Minicourse, led by Daniel Tubbenhauer (University of Sydney): From representations of monoids to representations of monoidal categories
Abstract: There is a MathOverflow question with title "Why aren't representations of monoids studied so much?"
Well, I do not know! But it can't be because the theory is bad. In fact, there is a well-developed about 80 years old theory, and this theory is almost as smooth and beautiful as group representation theory itself. There are even categorifications of it.
This series is a friendly overview of the representation theory of monoids and how to categorify bits and pieces of this story. Along the way we take a detour and discuss a linear version of monoid representation theory that goes under the name "cellular stuff."
Catharina Stroppel (University of Bonn): Motivic Representation theory
Abstract: In this talk I like to explain how motives naturally appear in geometric represenation theory. I will start with a brief summary of well-known comvolution algebra constructions and then explain how they can be interpreted in a motivic setting and will describe some advantages of this new approach.
Jose Simental (UNAM): Cluster structures on braid varieties
Abstract: Given a simple algebraic group G and an element β of its positive braid monoid we define a smooth, affine algebraic variety X(β), the braid variety of β. This is a locally closed subset of the Bott-Samelson variety associated to (any) braid word for β, and it generalizes well-known varieties in Lie theory, including open Richardson varieties. In joint work with R. Casals, E. Gorsky, M. Gorsky, I. Le and L. Shen we show that the algebra of functions on X(β) admits the structure of a Fomin-Zelevinsky cluster algebra, and explicitly construct several initial seeds, using combinatorial objects called weaves and tropicalization of Lusztig's coordinates. I will explain this construction (with several examples) and, time permitting, give properties of the corresponding cluster structure, including local acyclicity and the existence of a basis of ϑ-functions.
Melissa Sherman-Bennett (MIT): Type A braid varieties and cluster structures from 3D plabic graphs
Abstract: I'll discuss another way to identify the coordinate ring of a type A braid variety with a cluster algebra, inspired by a Deodhar-type stratification of the braid variety. The relevant combinatorial objects here are wiring diagrams and "3D plabic graphs," which generalize Postnikov's plabic (i.e. planar + bicolored) graphs. The relation with the cluster algebra from the previous talk is as yet unknown. This is joint work-in-progress with P. Galashin, T. Lam and D. Speyer.
Slava Krushkal (University of Virginia): q-series invariants and lattice cohomology of plumbed 3-manifold
Abstract: I will discuss a combinatorially defined invariant of negative definite plumbed 3-manifolds, equipped with a spin-c structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives certain q-series defined by Gukov-Pei-Putrov-Vafa which are conjectured to recover quantum invariants of 3-manifolds at roots of unity; there is also a Physics prediction for their categorification as BPS homology. (Joint work with Ross Akhmechet and Peter Johnson).
Ikshu Neithalath (University of Southern Denmark): Skein Lasagna modules of 2-handlebodies
Abstract: Morrison, Walker and Wedrich defined a generalization of Khovanov-Rozansky homology to links in the boundary of a 4-manifold. We will discuss joint work with Ciprian Manolescu on the "skein lasagna module," a basic part of MWW's invariant, for a certain class of 4-manifolds.
Paul Wedrich (University of Hamburg): On skein theory in dimension four
Abstract: The Temperley-Lieb algebra describes the local behaviour of the Jones polynomial and gives rise to the Kauffman bracket skein modules of 3-manifolds. Going up by one dimension, Bar-Natan's dotted cobordisms describe the local behaviour of Khovanov homology and, likewise, give rise to skein modules of 4-manifolds. I will describe how to compute such skein modules via a handle decomposition in terms of link homology in the 3-sphere. Based on joint work with Morrison--Walker and Manolescu--Walker.
Asilata Bapat (Australian National University): Contracting the stability manifold in type A
Abstract: Consider the 2-CY category corresponding to the type A quiver, which categorifies the Burau representation of the type A braid group. This category has an associated space of Bridgeland stability conditions. We exhibit an explicit contracting flow on this space, giving a direct proof of contractibility. As a result, we obtain a geometric proof of the faithfulness of the braid group action on the category. This talk is based on joint work with Anand Deopurkar and Anthony Licata.
Xinchun Ma (University of Chicago): Cherednik algebras, Hilbert schemes and Mirabolic D-modules
Abstract: We use the Hodge filtration on the cuspidal mirabolic D-modules to describe the coherent sheaves on the Hilbert scheme of n points on the plane corresponding to the finite-dimensional irreducible representations of type A Cherednik algebras via the Gordan-Stafford map. The result is a version of a conjecture of Gorsky and Negut.
Alistair Savage (University of Ottawa): The quantum isomeric supercategory
Abstract: We introduce the quantum isomeric supercategory and the quantum affine isomeric supercategory. These diagrammatically defined supercategories, which can be viewed as isomeric analogues of the HOMFLYPT skein category and its affinization, provide powerful categorical tools for studying the representation theory of the quantum isomeric superalgebras (commonly known as quantum queer superalgebras).
David Rose (University of North Carolina): A Kirby color for Khovanov homology
Abstract: We'll introduce the titular Kirby color, an object in (a completion of) the annular Bar-Natan category (ABN) that satisfies a handleslide property. Along the way, we'll review an equivalence (due to Russell) between ABN and the dotted Temperley-Lieb category as well as some facts about Ind-completion of categories. Our Kirby color models 2-handle addition in the Morrison-Walker-Wedrich "Khovanov skein lasagna module" (recalled in Ikshu's talk, which discusses work that motivated our construction). This is joint work with Matt Hogancamp and Paul Wedrich..
Matt Hogancamp (Northeastern University): A diagrammatic category for Kirby-colored Khovanov homology
Abstract: Colored Khovanov homology assigns homology groups to a pair (K, c) where K is a framed knot and c is a choice of color, which is to say an object of the annular Bar-Natan category or, equivalently, the dotted Temperley-Lieb category (dTL). In joint work with Rose and Wedrich we construct a Kirby color, living in an extension of dTL , whose associated colored homology satisfies the all-important handleslide relation. The goal of my talk will be to give a diagrammatic description of this Kirby color; precisely, I will introduce a monoidal diagrammatic category which is equivalent to dTL with the Kirby color adjoined. One interesting feature is the appearance of relations involving infinite sums of diagrams, reflecting the fact that the Kirby color is an Ind-object of dTL. Another interesting feature is that our Kirby color is a quasi-idempotent algebra object, in a sense I will make precise. All of this is joint with Rose and Wedrich.
Joanna Meinel (University of Bonn) Recipes for highest weight vectors in tensor products
Abstract: We describe a way to obtain highest weight vectors in tensor products of L(\lambda) with the fundamental representation which we apply in type A to obtain concrete formulas for the classical and quantum case. This is joint work in progress with Pablo Zadunaisky.
You Qi (University of Virginia): A categorification of the colored Jones polynomial at a root of unity
Abstract: We propose a categorification of the colored Jones polynomial evaluated at a 2p-th root of unity by equipping a p-differential discovered by Cautis on the triply graded Khovanov-Rozansky homology. This is based on joint work with Louis-Hadrien Robert, Joshua Sussan and Emmanuel Wagner.
Josh Sussan (CUNY): Non-semisimple Turaev-Viro invariants and Levin-Wen models
Abstract: Representations of quantum groups at roots of unity provide categories used by Turaev and Viro to construct 3-dimensional topological invariants. Using the same input categories, Levin and Wen described certain physical systems in condensed matter theory. Both of these constructions use a semisimplification of categories of representations. Geer, Patureau-Mirand, and Turaev defined Turaev-Viro type invariants without semisimplifying the categories involved. We will review these ideas and discuss non-semisimple versions of Levin-Wen models. This is joint work with Nathan Geer, Aaron Lauda, and Bertrand Patureau-Mirand.
Mee Seong Im (US Naval Academy): One-dimensional topological quantum field theories with zero-dimensional defects and finite state automata
Abstract: Quantum groups are related to 3-dimensional topological quantum field theories. Downsizing from three dimensions to one and from a ground field to a semiring, I will explain a surprising relation between topological theories for one-dimensional manifolds with defects and values in the Boolean semiring and finite-state automata and their generalizations. This is joint with Mikhail Khovanov.
Mikhail Khovanov (Columbia University): Annular foam evaluation
Abstract: We explain how to build state spaces for unoriented sl(3) webs in an annulus via foam evaluation. This is joint work with Ross Akhmechet.