Title:  Introduction to braid varieties

Abstract: Braids are familiar objects in topology, but they also build rich geometric spaces known as braid varieties. These varieties generalize many well-known spaces in Lie theory, such as open Richardson and positroid varieties, and possess a natural cluster algebra structure. This talk will introduce braid varieties through examples, explain how they encode both topological and algebraic information, and highlight their connections to cluster combinatorics.

Title:  2-dimensional Artin groups via quotients

Abstract: Two common families of groups in geometric group theory are the Artin groups and the Coxeter groups. Coxeter groups are very well-understood and well-studied. The Artin groups share many similarities, but very little is known about them in general. I will discuss recent work studying the "2-dimensional" Artin groups through their quotients to the so-called Shephard groups. A Coxeter group is a certain quotient of an Artin group, and these Shephard quotients can be thought of as interpolations between the Coxeter group quotient and the Artin group itself. The Shephard groups possess their own interesting and sometimes unintuitive (or even counterintuitive) geometry and topology, with some unexpected connections. We can leverage this divergence to prove new results for Artin groups themselves; as our main example, we apply a "pseudo-Dehn filling" procedure to show that the hyperbolic-type 2-dimensional Artin groups are fully residually hyperbolic, and if they are also FC-type, they are residually finite.

Title: A basis for the graph skein module of thickened surfaces

Abstract: The graph skein module of a 3-manifold is the quotient of the free algebra of embedded trivalent graphs by the Yamada local relations. It is closely related to the Kauffman bracket skein module. In this talk I will describe a basis for the graph skein module of thickened oriented surfaces. Time permitting, I will discuss the applications of this basis to finding generating sets for the graph skein algebras of small surfaces.

Title:  Steinberg skein relations at roots of unity 

Abstract: This talk discusses some of the relationships between skein theory and the representation theory of quantum sl2 when q is a root of unity. Specifically, I focus on the Frobenius pullback functor on Uq sl2 representations and see how this relates to Bonahon-Wong's Frobenius skein homomorphism between Kauffman bracket skein modules. I'll describe results from my joint work with Vijay Higgins in which we proved what we called Steinberg skein identities and used these in a new proof of the well-definition of the Frobenius skein homomorphism.

Title:  Exotic pseudo-mapping-classes for 4-manifolds

Abstract: A pseudo-isotopy is a weakening of the concept of isotopy, removing the insistence that it is level-preserving. Pseudo-isotopies play a key role in the study of mapping class groups in dimensions 4 and above. We investigate the question: if a self-diffeomorphism of a 4-manifold is topologically pseudo-isotopic to the identity, must it always be smoothly so? We produce the first examples where the answer is “no” -- the first exotic pseudo mapping classes. On the other hand, we derive a set of conditions on the fundamental group of a 4-manifold that identify a large class of examples where this topological-to-smooth upgrade can always be made. This is joint work with Mark Powell and Oscar Randal-Williams.

Title:  Fusion Trees in Non-semisimple Topological Quantum Computation

Abstract: Unitary (2+1)-dimensional topological quantum field theories (TQFTs) govern the mathematics of (2+1)-dimensional topological phases with rich properties that can host quasiparticle excitations called anyons. Freedman, Kitaev, Larsen, and Wang demonstrated that topological phases of matter supporting non-abelian anyons provide fault-tolerant quantum computation models via topological quantum computation (TQC). In these frameworks, modular categories (MCs) provide the algebraic theories of anyons, quantum information is encoded in the topologies of the system, quantum states are encoded as fusion trees in Hom-spaces, and quantum computation via unitary gates is performed by braiding anyons.

In recent years, there has been a gradual paradigm shift within the TQFT community toward exploring non-semisimple structures, which appear to provide several advantages over traditional semisimple TQFTs. In this talk, we discuss the mathematical foundations for non-semisimple topological quantum computation (NSS TQC) and their fusion trees. First, we explore the non-semisimple Ising category, which is an augmentation to the (traditional) Ising modular fusion category (based on joint work with Filippo Iulianelli, Aaron Lauda, and Joshua Sussan) [arXiv:2410.14860] [arXiv:2509.02843]. Next, we explain how a family of homological representations is related to the fusion trees that appear in NSS TQC [arXiv:2510.27218].

Title:  Cornered skein lasagna theory

Abstract: The Khovanov-Rozansky skein lasagna module was introduced by Morrison-Walker-Wedrich as an invariant of 4-manifold with a framed oriented link in the boundary. I will discuss an extension of the skein lasagna theory to 4-manifolds with codimension 2 corners, and its behavior under gluing. I will also talk about a categorical framework for a presentation of skein lasagna module of trisected closed 4-manifolds. This is joint work with Sarah Blackwell and Slava Krushkal.

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