Physics and knot homologies
Monday 10th April 2017 to Thursday 13th April 2017
09:00 to 09:50  Registration  
09:50 to 10:00  Welcome from Christie Marr (INI Deputy Director)  
10:00 to 11:00 
Cumrun Vafa (Harvard University) String Theory and Homological Invariants for 3Manifolds
In
this talk I review the recent progress made in defining homological invariants
for 3manifold using string theory constructions. This generalizes the
constructions of homological invariants for knots using M5 branes, to the case
of 3manifolds.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Daniel Roggenkamp (Universität Mannheim) Surface operators and categorification of quantum groups
In this talk I will discuss how certain categorifications of quantum groups arise from foams of surface operators in 4dimensional gauge theories. The talk is based on joint work with Sungbong Chun and Sergei Gukov. 
INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:30 to 14:30 
Andrea Brini (Imperial College London); (CNRS (Centre national de la recherche scientifique)) Mirror symmetry, integrable systems and the GopakumarVafa correspondence for CliffordKlein 3manifolds
I will report on recent progress on the GopakumarOoguriVafa correspondence, relating quantum (WittenReshetikhinTuraev) invariants of 3manifolds and knots therein with curvecounting theories (GromovWitten/DonaldsonThomas) of local CalabiYau threefolds, in the context of Seifertfibred 3manifolds. I will describe A and B model constructions for the correspondence in the broadest context where the standard form of the duality is expect to hold (spherical space forms), discuss the link with relativistic integrable systems and the EynardOrantin topological recursion, and present a rigorous proof of the Bmodel version of the correspondence via matrix model techniques. Implications for refined invariants, orbifold GW theory, and an allied class of Frobenius manifolds and 2DToda reductions will be also discussed time permitting. Based on joint work with G. Borot and further work in progress. 
INI 1  
14:30 to 15:30  Informal discussion and questions  INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Anna Beliakova (Universität Zürich) Unified invariants of homology 3spheres
In 2015 K. Habiro and T. Le defined unified quantum invariants of integral homology 3spheres associated with simple Lie algebras. These invariants dominate WittenReshetikhinTuraev invariants and belong to the Habiro ring of analytic functions at roots of unity. In the talk I will review the construction of unified invariants, discuss their properties and give few applications. Then I will mention our generalisations of the unified invariants for rational homology 3spheres. Joint work with T. Le, C. Blanchet and I. Buehler.

INI 1  
17:00 to 18:00  Welcome Wine Reception 
10:00 to 11:00 
Jørgen Andersen (Aarhus Universitet) The Verlinde formula for Higgs bundle moduli spaces
In this talk we will present a Verlinde formula for the quantization of the Higgs bundle moduli
spaces and stacks for any simple and simplyconnected group. We further present
a Verlinde formula for
the quantization of parabolic Higgs bundle moduli spaces and stacks. We will
explain how all these dimensions fit into a one parameter family of 2D TQFT's,
encoded in a one parameter family of Frobenius algebras, which we will
construct.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Matthew Hogancamp (University of Southern California) KhovanovRozansky homology and q,t Catalan numbers
I will discuss a recent proof of the
GorskyOblomkovRasmussenShende conjecture for (n,nm+1) torus knots, which
generally expresses the KhovanovRozansky homology of torus knots in terms of
representations of rational DAHA. The
proof is based off of a computational technique introduced by myself and Ben
Elias, using complexes of Soergel bimodules which categorify certain Young
symmetrizers. We will summarize this
technique and indicate how it results in a remarkably simple recursion which
computes the knot homologies in question.

INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:30 to 14:30 
Piotr Kucharski (University of Warsaw) Knots, (extremal) Apolynomials, and BPS invariants
Coauthor: Piotr Sulkowski (University of Warsaw, Caltech) In this talk I will introduce a new class of algebraic curves called extremal Apolynomials of a knot and use it to describe BPS invariants introduced by Labastida, Marino, Ooguri, and Vafa. I will present results obtained from the analysis of both classical and quantum extremal Apolynomials. The first lead to exact formulas for BPS invariants imposing nontrivial integrality statements in number theory. The latter enabled us to construct the combinatorial model for calculating BPS invariants. I will also indicate how these results relate to the formalism of quivers introduced in the talk by Piotr Sulkowski. 
INI 1  
14:30 to 15:30 
Ramadevi Pichai (Indian Institute of Technology) Arborescent knots, mutants  current status on their invariants
Computation of colored HOMFLYPT polynomials for knots carrying arbitrary representations is still a challenging problem. First I will recapitulate the necessary tools for determining the colored knot invariant within ChernSimons theory. Then, I will present our results on quantum Wigner 6j useful for writing polynomial form of the knot invariant. Further, we will discuss our results for mutant knot pairs. Finally, we summarize the current status on these polynomials which we periodically update on the website http://knotebook.org

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Paul Wedrich (Imperial College London) 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)
KhovanovRozansky link homologies, which are functorial under link cobordisms,
as well as a study of their deformations and stability properties. I will
finish by discussing some implications for colored, triplygraded HOMFLYPT
homologies, including an exponential growth property conjectured by Gorsky,
Gukov and Stosic.

INI 1 
10:00 to 11:00 
Pavel Putrov (Institute for Advanced Study, Princeton) Integrality in analytically continued ChernSimons theory
Physics predicts existence of homological invariants of closed oriented 3manifolds similar to KhovanovRozansky homology of knots in a 3sphere. The decategorified version of such invariants are qseries with integer coefficients. In my talk I will discuss properties of such invariants, how they are related to ChernSimons partition function (WRT invariant) analytically continued w.r.t. level, and give some examples. If time permits I will also discuss how resurgence theory can be used to construct such invariants and relation to open topological strings.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Tobias Ekholm (Uppsala Universitet); (Institut MittagLeffler) Higher genus knot contact homology and recursion for the colored HOMFLY polynomial
We present a conjectural description of Legendrian symplectic field theory for the conormal of a knot ("higher genus knot contact homology") and discuss its relation to the recursion relation for the colored HOMFLY polynomial. This reports on joint work with Lenny Ng.

INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:30 to 17:00  Free Afternoon  
19:30 to 22:00  Formal Dinner at Trinity College 
10:00 to 11:00 
Mohammed Abouzaid (Columbia University) Towards a symplectic model of odd Khovanov homology
I will report on joint work in progress with Ivan Smith
which
combines ideas of Lawrence and Bigelow and GaiottoWitten
with motivation
from homological mirror symmetry to propose a symplectic
construction of a
pair of knot invariants which are expected to correspond
to the odd and
even Khovanov homologies. I will mostly focus on the only
computation
which is fully understood: the trivial diagram of the
unknot.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Piotr Sulkowski (Uniwersytet Warszawski); (CALTECH (California Institute of Technology)) BPS states, knots and quivers
I will present a surprising relation between knot invariants and quiver representation theory, motivated by various string theory constructions involving BPS states. Consequences of this relation include the proof of the famous LabastidaMarinoOoguriVafa conjecture, explicit (and unknown before) formulas for colored HOMFLY polynomials for various knots, new viewpoint on knot homologies, new dualities between quivers, and many others.

INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:30 to 14:30 
Alexey Sleptsov (ITEP (Institute for Theoretical and Experimental Physics)) Colored knot invariants from ReshetikhinTuraev approach
I will discuss ReshetikhinTuraev approach for construction of colored quantum link invariants, which are colored HOMFLY polynomials in the case of sl(N). This approach involves quantum Rmatrices and inclusive quantum Racah coefficients also known as 6j symbols and provides a systematic way for calculation of colored invariants. I will present our recent results for threestrand knots and relation with an alternative approach coming from WZW CFT. 
INI 1  
14:30 to 15:30 
Amer Iqbal (Abdus Salam School of Mathematical Sciences GC University) BPS states of M5brane on T^3
We will discuss a subclass of BPS states in the M5brane
theory on T^3 x R^3 which are related to little strings and whose
degeneracies can be worked out exactly. The generating function of these BPS
states has interesting modular properties and seems to have the structure
expected of the partition function with target space a symmetric product.

INI 1  
15:30 to 16:00  Afternoon Tea 