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Christian Blanchet (Université de Paris)

Heisenberg homology of surface configurations We study the action of the mapping class group of Σ=Σ_{g,1} on
the homology of configuration spaces with coefficients twisted by the discrete Heisenberg group H=H(Σ).
We will specialise to the Schrödinger representation and its finite
dimensional analogue in which case we obtain representations of a
central extension of the mapping class group.

Sabin Cautis* (University of British Columbia)

Vertex
operators in knot homology Vertex
operators can inspire some fairly explicit
constructions of homological knot invariants. To
explain this statement we survey some categorical
constructions from the past 10-15 years.

Nicolle Gonzalez* (UCLA)

A skein theoretic Carlsson–Mellit algebra 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.

The Trace of the affine Hecke category Morton and Samuelson related the skein algebra of
the torus to a certain specialization of the elliptic
Hall algebra. I will report on progress towards the
categorification of their result, replacing the skein
algebra by the horizontal trace of the affine Hecke
category. This is a joint work with Andrei Negut.

Decorated TQFTs: from q-series to non-semisimple invariants
Invariants of knots and 3-manifolds in
Quantum Topology are usually constructed from basic
ingredients of algebraic origin. Such algebraic
structures, in turn, describe symmetries of integrable
lattice models (as in the case of quantum groups) or
vertex operator algebras (as in the case of tensor
categories). In simple instances, as e.g. in the WRT
invariants of 3-manifolds and in the corresponding knot
invariants, these symmetries (algebraic structures) do
not have interesting symmetries of their own,
i.e. automorphisms. When they do, however, the
cutting-and-gluing rules in the corresponding TQFT
become more interesting and non-trivial. The goal of
this talk, based on the recent work with François
Costantino and Pavel
Putrov arXiv:2107.14238,
is to describe a unified framework ("decorated TQFT")
that can put under the same roof additional structures /
decorations that appear e.g. in the study
of G-crossed modules, sutured and bordered Floer
homology, Ẑ and BCGP invariants, and
various other TQFT-like structures that until recently
have been developed independently. In other words, the
goal is to build new bridges that can facilitate
translation between different ways decorations are
described in these theories, thereby allowing to explore
relations among them and building new decorated
TQFTs.

Topological theories and
automata We'll explain the connection
between topological theories for one-manifolds with
defects and values in the Boolean semiring and automata
and their generalizations. A finite state automaton
accepts a regular language. To each pair of a regular
language and a circular regular language we associate a
topological theory for one-dimensional manifolds with
zero-dimensional defects labelled by letters of the
language. This theory takes values in the Boolean
semiring B. Universal construction gives rise to
a monoidal category of B-semilinear combinations
of one-dimensional cobordisms with defects modulo skein
relations. The latter category can be interpreted as a
semilinear rigid monoidal closure of standard structures
associated to a regular language, including minimal
deterministic and nondeterministic finite state automata
for the language and the syntactic monoid. The circular
language plays the role of a regularizer, allowing to
define the rigid closure of these structures.
The talk is based on a joint paper in progress with Mee Seong Im.

Khovanov
homology and rational unknotting In this
talk, we will see a new geometric application of
Khovanov homology, generalizing work by Alishahi and
Dowlin. We'll use a universal Khovanov homology theory
that associates to a knot diagram a ℤ[G]-complex C,
where G is a formal variable, such that C/(G=1) has
homology ℤ. We'll define a metric on homotopy classes
of such ℤ[G]-complexes. This metric turns out to
provide a lower bound for the proper rational
unknotting number, i.e. the minimal number of
connectivity preserving rational tangle replacements
needed to make a knot trivial. This talk is based on
joint work with Damian Iltgen and Laura Marino (see
https://arxiv.org/abs/2110.15107).

Khovanov homology and non-orientable surfaces The first half of the talk will introduce an
invariant of non-orientable knot cobordisms, coming
from deformed Khovanov homology, imitating the
construction of the Heegaard Floer mixed invariant
(i.e., the Seiberg–Witten invariant). The second half
will discuss properties and computations of Khovanov
homology invariants of cobordisms, including
leveraging work of Hayden–Sundberg to show that they
distinguish some exotic pairs of non-orientable
surfaces. This is joint with Sucharit
Sarkar.

A knot Floer stable homotopy type Given a grid diagram for a knot or link K in 𝕊^{3},
we construct a spectrum whose homology is the knot
Floer homology of K. We conjecture that the homotopy
type of the spectrum is an invariant of K. Our
construction does not use holomorphic geometry, but
rather builds on the combinatorial definition of grid
homology. We inductively define models for the moduli
spaces of pseudo-holomorphic strips and disk bubbles,
and patch them together into a framed flow
category. The inductive step relies on the vanishing
of an obstruction class that takes values in a complex
of positive domains with partitions. (This is joint
work with Sucharit Sarkar.)

Anton Mellit (Universität Wien)

Knot homology, tautological classes and 𝔰𝔩_{2} I will talk about a new kind of structure we
discovered on the triply graded Khovanov–Rozansky link
homology, motivated by a study of character
varieties. The complexes whose homology is the link
homology are endowed with an action of a certain DG
algebra. Applying the Koszul duality we pass to the
y-ified Khovanov–Rozansky homology, which now acquires
an action of a commuting family of operators we call
tautological classes. Lefschetz property satisfied by
the second tautological class implies the symmetry
conjectured by Dunfield, Gukov and Rasmussen. This is
a joint work with Eugene Gorsky and Matt
Hogancamp

Annular
link Floer homology and 𝖌𝖑(1|1) The Reshetikhin–Turaev
construction for the quantum
group U_{q}(𝖌𝖑(1|1)) sends tangles to
ℂ(q)-linear maps in such a way that a
knot is sent to its Alexander polynomial. Tangle Floer
homology is a combinatorial generalization of knot
Floer homology which sends tangles to (homotopy
equivalence classes of) bigraded dg bimodules. In
earlier work with Ellis and Vertesi, we show that
tangle Floer homology categorifies a
Reshetikhin–Turaev invariant arising naturally
in the representation theory
of U_{q}(𝖌𝖑(1|1)); we further construct
bimodules E and F corresponding
to E, F in U_{q}(𝖌𝖑(1|1)) that
satisfy appropriate categorified relations. After a
brief summary of this earlier work, I will discuss how
the horizontal trace of the E and F
actions on tangle Floer homology gives a 𝖌𝖑(1|1)
action on annular link Floer homology that has an
interpretation as a count of certain holomorphic
curves. This is based on joint work in progress with
Andy Manion and Mike Wong.

Jacob Rasmussen* (University of Cambridge)

Knot Floer homology, sutures, and the solid torus Over the last few years there has been a lot of
progress on categorified invariants for knots in the
solid torus. In the hope that comparison with knot
Floer homology may still be a fruitful exercise, I'll
give a graphical description of HF̂K for such
knots. This is an extension of earlier work with
Jonathan Hanselman and Liam Watson.

Marko Stosic* (Universidade de Lisboa)

Knot
invariants, knot complements and quivers
In this talk I will give an overview of the
knots-quivers correspondence, from the first
formulation till some of the recent results.
Originally knots-quivers correspondence was introduced
by rewriting colored HOMFLY-PT invariants in the form
of quiver generating series for suitable quivers.
Recently, other knot invariants have also been shown
to be re-writable in a "quiver form", namely
Gukov–Manolescu Ẑ-invariants of
knot complements. I will present some basic facts
about these correspondences, as well as relations
between them.

Joshua Sussan (CUNY)

p-DG structures in higher representation theory One of the goals of the categorification program
is to construct a homological invariant of 3-manifolds
coming from the higher representation theory of
quantum groups. The WRT 3-manifold invariant uses
quantum groups at a root of unity. p-DG theory was
introduced by Khovanov as a means to categorify
objects at prime roots of unity. We will review this
machinery and show how to construct categorifications
of certain representations of quantum 𝖘𝖑(2) at prime
roots of unity.

Pedro Vaz (Université Catholique de Louvain) Slides.

2-Verma
modules and link homologies In this talk
I will explain how 2-Verma modules can be used to
produce several link homologies for knots in the
3-space and in the solid torus. The various
constructions go through a categorification of
(parabolic) Verma modules and its tensor products with
finite-dimensional irreducibles of 𝖌𝖑(n). The
material presented spreads along several
collaborations with Abel Lacabanne, and Grégoire
Naisse.

Emmanuel Wagner (Université de Paris)

Categorification of
colored Jones polynomial at root of unity
In the first part of the talk, we will define colored
symmetric 𝖌𝖑(2) homologies. These are
homological link invariants. We'll focus on a
combinatorial and elementary approach of this
construction and explain how it is related to Soergel
bimodules and to triply graded homology. In the second
part, we'll see how to endow them with pDG structures
and providing so a categorification of colored Jones
polynomial at root of unity. This is a joint work with
You Qi, Louis-Hadrien Robert and Joshua Sussan.

A skein relation for singular Soergel bimodules Soergel bimodules categorify Hecke algebras and
lead to invariants of braids that take values in
monoidal triangulated categories. In this process, the
quadratic `skein relation' on Artin generators is
promoted to a distinguished triangle. I will talk
about an analog of this relation in the setting of
singular Soergel bimodules and Rickard complexes, in
which the distinguished triangle gets replaced by a
longer one-sided twisted complex. Joint work with
M. Hogancamp and D.E.V. Rose.

All talks will be accessible by zoom in the meeting room 641 4366 0478. The password is the last name of the mathematician who discovered a polynomial knot invariant distinguishing left and right trefoils.