Institute for Computational and Experimental Research in Mathematics (ICERM)

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.

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.

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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.

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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.

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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.

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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.

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There is a well-known homomorphism from Artin's braid group to (the group of invertible elements of the) Iwahori-Hecke algebra of the symmetric group, or more generally from any Artin-Tits group to the corresponding Hecke algebra. Consider the positive lifts of the elements of the Coxeter group in the Artin-Tits group. Then their images in the Hecke algebra yield the so-called standard basis of the Hecke algebra. Elements of the standard basis have a positive expansion in one of Kazhdan and Lusztig's canonical bases, i.e., have coefficients which are Laurent polynomials with nonnegative coefficients.
In the case where the Coxeter group is finite, the positive lifts of the elements of the Coxeter group in the Artin-Tits group are the so-called simple elements of the classical Garside structure. An alternative Garside structure, called dual Garside structure, was introduced for spherical type Artin-Tits groups. One can wonder if the images of these elements in the Hecke algebra still have a positive KL expansion or not. This is especially interesting in type A, as simple dual braids yield a basis of the Temperley-Lieb quotient of the Hecke algebra.
We will explain how positivity of images of simple dual braids can be obtained in spherical type using a generalization of Kazhdan and Lusztig's inverse positivity, which predicts that certain elements of Artin-Tits groups, which we call ""Mikado braids"", have a positive Kazhdan-Lustig expansion, together with the fact that simple dual braids are Mikado braids. The positivity of the KL expansion of Mikado braids, shown for finite Weyl groups by Dyer and Lehrer, can be generalized to arbitrary Coxeter systems by adapting a result of Elias and Williamson on the perversity of minimal Rouquier complexes of positive simple braids to a ""twisted"" setting as introduced by Dyer, and asks the question of determining which braids have a minimal braid complex which is perverse.

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Artin (or Artin-Tits) groups are generalizations of braid groups that are defined using a finite set of generators $S$ and relations $abab\cdots=baba\cdots$, where both words of the equality have the same length. Although this definition is quite simple, there are very few results known for Artin groups in general. Classic problems as the word problem or the conjugacy problem are still open. In this talk, we study a problem concerning a family of subgroups of Artin groups: parabolic subgroups. These subgroups have proven to be useful when studying Artin groups --for example, they are used to build interesting simplicial complexes--, but again, we do not know much about them in general. Our problem will be the following: Given two conjugate elements of a parabolic subgroup $P$ of an Artin group $A$, are they conjugate via an element of $P$? This is called the conjugacy stability problem. In 2014, González-Meneses proved that this is always true for braids, that is, geometric embedding of braids do not merge conjugacy classes. In an article with Calvez and Cisneros de la Cruz, we gave a classification for spherical Artin groups an proved that the answer to the question is not always affirmative. In this talk, we will explain how to give an algorithm to solve this problem for every Artin group satisfying three properties that are conjectured to be always true.

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Braid groups belong to a broad class of groups known as Artin groups, which are defined by presentations of a particular form. These groups fall into two classes, finite-type and infinte-type Artin groups. The former come equipped with a powerful combinatorial structure, known as a Garside structure, while the latter are much less understood and present many challenges. However, if one restricts to the Artin monoid, a submonoid of the Artin group, then some aspects of Garside theory still apply in the infinite-type case. I will talk about joint work with Rachael Boyd and Rose Morris-Wright on geometric relations between Artin monoids and Artin groups.

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The braid groups have two well known Garside presentations. The elegant minimal standard presentation is closely related to the Salvetti complex, a cell complex derived from the complement of the complexification of the real braid arrangement. The dual presentation, introduced by Birman, Ko and Lee, leads to a second Garside structure and a second classifying space, but it has been less clear how the dual braid complex is related to the (quotient of the) complexified hyperplane complement, other than abstractly knowing that they are homotopy equivalent. In this talk, I will discuss recent progress on this issue. Following a suggestion by Daan Krammer, Michael Dougherty and I have been able to embed the dual braid complex into the complement of the complex braid arrangement. This leads in turn to a whole host of interesting complexes, combinatorics, and connections to other parts of the field. This is joint work with Michael Dougherty.

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Since the pioneering work of Chuang and Rouquier, the construction of highly nontrivial derived equivalences has been one of the most powerful tools resulting from higher representation theory. Cautis-Kamnitzer-Licata showed these derived equivalences arising from categorified quantum groups gave rise to categorical actions of braid groups of the corresponding Lie type with Chuang-Rouquier's equivalences corresponding to the elementary braid generators. In 2011, motivated by the discovery of odd Khovanov homology, Ellis-Khovanov-Lauda proposed a new `odd' categorification of sl2. At the same time, this `odd sl2' was independently discovered by Kang-Kashiwara-Tsuchioka who were investigating super categorifications of Kac-Moody algebras. In this talk we will explain joint work with Mark Ebert and Laurent Vera giving new super analogs of the derived equivalences studied by Chuang and Rouquier coming from the odd categorification of sl2. Just as Chuang and Rouquier used their equivalences to achieve new results on the modular representation theory of the symmetric group, we will discuss how our new super equivalences can be applied to the spin symmetric group.

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How does an object of a triangulated category evolve under repeated applications of an auto-equivalence? I will describe how this amorphous question can be made precise using a Bridgeland stability condition. For 2-CY categories associated to A_n quivers, I will describe how this investigation turns out to be a categorified version of well-studied notions in combinatorial geometry.

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Categorifying Burau Representations and Fusion Categories
Edmund Heng, The Australian National University
In this talk, we will look at a categorification of the Burau representations for the non-simply-laced type braid groups, generalising a construction given by Khovanov-Huefarno and Rouquier-Zimmermann. This will involve building certain algebra objects in the fusion categories associated to the quantum group sl2.

Census L-space knots are braid positive, except for one that is not
Marc Kegel, Humboldt-Universität zu Berlin
I will explain and prove the statement in the title. This is based on joint work with Ken Baker.

Kazhdan-Laumon Categories and the Symplectic Fourier Transform
Calder Morton-Ferguson, MIT
In 1988, Kazhdan and Laumon defined a “glued category” of perverse sheaves on the basic affine space. The key ingredient in their construction was the symplectic Fourier transform, which gives an action of the braid group on the category of perverse sheaves. They proposed a new construction of representations of Chevalley groups using this category, but this proposed construction depended on a conjecture which was later shown to be false. In this talk, we will discuss the action of the symplectic Fourier transform as a representation of the braid group. We will then discuss progress toward reworking Kazhdan-Laumon’s construction in the context of braids.

A hooking conjecture on circle graphs motivated by Khovanov homology
Marithania Silvero Casanova, Universidad de Sevilla
We present a conjecture stating that the independence complex of any circle graph is homotopy equivalent to a wedge of spheres. This conjecture is motivated by the fact that extreme Khovanov homology of a link diagram $D$ coincides with the cohomology of the independence complex associated to its Lando graph (Lando graphs are bipartite circle graphs). We also give some advances on the proof of this conjecture; in particular, we prove it for permutation graphs, non-nested graphs, and graphs associated to closed braids with less than 5 strands.

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Let \lambda be a partition of n. We consider the Kazhdan-Lusztig basis of the corresponding Specht module, which is indexed by standard Young tableau of shape \lambda. One of the amazing features of this basis is that it can be used to relate representation theoretic properties of Specht modules to combinatorial properties of tableau. For example, in the 90s Berestein-Zelevinsky and Stembridge showed that the long element of the symmetric group acts on the Kazhdan-Lusztig basis by the Schutzenberger involution on tableau. Similarly, in 2010 Rhoades showed that the long cycle (1,2,...,n) acts by the jeu de taquin promotion operator when \lambda is rectangular. In this talk we will explain how to use braid groups acting on triangulated categories to generalize Rhoades' result in three directions: we lift the condition on the shape of the partition, we greatly enlarge the class of permutations for which the result holds, and we prove analogs in other Lie types. This is based on joint work with Martin Gossow.

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Topological quantum field theories coming from semisimple categories build upon interesting structures in representation theory and have important applications in low dimensional topology and physics. The construction of non-semisimple TQFTs is more recent and they shed new light on questions that seem to be inaccessible using their semisimple relatives. In order to have potential applications to physics, these non-semisimple categories and TQFTs should possess Hermitian structures. We will define these structures and give some applications.

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Associated to a positive braid, we define an affine algebraic variety via an explicit set of polynomial equations. I will give properties of these varieties, including their dimension, smoothness properties and a realization as a moduli space of chains of flags. I will also explain how some classical varieties in Lie theory, such as positroid and more generally Richardson varieties, appear in this way, as well as a connection to the computation of the Khovanov-Rozansky homology of the link obtained by closing the braid. This is joint work with Roger Casals, Eugene Gorsky and Mikhail Gorsky.

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In the talk I will define braid varieties, a class of affine algebraic varieties associated to positive braids. I will discuss their relation to Richardson and positroid varieties, HOMFLY polynomial and HOMFLY homology, and Legendrian link invariants. This is a joint work with Roger Casals, Mikhail Gorsky and Jose Simental Rodriguez.

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In 1970, Arnold proved that the homology groups of the braid groups on n strands stabilizes as n tends to infinity, a phenomenon called "homological stability". The pure braid groups, in contrast, are not homologically stable. In this (partly expository) talk I will describe a sense in which (co)homology groups of the pure braid groups do stabilize when we take into account the natural symmetric group actions. We will use tools from "representation stability" to shed light on the structure of the (co)homology of the pure braid groups, and many of their generalizations. This talk will survey work of Church, Ellenberg, and Farb, and joint work with Miller. 

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Khovanov and Seidel defined an action of the braid group by autoequivalences of a certain category of projective modules over the so-called zigzag algebra. Taking the Grothendieck group, one recovers the famous Burau representation, but unlike the latter, Khovanov-Seidel representation is faithful. In work with Licata, I showed how to use Khovanov-Seidel representation to extract metric data on braids. Building upon this idea, I'll try to convince the audience that such categorical tools should play in the larger context of geometric group theory.

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We will discuss new categorical interpretations of two distinct $q$-deformations of the rational numbers. The first one was introduced in a different context by Morier-Genoud and Ovsienko, and enjoys fascinating combinatorial, topological, and algebraic properties. The second one is a natural partner to the first, and is new. We obtain these deformations via boundary points of a compactification of the space of Bridgeland stability conditions on the 2-Calabi--Yau category of the $A_{2}$ quiver. The talk is based on joint work with Louis Becker, Anand Deopurkar, and Anthony Licata.

• Video Available Soon

Abstract

The configuration space of particles on a graph is a classifying space for the graph's braid group and thus computes the group cohomology. If instead one considers compactly supported cohomology the resulting groups depend only on the genus of the graph, or "loop order", and admit a particularly interesting action by Out(F_g). In this talk I will explain how tropical geometry relates these latter representations to the cohomology of the moduli spaces M_{g,n} and discuss computational approaches.

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