## Abstracts

Ivan Cherednik (UNC Chapel Hill)
Title: Riemann hypothesis for plane curve singularities

Laurent Cote (Stanford University)
Title: A sheaf-theoretic SL(2, C) Floer homology for knots
Abstract: I'll outline the construction of an invariant for knots in homology 3-spheres which can be thought of as an SL(2,C) analog of Kronheimer and Mrowka's singular knot instanton homology. This invariant is similar to an invariant of 3-manifolds introduced earlier by Abouzaid and Manolescu, using tools from derived algebraic geometry. I'll therefore begin by summarizing their work and explaining the modifications needed to build this knot invariant. I will then outline some computations of the invariant for various families of knots, and discuss some of its (partly conjectural) properties. This talk is based entirely on joint work with Ciprian Manolescu.

Boris Feigin (HSE, Moscow)
Title: Introduction to Logarithmic VOAs
Abstract: Logarithmic vertex algebras and the corresponding non-semisimple tensor categories enter topology in a number of different ways. This talk, intended for non-experts, will offer an introduction into this subject.

Francesca Ferrari (SISSA Trieste)
Title: Mock modular forms and 3-manifolds
Abstract: Recently, a new homological invariant that categorifies the Witten-Reshetikhin-Turaev invariant has been discovered. This is known as the homological block. When the 3-manifold is a Seifert manifold given by a negative-definite plumbing the homological block turned out to be related to false theta functions and characters of logarithmic CFTs. Little is known, however, about the homological block of the oppositly oriented 3-manifold. In this talk, I introduce the concept of mock modular forms and describe their role in the study of these invariants.

Bostjan Gabrovsek (University of Ljubljana)
Title: Skein modules of lens spaces and stratified Khovanov homology
Abstract: We will show that the HOMFLYPT skein module of lens spaces of type $L(p,1)$ is a free module with an infinite basis and present arguments that this should also be the case for $L(p,q), q>1$. By using similar methods, skein modules of other classes of Seifert manifolds can be computed. We will also take a glance at the stratified Khovanov homology categorifying the Kauffman bracket skein module of the projective space, where we have to, in order for it to work, replace the usual symmetric product of the tensor spaces with the exterior one.

Aaron Lauda (USC)
Title: The Howe duality revolution in link homology and unexplored frontiers
Abstract: The categorical braid group actions used in many link homology theories can all be understood as arising from a single source using the language of higher representation theory. Within the theory of categorified quantum groups in type A one can define Rickard complexes that categorify the quantum Weyl group (braid group) action on representations. Via 2-representations of categorified quantum groups the quantum Weyl group action is transported to give the braiding in sl(n) homology, (singular) Soergel bimodules, and many other settings. Even the elementary foam presentation of Khovanov homology and its sl(n) variants can be understood as 2-representations where the foam relations arise from relations in the categorified quantum group. Many of the structural properties of these homology theories can naturally be understood in this framework. From the higher representation theoretic perspective, the Rickard complexes giving these categorical braidings appear to admit a great deal more structure, the implications of which have largely been unexplored in the context of link homology. Here we present one such example showing that natural structures on Rickard complexes give rise to curved braiding complexes and deformed link homologies generalizing those of Batson-Seed for sl(2), Gorsky-Hogancamp for HOMFLYPT homology, and the algebo-geometric deformations of sl(n) homology defined by Cautis and Kamnitzer. This is joint work with Sabin Cautis and Joshua Sussan.

Sven Meinhardt (U. of Sheffield)
Title: Donaldson-Thomas theory of oriented compact 3-manifolds
Abstract: Donaldson-Thomas theory has been invented around 2000 as an alternative way to count curves in Calabi-Yau 3-folds. In the subsequent years the theory has been extended in many directions and todays foundation of Donaldson-Thomas theory has very little in common with its original approach. In particular, the theory is not restricted to curve counting in complex geometry. Instead, there are applications in representation theory as well as in low dimensional topology. My talk should provide a gentle introduction into the principles and main ideas of Donaldson-Thomas theory with an eye towards applications in topology. More precisely, we will illustrate the construction of some infinite dimensional Cohomological Hall algebra as well as some graded Lie algebra whose (graded) dimension provides a sophisticated count of flat complex connections on an oriented compact 3-manifold.

Du Pei (Caltech)
Title: 4-manifolds and topological modular forms
Abstract: Quantum field theories in higher dimensions predict many hidden algebraic structures in low-dimensional topology. In this talk, I will survey some recent developments in this direction and introduce a new class of invariants of 4-manifolds that are topological modular forms.

Raphael Rouquier (UCLA)
Title: Higher Lie structures
Abstract: I will explain how higher-dimensional structures (within homotopical algebraic geometry) arise from categories, creating non-linear versions of Lie algebras and vertex operators.

Title: Skein modules and algebras.
Abstract: A skein module of a 3-manifold M is built of R[q^{\pm 1}]-linear combinations of links (or, more generally, graphs) in M modulo certain local "skein" relations. Stacking links in skein modules of thickened surfaces makes them into associative "skein algebras". The skein module of a manifold M is a module over the skein algebra of its thickened boundary. Skein modules and algebras are usually q-deformations of coordinate rings of certain moduli spaces. For example, the skein module based on the Kauffman bracket quantises the SL(2)-character varieties and the corresponding skein algebra of a surface is a version of the quantum Teichmuller space of Checkhov-Fock. We will survey the theory of skein modules and algebras and its applications.

Yuuji Tanaka (Oxford)
Title: On orientations for gauge-theoretic moduli spaces
Abstract: This talk is based on joint work with Dominic Joyce and Markus Upmeier. The first half of the talk will be a mild introduction of the anti-self dual instanton moduli spaces, aiming to deliver backgrounds and motivation of problems on orientations for gauge-theoretic moduli spaces for general audience in the seminar. I then move on to telling more general framework and techniques dealing with the problems, and describe new results on the orientation problems for the anti-self-dual instanton moduli spaces and other interesting gauge-theoretic moduli spaces.

Paul Wedrich (Australian National U.)
Title: On categorification of skein modules and algebras
Abstract: Khovanov homology and its cousins are usually defined as functorial invariants of links in R^3. Embracing their reliance on link projections as a virtue, they admit extensions to links in thickened surfaces, and, thus, categorify surface skein modules and, conjecturally, their algebra structures. Skein algebras are related to character varieties and quantum Teichmüller theory, and are the subject of positivity conjectures that appear in reach of categorification techniques. The focus of this talk will be recent joint work with Hoel Queffelec on functorial gl(2) surface link homologies.

Schedule