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Timetable (HTLW04)

Quantum topology and categorified representation theory

Monday 26th June 2017 to Friday 30th June 2017

Monday 26th June 2017
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director)
10:00 to 11:00 Christian Blanchet
Non semisimple TQFTs from quantum sl(2)
We will describe quantum invariants and TQFTs extracted from quantum sl(2) at root of unity, focusing on new features related with the non semisimplicity of the quantum group at root of 1.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Zsuzsanna Dancso
Lattices and Homological Algebra
Co-author: Anthony Licata (Australian National University)

I'll begin with some dreams and motivation regarding lifting notions of lattice theory to homological algebra. For most of the talk, we'll study a concrete example: the lattices of integer cuts and flows associated to a finite graph. Given a graph G and choice of spanning tree T, we construct an algebra A(G,T) such that the Groethendieck group of the category of finitely generated A(G,T)-modules with the Euler (Ext) pairing contains the cut and flow lattices of G as orthogonal complements. We'll discus many open problems regarding generalisations and possible uses for the extra structure that is present at the category level, such as gradings. Joint work in progress with Anthony Licata.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Peter Samuelson
Hall algebras and Fukaya categories
(joint with Ben Cooper)

The multiplication in the Hall algebra of an abelian category is defined by "counting extensions of objects," and the representation theory of this algebra tends to be quite interesting (e.g. Ringel showed the Hall algebra of modules over a quiver is the quantum group). Recently, Burban and Schiffmann explicitly described the Hall algebra of coherent sheaves over an elliptic curve, and various authors have connected this algebra to symmetric functions, Hilbert schemes, torus knot homology, the Heisenberg category, and the skein algebra of the torus. Motivated by this last connection and homological mirror symmetry, we discuss some computations in progress involving the Hall algebra of the Fukaya category of a (topological) surface.
14:30 to 15:30 David Rose
Traces, current algebras, and link homologies
We'll show how categorical traces and foam categories can be used to define an invariant of braid conjugacy, which can be viewed as a "universal" type-A braid invariant. Applying various functors, we recover several known link homology theories, both for links in the solid torus, and, more-surprisingly, for links in the 3-sphere. Variations on this theme produce new annular invariants, and, conjecturally, a homology theory for links in the 3-sphere which categorifies the sl(n) link polynomial but is distinct from the Khovanov-Rozansky theory. Lurking in the background of this story is a family of current algebra representations.

This is joint work with Queffelec and Sartori.

15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Hoel Queffelec
Around Chebyshev's polynomial and the skein algebra of the torus
The diagrammatic version of the Jones polynomial, based on the Kauffman bracket skein module, extends to knots in any 3-manifold. In the case of thickened surfaces, it can be endowed with the structure of an algebra by stacking. The case of the torus is of particular interest, and C. Frohman and R. Gelca exhibited in 1998 a basis of the skein module for which the multiplication is governed by the particularly simple "product-to-sum" formula.
I'll present a diagrammatic proof of this formula that highlights the role of the Chebyshev's polynomials, before turning to categorification perspectives and their interactions with representation theory.

Joint work with H. Russell, D. Rose and P. Wedrich.

17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 27th June 2017
10:00 to 11:00 Mikhail Khovanov
How to categorify the ring of integers localized at two
We construct a triangulated monoidal idempotent complete category with the
Grothendieck ring naturally isomorphic to the ring of integers with two inverted.
This is a joint work with Yin Tian.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Vanessa Miemietz
Introduction to p-dg 2-representation theory
I will report on joint work with Robert Laugwitz developing an abstract 2-representation theory for p-dg 2-categories, in the spirit of categorification of quantum groups at prime roots of unity by Khovanov, Qi and Elias.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Lukas Lewark
An Upsilon-like invariant from Khovanov-Rozansky homology
Co-author: Andrew Lobb (Durham University)

Khovanov-Rozansky homology in its most general form (so-called equivariant homology) associates to a knot a chain complex (invariant up to homotopy equivalence) over a certain polynomial ring. Equivariant homology yields various lower bounds to the slice genus, some of them concordance homomorphisms, some not; and also a piecewise linear function which has much resemblance with the recently introduced Upsilon-invariant from Heegaard-Floer homology.
14:30 to 15:30 Ben Webster
Representation theory and the Coulomb branch
For many years, my collaborators and I tried to understand the Coulomb branches of certain field theories from physics and failed miserably. Luckily, recent work of Braverman-Finkelberg-Nakajima gives a mathematical construction of these spaces, and algebras quantizing them. I'll discuss an approach to the representation theory of these algebras (building on joint work with Braden-Licata-Proudfoot and many other authors). Applications include a version of the Koszul duality between the Higgs and Coulomb branches of such a theory, a new perspective on category O for Cherednik algebras, and a new understanding of coherent sheaves on Coulomb branches in terms of KLR algebras.

Related Links
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Paul Wedrich
On colored link homologies
I will talk about recent progress in understanding the structure of type A link homologies. This includes the definition of integral, equivariant, colored sl(N) Khovanov-Rozansky link homologies, which are functorial under link cobordisms, a study of their deformations and stability properties and the question of how these invariants extend to links in thickened surfaces. 

Joint work with M. Ehrig, H. Queffelec, D. Rose and D. Tubbenhauer.
Wednesday 28th June 2017
10:00 to 11:00 Andrei Negut
Categorified knot invariants and algebraic geometry
In this talk, we will survey some recent progress in a broad and developing framework that seeks to connect categorified knot invariants, geometric representation theory, Hilbert schemes and matrix factorizations. The contributions discussed come from the work of many mathematicians, and I hope to also convey some interesting future perspectives on the topic.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Matthew Hogancamp
Categorical diagonalization
It goes without saying that diagonalization is an important tool in linear algebra and representation theory.  In this talk I will discuss joint work with Ben Elias in which we develop a theory of diagonalization of functors, which has relevance both to higher representation theory and to categorified quantum invariants.  For most of the talk I will use small examples to illustrate of components of the theory, as well as subtleties which are not visible on the linear algebra level.  I will also state our Diagonalization Theorem which, informally, asserts that an object in a monoidal category is diagonalizable if it has enough ``eigenmaps''.  Time allowing, I will also mention our main application, which is a diagonalization of the full-twist Rouquier complexes acting on Soergel bimodules in type A.  The resulting categorical eigenprojections categorify q-deformed Young idempotents in Hecke algebras, and are also important for constructing colored link homology theories which, conjecturally, are functorial under 4-d cobordisms.  
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 17:00 Free Afternoon
19:30 to 22:00 Formal Dinner at Christ's College
Thursday 29th June 2017
10:00 to 11:00 Pedro Vaz
2-Verma modules and the Khovanov-Rozansky link homologies
In this talk I will describe a construction of Khovanov and Rozansky's HOMFLY-PT and sl(N)-link homologies using a categorification of certain parabolic Verma modules for gl(2k), the latter based on a generalization of Khovanov-Lauda-Rouquier algebras.I will also explain how to prove a conjecture about the spectral sequence from the HOMFLY-PT-link homology to the sl(N)-link homology (due to Dunfield, Gukov and Rasmussen) using our version of these link homologies.This is joint work with G. Naisse.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Radmila Sazdanovic
Non-multiplicative TQFTs and diagrammatic categorifications of the polynomial ring
Co-author: Mikhail Khovanov (Columbia University)

We focus on non-multiplicative TQFTs via representations of non-unital rings built out of cobordism categories modulo various relations. Examples include diagrammatic categorifications of the polynomial ring Z[x] and several classes of orthogonal polynomials.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Daniel Tubbenhauer
Some webs and q-Howe dualities in types BCD
Co-author: Antonio Sartori (University of Freiburg)

In seminal work, Cautis-Kamnitzer-Morrison gave a diagrammatic presentation, via a so-called web calculus, of the category of quantum gln modules tensor generated by the exterior powers. Their novel observation was that a classical tool from representation and invariant theory, known as skew Howe duality, can be quantized and used to construct the corresponding diagrammatic presentation.

The work of Cautis-Kamnitzer-Morrison was then extended to various other instance (and even categorified), but all of these have in common that they stay in type A.

In this talk I will describe the first steps towards generalizing Cautis-Kamnitzer-Morrison's results to other types, where several new features (or flaws?) appear.
14:30 to 15:30 Yian Tian
Towards a categorical boson-fermion correspondence
The celebrated boson-fermion correspondence is an isomorphism between the bosonic Fock space and the fermionic Fock space. We present  categorification of the bosonic Fock space and the Heisenberg algebra which is a modification of Khovanov's Heisenberg category. The categorifcation of the fermionic Fock space is based on Honda's category studying contact topology in dimension three. This is a joint work with Mikhail Khovanov.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Anna Beliakova
Quantum Annular Link Homology via Trace Functor
In this talk I will construct  a new triply graded
invariant of links in a solid torus, which is functorial with respect to
annular link cobordisms and admits a quantum sl(2) action.
We discuss several properties of this invariant.
This is a joint work with Kris Putyra and Stephan Wehrli.
Friday 30th June 2017
11:00 to 11:30 Morning Coffee
12:30 to 13:30 Lunch @ Wolfson Court
15:00 to 16:00 Cameron Gordon
Left-orderability and 3-manifold groups: Rothschild Lecture
The fundamental group is a more or less complete invariant of a 3-dimensional manifold. We will discuss how the purely algebraic question of whether or not this group has a left-invariant total order appears to be related to two other, seemingly quite different, properties of the manifold, one geometric and the other essentially analytic.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons