Categorification in quantum topology
and beyond

Quantised lattices, or q-lattices, appear naturally through categorification constructions - for example from zigzag-algebras - but they haven't been studied from a lattice theory point of view. After establishing the necessary background, we'll explain the q-versions of various lattice theory concepts and results, illustrated with examples from combinatorics and graph theory. We'll also list open problems on the q-lattice and categorical level. (With Tony Licata and Leo Jiang.)

Bernstein operators arise as vertex operators that create and annihilate Schur polynomials. These operators play a significant role in the mathematical formulation of the Boson-Fermion correspondence due to Kac and Frenkel. The role of this correspondence in mathematical physics has been widely studied as it bridges the actions of the infinite Heisenberg and Clifford algebras on Fock space. Cautis and Sussan conjectured a categorification of this correspondence within the framework of Khovanov's Heisenberg category. I will discuss how to categorify the Bernstein operators and settle the Cautis-Sussan conjecture, thus proving a categorical Boson-Fermion correspondence.

The skein algebra of a surface has relations to the character variety, the Jones polynomial and its TQFT, the (quantum) Teichmuller space and the (quantum) cluster algebra of surfaces.It serve as a bridge between classical and quantum topology.
This minicourse is an introduction to the skein algebra. We will cover:
+ basic definitions and facts, relation to character variety.
+ triangular decomposition -- quantization of Thurston shear coordinates-- and relation to quantum special linear group SL_2(q).
+ quantum Teichmuller space and quantum trace map. Positivity.
+ Representations of skein algebras and hyperbolic TQFT.

The factorisation homology of a surface with coefficients in a quantum group gives a quantisation of the character stack and its algebra of invariants gives a quantisation of the character variety of the surface. In this talk I show that for two simple surfaces, the punctured torus and four-punctured sphere with coefficients in quantum SL_2, the algebra of invariants obtained is isomorphic to the Kauffman bracket skein algebra of the same surface and hence in these cases one obtains the same quantised character variety via both factorisation homology and skein algebras.

The skein algebra of a surface has relations to the character variety, the Jones polynomial and its TQFT, the (quantum) Teichmuller space and the (quantum) cluster algebra of surfaces.It serve as a bridge between classical and quantum topology.
This minicourse is an introduction to the skein algebra. We will cover:
+ basic definitions and facts, relation to character variety.
+ triangular decomposition -- quantization of Thurston shear coordinates-- and relation to quantum special linear group SL_2(q).
+ quantum Teichmuller space and quantum trace map. Positivity.
+ Representations of skein algebras and hyperbolic TQFT.

Skein modules are a natural extension of the Jones polynomial to 3-manifolds. In the case where the manifold is a thickened surface, they naturally come with a (usually) non-commutative multiplicative structure. I'll present recent advances in the categorification by foams of these structures and related knot invariants, before discussing open questions and conjectures. (joint work with Paul Wedrich)

The skein algebra of a surface has relations to the character variety, the Jones polynomial and its TQFT, the (quantum) Teichmuller space and the (quantum) cluster algebra of surfaces.It serve as a bridge between classical and quantum topology.
This minicourse is an introduction to the skein algebra. We will cover:
+ basic definitions and facts, relation to character variety.
+ triangular decomposition -- quantization of Thurston shear coordinates-- and relation to quantum special linear group SL_2(q).
+ quantum Teichmuller space and quantum trace map. Positivity.
+ Representations of skein algebras and hyperbolic TQFT.

The Khovanov–Rozansky (KhR) polynomials are topological invariants, associated with a knot, with a quantum group, and with certain homologies at the same time. The last ingredient makes these invariants really hard to study, and especially to compute explicitly. In particular, general KhR invariants are not expressed by the analytic formulas, unlike, e.g., the HOMFLY polynomials, which are the Euler characters of the same complex. Yet, there are analytic formulas for the KhR polynomials for certain knot families (the most common case includes the positive torus knots). The peculiarities of the KhR polynomials, associated with the homology calculus, are then revealed in the non-trivial (non-analytic) behavior of the KhR invariants on the boundaries of the stability regions in the space of knot parameters. Similar phenomenon happens for particular values of N associated with the quantum group SL_N. The talk is based on the joint work with A. Morozov, P. Dunin-Barkowski, and A. Popoliov.

The skein algebra of a surface has relations to the character variety, the Jones polynomial and its TQFT, the (quantum) Teichmuller space and the (quantum) cluster algebra of surfaces.It serve as a bridge between classical and quantum topology.
This minicourse is an introduction to the skein algebra. We will cover:
+ basic definitions and facts, relation to character variety.
+ triangular decomposition -- quantization of Thurston shear coordinates-- and relation to quantum special linear group SL_2(q).
+ quantum Teichmuller space and quantum trace map. Positivity.
+ Representations of skein algebras and hyperbolic TQFT.

I'll give a combinatorial and down-to-earth definition of the symmetric gl(1) homology. It is a (non-trivial) link homology which categorifies the trivial link invariant (equal to 1 on every link). Then I'll explain how to use this construction to categorify the Alexander polynomial. Finally, I will relate these construction to the Hochschild homology of Soergel bimodules. (joint with E. Wagner)

Khovanov and Rozansky defined a link invariant called triply graded homology. It is conjectured by Gorsky, Negut and Rasmussen that this invariant can be expressed geometrically by a functor from complexes of Soergel bimodules to the derived category of coherent sheaves on the dg flag Hilbert scheme followed by taking cohomology. A functor with similar properties has been constructed by Oblomkov and Rozansky using matrix factorizations and it is believed that this functor solves the conjecture. The aim of this joint work in progress with Roman Bezrukavnikov is to relate the two constructions using previous work of Arkhipov and Kanstrup.

I will outline a construction of a 3D TQFT whose algebraic output is a Hopf algebra rather then its category of modules, and discuss examples where this category of modules is not semisimple.

The bordered Floer homology of Lipshitz, Ozsvath and Thurston (partly) extends Heegaard Floer homology to a 2+1+1 dimensional TQFT. Its decategorification is a 2+1 dimensional TQFT previously studied by Donaldson (in the context of Seiberg-Witten theory) and Petkova. In this talk I will explain how this TQFT can be usefully revisited from the perspective of covering spaces of the Jacobian. Despite the potentially scary title, the talk is aimed at categorifiers and will not assume any knowledge of Floer homology.

This talk will be an introduction to the field of what we call finitary 2-representation theory (which is a categorical lift of the representation theory of a finite-dimensional algebra; hence the word finitary), as well as a survey of the state of the arts. I will explain the main ideas and prototypical examples.

I will present DG-enhanced versions of cyclotomic Khovanov-Lauda-Rouquier algebras and explain how to use them to categorify parabolic Verma modules for (symmetrizable) quantum Kac-Moody algebras. I also present some ramifications of the program of categorification of Verma modules.

A basic tool used in the study of Kac-Moody Weyl groups is the linear action of the Weyl group on the root lattice, where the the generators act by reflections. The braid groups of these Weyl groups are not nearly as well understood, but they have a similar linear incarnation, where the braid generators act by spherical twists on a triangulated category. In much the same spirit as for Weyl groups, it is tempting to study group theoretic questions about braid groups (e.g. dynamics, word and conjugacy problems) by using the action on a triangulated category and the tools of higher representation theory. The goal of this talk will be to highlight some of the group theoretic questions that one can (or should be able to) answer in this way.

Fusion categories and their fusion algebras arising from quantum groups at roots of unities give for instance rise to 3-manifold invariants, are interesting from a more representation theoretic point and were partly also studied via methods from integrable systems. In this talk I like to revisit some of the classical theory and study the fusion algebras for classical Lie algebras again in more detail from a new perspective. In particular I like to connect it with Cherednik's double affine Hecke algebras (DAHAs).

The Witten-Reshetikhin-Turaev invariants and TQFTs are constructed from a weak version of quantum sl(2) at root of 1. There are variants of quantum sl(2) producing other families of quantum invariants. We will focus on unrolled and non restricted versions and discuss the corresponding invariants. This is based on work with Costantino-Geer-Patureau and Geer-Patureau-Reshetikhin respectively.

I will survey the program aiming to construct 4-dimensional field theories via higher representation theory. I will discuss both the usual Lie-theoretic setting (for example, sl2) and the case of gl(1|1) and its connection to Heegaard-Floer theory.

I'll present a recent result with Alexei Oblomkov that realizes some ideas of Hikita and others, connecting the diagonal coinvariant algebra DR_n with the homology of a certain Springer fiber in the affine flag manifold (for instance, they are isomorphic as abstract vector spaces). One consequence of our theorem is the existence of an explicit monomial basis of DR_n, which categorifies the Haglund-Loehr formula for its Hilbert series. We hope that applications to the perverse filtration will be forthcoming.

Rouquier categorified the braid group using Soergel bimodules. I will describe various properties of the category generated by Rouquier complex of the full twist, and its relation to the algebraic geometry of the Hilbert scheme of points. The connections to Khovanov-Rozansky homology will be also outlined.

This talk is about unexplained coincidences of decomposition numbers between seemingly unrelated objects. The decomposition matrix of a unipotent block of a finite group of Lie type in cross characteristic has a square submatrix indexed by the unipotent characters. Many low-rank examples of these decomposition matrices were computed in recent years by Dudas and Malle. In many cases, the matrices obtained are identical on the principal series characters, which are indexed by the irreducible characters of the Weyl group, to decomposition matrices I computed for the rational Cherednik algebra at a corresponding parameter. I will explain structural parallels and differences between the two theories and summarize the numerical data. Then I will provide examples showing that in general the decomposition matrix of the Cherednik algebra is not submatrix of the decomposition matrix of the finite group -- although it appears in all known cases that its decomposition numbers are lower bounds for a submatrix. I will also conjecture the full branching rule for Harish-Chandra induction in classical types, extending a conjecture of Gerber-Hiss-Jacon which is now a theorem of Gerber-Hiss and Dudas-Varagnolo-Vasserot.

There are two famous geometric construction of representations of a simply-laced simple algebraic group. One uses quiver varieties and the other uses affine Grassmannians. We use the ideas of categorical symplectic duality to explain the link between these two constructions. I will explain that the category of highest weight modules for truncated shifted Yangians (which quantize affine Grassmannian slices) is equivalent to the category of modules for Khovanov-Lauda-Rouquier-Webster algebras. Modules for KLRW algebras were earlier related to quantized quiver varieties.

The study of p-DG algebras arises naturally when trying to category quantum groups at prime roots of unity. We will review this framework and construct some categorical representations for the case of sl(2).

Many applications of ideas from physics to categorification, and more generally to produce invariants in geometry and topology, rely on constructions involving twists of supersymmetric field theories. To begin, I'd like to say a few words about this relationship and give a general overview of the twisting procedure, aimed at those in the audience to whom "supersymmetric field theory" may be unfamiliar. I'd also like to spend some time discussing a perspective according to which the classification of all possible twists of supersymmetric theories, the understanding of relationships between those twists, and the construction of the supersymmetric theories themselves can be thought of in terms of the algebraic geometry of certain varieties, in any dimension and with any amount of supersymmetry. The last of these points hinges on a close relationship between twisting and the pure spinor superfield formalism. This is joint work with Johannes Walcher and Richard Eager.

An Azumaya algebra A is a non-commutative algebra with a large center Z, such that etale locally over spec(Z), A is simply a matrix algebra over the base field. In this talk I will explain that quantum character varieties, in the form we introduced them with Ben-Zvi and Brochier, are Azumaya algebras over their classical degenerations, when the quantum parameter q is a root of unity. The technique involve the quantum Frobenius homomorphism, quantum Hamiltonian reduction, and the remarkable technique of Poisson orders. This is joint work with Iordan Ganev and Pavel Safronov.
Beautiful results of Bonahon--Wong and Frohman--Kania-Bartoszynksa--Le state that the Kauffman bracket skein algebra is Azumaya (over an open subset). I will outline a hope to connect the two pictures, using work of Cooke, and work in progress with Brochier and Snyder.

In the first part of this talk we will review the interplay between the quantum sl2 invariant for tangles and Springer theory for the symmetric group. We will explain how one can study the topology of certain Springer fibers using the quantum group combinatorics and how the Springer fibers can be used to categorify certain representations of the quantum group. A key ingredient is given by Schur-Weyl duality. In the second part of the talk we will exchange the symmetric group for a Weyl group of type B/C. In this case certain exotic Springer fibers and tangles in a thickened disc with two punctures naturally make their appearance.