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Organizing Committee
Universität Zürich
University of Oregon
Universidad de Sevilla
Australian National University
Abstract
Braid groups and their generalizations play a central role in a number of places in 21st-century mathematics. In modern representation theory, braid groups have come to play an important organizing role, somewhat analogous to the role played by Weyl groups in classical representation theory. Recent advances have established strong connections between homological algebra (t-structures and stability conditions), geometric representation theory (Hilbert schemes, the Hecke category, and link homologies), and algebraic combinatorics (shuffle algebras, symmetric functions, and also Garside theory). Braid groups appear prominently in many of these connections. The goal of this workshop will be to bring experts in these different areas together to both communicate recent advances and also to formulate important questions for future work.
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Workshop Schedule
Monday, February 14, 2022
8:50 - 9:00 AM EST
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM EST
Higher structure and symmetry in Khovanov-Rozansky homology
11th Floor Lecture Hall
Speaker
Matt Hogancamp, Northeastern University
Session Chair
Anthony Licata, Australian National University
Abstract
In this talk I will show how one constructs the action of a certain commutative dg algebra on the Khovanov-Rozansky complex of a link. The central application is a proof of the "mirror symmetry" property of triply graded Khovanov-Rozansky homology of a knot, originally conjectured in 2005 by Dunfield-Gukov-Rasmussen. This was proven first by Oblomov-Rozansky using their geometric link homology, but I will discuss an independent proof developed in joint work with Gorsky and Mellit.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
A skein theoretic Carlsson-Mellit algebra
11th Floor Lecture Hall
Speaker
Nicolle Gonzalez, UCLA
Session Chair
Anthony Licata, Australian National University
Abstract
The shuffle theorem gives a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. In their proof, Carlsson and Mellit introduce a new interesting algebra denoted $A_{q,t}$. This algebra arises as an extension of the affine Hecke algebra by certain raising and lowering operators and acts on the space of symmetric functions via certain complicated plethystic operators. Afterwards Carlsson, Mellit, and Gorsky showed this algebra and its representation could be realized using parabolic flag Hilbert schemes and in addition to containing the generators of the elliptic Hall algebra. In this talk I will discuss joint work with Matt Hogancamp where we construct skein theoretic formulations of the representations of $A_{q,t}$ that arise in the proofs of the shuffle theorems and how this framework enables difficult computations to become simple diagrammatic manipulations as well as sheds light on potential applications to combinatorics and link homology.
11:30 AM - 12:15 PM EST
Very positive braids are parity braids?
11th Floor Lecture Hall
Speaker
Alexei Oblomkov, UMASS Amherst
Session Chair
Anthony Licata, Australian National University
Abstract
Based on joint work with Lev Rozansky. A braid is a parity braid if Khovanov-Rozansky homology of the closure of the braid has only odd or only even homological grading.
It is expected that algebraic braids are parity, but probably there are more. It also seems to be natural to conjecture that after twisting by a very large power of the full twist any braid becomes parity. In our work we computed homology of the closure of composition of a quasi-Coxeter braid and a Jucys-Murphy braids. For these braids the answer to question in the title is yes.
12:30 - 2:30 PM EST
Lunch/Free Time
2:30 - 3:15 PM EST
A categorification of colored Jones polynomial at prime roots of unity
11th Floor Lecture Hall
Speaker
You Qi, University of Virginia
Session Chair
María Cumplido Cabello, University of Seville
Abstract
We propose a categorification of the colored Jones polynomial evaluated at a 2pth root of unity by equipping a p-differential discovered by Cautis on the triply graded Khovanov-Rozansky homology.
3:30 - 4:00 PM EST
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EST
Braids: Classical, Virtual and Welded, Oh my!
11th Floor Lecture Hall
Speaker
Nancy Scherich, University of Toronto
Session Chair
María Cumplido Cabello, University of Seville
Abstract
We will discuss the difference between the classical, virtual, and welded braid groups from an algebraic and topological perspective. We will discuss techniques to extend representations of classical braid groups to the virtual and welded settings.
5:00 - 6:30 PM EST
Reception
11th Floor Collaborative Space
Tuesday, February 15, 2022
9:00 - 9:45 AM EST
Virtual Artin groups
11th Floor Lecture Hall
Virtual Speaker
Luis Paris, University of Burgundy
Session Chair
You Qi, University of Virginia
Abstract
This talk concerns a joint work with Paolo Bellingeri and Anne-Laure Thiel.
Starting from the observation that the standard presentation of a virtual braid group mixes the presentations of the corresponding braid group and the corresponding symmetric group together with the action of the symmetric group on its root system, we define a virtual Artin group ${\rm VA}[\Gamma]$ with a presentation that mixes the standard presentations of the Artin group $A[\Gamma]$ and of the Coxeter group $W[\Gamma]$ together with the action of $W[\Gamma]$ on its root system.
By definition we have two epimorphisms $\pi_K:{\rm VA}[\Gamma]\to W[\Gamma]$ and $\pi_P:{\rm VA}[\Gamma]\to W[\Gamma]$ whose kernels are denoted by ${\rm KVA}[\Gamma]$ and ${\rm PVA}[\Gamma]$, respectively.
In this talk we will focus on ${\rm KVA}[\Gamma]$.
We will show that this group is an Artin group whose standard generating set is in one-to-one correspondence with the root system of $W[\Gamma]$.
Afterwards, we use this presentation to show that the center of ${\rm VA}[\Gamma]$ is always trivial, and to show that ${\rm VA}[\Gamma]$ has a solvable word problem and finite virtual cohomological dimension when $\Gamma$ is of spherical type or of affine type.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
