The Topology Seminar
Organizers: Melody Molander and Rhea Palak Bakshi
2024-2025 School Year Schedule:
(All talks 3:30-4:30 in SH 6635 unless otherwise stated)
- Tuesday, 11/19: Vijay Higgins (UCLA) "Annular webs, central elements of skein algebras, and miraculous cancellations"
- Monday, 12/02, 4-5 PM: Qiuyu Ren (UC Berkeley) "Lasagna s-invariant detects exotic 4-manifolds"
- Tuesday, 01/14: Jennifer Schultens (UC Davis) "Flipping Heegaard splittings"
- Friday, 01/31, 12-1 PM: Aaron Lauda (USC) "Nonsemisimple Topological Quantum Computation"
- Tuesday, 02/04: Jon McCammond (UCSB) "The geometric combinatorics of polynomials and braids"
- Monday, 02/10, 4-5 PM: Peter Samuelson (UC Riverside) "Skein algebras and Hall algebras"
- Tuesday, 02/11: Anup Poudel (The Ohio State University) "Categories generated by 4-valent vertices"
- Tuesday, 02/18: Agustina Czenky (USC) "Unoriented 2-dimensional TQFTs and the category Rep(\(S_t\))"
- Tuesday, 02/25: Christine Ruey Shan Lee (Texas State) "A topological model for the HOMFLY-PT polynomial"
- Tuesday, 03/04: Thang Lê (Georgia Tech) "Stated skein algebras and quantum groups"
- Monday, 03/10, 4-5 PM Chloé Postel-Vinay (University of Chicago) "\(k\)-Shuffle Braid Groups"
- Tuesday, 03/11: Corey Jones (NC State) "Holography for topological codes"
- Monday, 03/17, 4-5 PM: Nathan Geer (Utah State) TBA
- Tuesday, 03/18: Colleen Delaney (Purdue University) TBA
- Tuesday, 04/01: David Penneys (The Ohio State University) TBA
- Tuesday, 04/08: Melissa Zhang (UC Davis) TBA
- Tuesday, 04/15: Józef Przytycki (The George Washington University) TBA
- Tuesday, 04/22: Kevin Walker TBA
- Tuesday, 04/29: Sujoy Mukherjee (University of Denver) TBA
- Tuesday, 05/06: Sunghyuk Park (Harvard) TBA
- Tuesday, 05/13: Tian Yang (Texas A&M) TBA
- Tuesday, 05/20, on Zoom: Jennifer Taback (Bowdoin College) TBA
Abstract
The skein algebra of a surface is spanned by links in the thickened surface, subject to skein relations which diagrammatically encode the data of a quantum group. The multiplication in the algebra is induced by stacking links in the thickened surface. This product is generally noncommutative. When the quantum parameter q is generic, the center of the skein algebra is essentially trivial. However, when q is a root of unity, interesting central elements arise. When the quantum group is quantum SL(2), the work of Bonahon-Wong shows that these central elements can be obtained by a topological operation of threading Chebyshev polynomials along knots. In this talk, I will discuss joint work with Francis Bonahon describing how the threading operation extends to SL(n) skein algebras, including a method for verifying the centrality of the elements by studying webs in the annulus. Then in the case of SL(3), I will discuss how 'stated skein algebras' can be used to show that the threading operation yields a well-defined algebra homomorphism, analogous to the Frobenius homomorphism for quantum groups.
Abstract
We introduce a lasagna version of Rasmussen's s-invariant coming from the study of Khovanov/Lee skein lasagna modules, which assigns either an integer or \(-\infty\) to each second homology class of a given smooth 4-manifold. After presenting some properties of the lasagna s-invariant, we show that it detects the exotic pair of knot traces \( X_{-1}(-5_2) \) and \(X_{-1}(P(3,-3,-8))\). This gives the first gauge/Floer-theory-free proof of the existence of exotic compact orientable 4-manifolds. Time permitting, we mention some other applications of lasagna s-invariants. This is joint work with Michael Willis.
Abstract
Heegaard splittings provide natural decompositions of 3-manifolds into two symmetric pieces. We provide examples of such decompositions and delve into the question of when the two pieces can be interchanged, or flipped, via an isotopy.
Abstract
Since the foundational work of Freedman, Kitaev, Larsen, and Wang, it has been understood that 3-dimensional topological quantum field theories (TQFTs), described via modular tensor categories, provide a universal model for fault-tolerant topological quantum computation. These TQFTs, derived from quantum groups at roots of unity, achieve modularity by semisimplifying their representation categories—discarding objects with quantum trace zero. The resulting semisimple categories describe anyons whose braiding enables robust quantum computation.
This talk explores recent advances in low-dimensional topology, focusing on the use of nonsemisimple categories that retain quantum trace zero objects to construct new TQFTs. These nonsemisimple TQFTs surpass their semisimple counterparts, distinguishing topological features inaccessible to the latter. For physical applications, unitarity is essential, ensuring Hom spaces form Hilbert spaces. We present joint work with Nathan Geer, Bertrand Patureau-Mirand, and Joshua Sussan, where nonsemisimple TQFTs are equipped with a Hermitian structure. This framework introduces Hilbert spaces with possibly indefinite metrics, presenting new challenges.
We further discuss collaborative work with Sung Kim, Filippo Iulianelli, and Sussan, demonstrating that nonsemisimple TQFTs enable universal quantum computation at roots of unity where semisimple theories fail. Specifically, we show how Ising anyons within this framework achieve universality through braiding alone. The resulting braiding operations are deeply connected to the Lawrence-Krammer-Bigelow representations, with the Hermitian structure providing a nondegenerate inner product grounded in quantum algebra.
Abstract
Braid groups are the original Artin groups and they have many different classifying spaces: (1) quotient of a complex hyperplane complement, (2) monic (centered) polynomials with distinct roots, (3) Salvetti complex from face-identifying a permutahedron, and (4) dual braid complex from face-identifying the order complex of the noncrossing partition lattice. This talk connects the hyperplane complement and monic polynomials to the noncrossing partitions in the dual structure using Morse theory and the Lyashko-Looijenga map. It is based on joint work with Michael Dougherty (https://arxiv.org/abs/2410.03047).Abstract
Skein algebras are noncommutative algebras associated to surfaces which can be viewed as quantizations of character varieties of surfaces. The Hall algebra of a category has a basis given by isomorphism classes of objects, and the product "counts extensions" in the category. In this talk, we discuss work with Morton that shows that the Homflypt skein algebra of the torus is isomorphic to the Hall algebra of the category of sheaves over an elliptic curve (which itself was described by Burban and Schiffmann). If time permits, we will briefly discuss followup work with Cooper connecting this to Fukaya categories of surfaces. This talk will be introductory, and, in particular, it will not assume familiarity with the objects mentioned above.
Abstract
We work with a category with object an oriented marking and morphisms generated by tagged and untagged 4-valent vertices. The category is defined combinatorially in terms of diagrammatic generators and relations. We use linear algebraic and skein theoretic methods to explore topological invariants coming from such a category. As a consequence, we show that a specialization of our parameters provides a 4-valent category that is equivalent to the SL(4) representation category. We further provide a topological evaluation algorithm of closed webs providing a (topological) criterion for reducible webs. We also show that certain HOMFLY relations exist in our category. Our evaluation algorithm works at a very abstract level and doesn’t use any algebraic constraints coming from the representation theory. This is a joint work with Giovanni Ferrer and Jiaqi Lu.
Abstract
Let \(\mathbb{k}\) be an algebraically closed field of characteristic zero. The category of oriented 2-dimensional cobordisms can be understood in purely algebraic terms via a description by generators and relations; moreover, it is possible to recover from it the Deligne category Rep(\(S_t\)), which interpolates the category of finite-dimensional representations of the symmetric group \(S_n\) from \(n\) a positive integer to any parameter \(t\) in \(\mathbb{k}\). We show that an analogous story happens in the unoriented case: via a description by generators and relations of the unoriented 2-bordism category, we recover the generalized Deligne category Rep(\(S_t \wr Z_2\)), which interpolates the category of finite-dimensional representations of the wreath product \(S_t \wr Z_2\).Abstract
A topological model for a knot invariant is a realization of the invariant as graded intersection pairings on coverings of configuration spaces. In this talk I will describe a topological model for the HOMFLY-PT polynomial. I plan to discuss the motivation from previous work by Lawrence and Bigelow giving topological models for the Jones and \(SL_n\) polynomials, and our construction, joint with Cristina Anghel, which uses a state sum formulation of the HOMFLY-PT polynomial to construct an intersection pairing on the configuration space of a Heegaard surface of the link.
Abstract
We will introduce the theory of stated skein algebras of surfaces and demonstrate how various algebraic properties of quantum groups have natural geometric interpretations in terms of skeins. In particular, we will show how the dual canonical basis of the quantized coordinate rings of \(SL_2\) and \(SL_3\) can be expressed using simple stated skeins. The talk is based on joint work with F. Costantino and A. Sikora.
Abstract
The study of Braid groups through the combinatorics of non-crossing partitions has been extremely fruitful in understanding Artin groups. In particular, they give to Garside structures in the case of braid groups, which in turn allows one to build a \(K(B_n,1)\). In this talk I will talk about certain classes of groups defined by \(k\)-shuffle partitions, which arise as Kreweras complements of \(k\)-divisible partitions. While these do not possess Garside structures, they still share many similarities with braid groups. We will talk about the applicability of the Artin group techniques, how they still give rise to classifying spaces in some cases, as well as the limitations posed by the lack of a Garside structure.Abstract
We present a precise version of the holographic principle for topological quantum error correcting codes by constructing a discrete algebraic quantum field theory living at the boundary. In 2 spatial dimensions, these 1D boundary objects have a braided tensor category of DHR bimodules, which we show recovers the bulk topological order in the case of standard string net models (e.g. Levin-Wen models). Based on joint work with Pieter Naaijkens, David Penneys, and Daniel Wallick.