

The seminar meets from 2-3 PM on Thursdays in 732 Evans.

For questions, or if you'd like to give a talk or be added to the mailing list, contact Calvin via email at cmcs, with domain math.berkeley.edu. You might also be interested in the Representation Theory and Mathematical Physics research seminar.


21 September
Low-dimensional Dijkgraaf-Witten theory, Aaron Brookner
Abstract: DW theory is like Chern-Simons theory, but with a finite gauge group. This means our "states" are (disconnected) normal covering spaces. In this context we can recover the orbit-stabilizer theorem.
We'll then quantize the DW theory, and see how this is the same information as the character table of G; by applying this theory to surfaces we can recover the convolution product and a homomorphism-counting formula related to Burnside's lemma.
The only background needed is a knowledge of covering spaces.
References: Notes on topological quantum field theory, D. Freed.
arXiv:hep-th/9212115, "Higher Algebraic Structures and Quantization," D. Freed.
28 September
Finding Topology in Entanglement, Ryan Thorngren
Abstract: I will describe an important classification problem in many-body quantum mechanics which will lead us to consider Dijkgraaf Witten theory. I hope to convince you that tensor networks are a computable platform for formulating higher representation theory.
References: arXiv 1607.06766, 1412.S604, 1409.S436, 0701055, 0601075
5 October
Chromatic polynomials and the Temperley-Lieb algebra, Calvin McPhail-Snyder
Abstract: The Temperley-Lieb algebra shows up in a variety of contexts, for example as a diagrammatic calculus for the representation category of quantum sl_2. We show that it also has connections to chromatic polynomials of planar graphs, and use this connection to prove some remarkable identities.
References: arXiv:0711.0016v3, "Tutte Chromatic Identities from the Temperley-Lieb Algebra," P. Fendley and V. Krushkal.
12 October
Categorification of HOMFLY-PT polynomials via Soergel bimodules, Tao Su
Abstract: Associated to oriented links represented by braid closures, one way to define the HOMFLY-PT polynomials is via the associated Hecke algebras and the Jones-Ocneanu trace. In this talk, I will introduce the categorification of the previous construction via Soergel bimodules and Hochschild homology, leading to a natural categorification (also known as the triply graded Khovanov-Rozansky homology) of HOMFLY-PT polynomials. Time permitting, I will also mention a few words about the geometric interpretation of the categorification via perverse sheaves.
References: Annals of Math,1987, "Hecke algebra representations of braid groups and link polynomials", V.Jones
arXiv:0510265, "Triply-graded link homology and Hochschild homology of Soergel bimodules", M.Khovanov.
arXiv:0505056, "Matrix factorizations and link homology II", M.Khovanov, L.Rozansky.
arXiv:0905.0486, "A geometric construction of colored HOMFLY-PT homology", B.Webster, G.Williamson.
19 October
Calabi-Yau categories and 2D TCFT, Alex Takeda
Abstract: In this talk I will review K. Costello's definition of a 2d TCFT and how it is supposed to elucidate the topological field-theoretic structures underlying the models that calculate some quantities of mathematical importance such as the Gromov-Witten invariants of a Calabi-Yau manifold. Time allowing I will describe some developments related to my own work with V. Shende, which includes among the possible targets of such a model the topological Fukaya categories of a Weinstein manifold.
References: arXiv:0807.3052
26 October
Representations of Loop Groups, Kevin Donoghue
Abstract: A loop group is the (infinite dimensional) group of maps of a circle into a target group G. Surprisingly, the representation theory of loop groups is very similar to the representation theory of compact groups. I will describe the representations of loop groups (weights, Weyl group, characters, Borel-Weil, etc) mainly via comparison to the compact case. If time permits, I will say something about how the loop group fits into Chern-Simons theory.
2 November
Central Extensions of Loop Groups, Kevin Donoghue
Abstract: Last time we saw the construction of a central extension of the loop Lie algebra. The constructions of the corresponding group are considerably thornier. The subject of this talk is one construction that dips its toes into the world of 2- and 3-manifolds.
9 November
Perturbative Chern-Simons invariants from quantum BV-BFV formalism, Konstantin Wernli, University of Zurich (Joint with RTMP seminar)
Abstract: I will report on recent developments in the computation of Perturbative Chern-Simons invariants via cutting and gluing in the quantum BV-BFV formalism. In particular, I will present results on theta-invariants of lens spaces that agree with earlier works of Kuperberg, Thurston and Lescop. This is ongoing joint work with A. Cattaneo and P. Mnev.
16 November
Pfaffian Sign Theorem for the Dimer Model on a Triangular Lattice, Drazen Petrovic, IUPUI (Joint with RTMP seminar)
Abstract: We prove the Pfaffian Sign Theorem for the dimer model on a triangular lattice embedded in the torus. More specifically, we prove that the Pfaffian of the Kasteleyn periodic-periodic matrix is negative, while the Pfaffians of the Kasteleyn periodic-antiperiodic, antiperiodic-periodic, and antiperiodic-antiperiodic matrices are all positive. The proof is based on the Kasteleyn identities and on small weight expansions. As an application, we obtain an asymptotics of the dimer model partition function with an exponentially small error term. This is a joint work with Pavel Bleher and Brad Elwood.
23 November
Thanksgiving. No seminar.
30 November
Gaussian Free Field in Random Tilings, Vadim Gorin, MIT (Joint with RTMP seminar)
Abstract: The fluctuations of the height function of random lozenge and domino tilings of polygonal domains are believed to be governed by the 2d Gaussian Free Field in an appropriate complex structure. This was conjectured by Kenyon and Okounkov and they also suggested a neat geometric procedure for defining the corresponding complex structure.
I will present a new approach to this conjecture for a class of domains through gluings of Gelfand-Tsetlin patterns. It combines discrete Dyson-Schwinger equations with the method of Schur generating functions to yield the result.
7 December
A-infinity extended TQFT, Maxim Jeffs
Abstract: Physical considerations suggest that some important examples of gauge-theoretic TQFTs can be (conjecturally) extended to lower dimensions using the Lagrangian Floer homology of certain representation varieties. I will explain how the the additional A-infinity structures (Fukaya categories) on these groups can be incorporated into the categorical extended TQFT picture. This talk will be mostly expository and no prior knowledge of Floer theory will be assumed.
14 December