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Timetable (HTLW03)

Physics and knot homologies

Monday 10th April 2017 to Thursday 13th April 2017

Monday 10th 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
String Theory and Homological Invariants for 3-Manifolds
In this talk I review the recent progress made in defining homological invariants for 3-manifold using string theory constructions.  This generalizes the constructions of homological invariants for knots using M5 branes, to the case of 3-manifolds.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Daniel Roggenkamp
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 4-dimensional gauge theories. The talk is based on joint work with Sungbong Chun and Sergei Gukov.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Andrea Brini
Mirror symmetry, integrable systems and the Gopakumar--Vafa correspondence for Clifford--Klein 3-manifolds
I will report on recent progress on the Gopakumar--Ooguri--Vafa correspondence, relating quantum (Witten--Reshetikhin--Turaev) invariants of 3-manifolds and knots therein with curve-counting theories (Gromov--Witten/Donaldson--Thomas) of local Calabi--Yau threefolds, in the context of Seifert-fibred 3-manifolds. 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 Eynard--Orantin topological recursion, and present a rigorous proof of the B-model version of the correspondence via matrix model techniques. Implications for refined invariants, orbifold GW theory, and an allied class of Frobenius manifolds and 2D-Toda reductions will be also discussed time permitting.

Based on joint work with G. Borot and further work in progress.

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
Unified invariants of homology 3-spheres
In 2015 K. Habiro and T. Le defined unified quantum invariants of integral homology 3-spheres associated with simple Lie algebras. These invariants dominate Witten-Reshetikhin-Turaev  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 3-spheres. Joint work with T. Le, C. Blanchet and I. Buehler.
17:00 to 18:00 Welcome Wine Reception
Tuesday 11th April 2017
10:00 to 11:00 Jørgen Andersen
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 simply-connected 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.​
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Matthew Hogancamp
Khovanov-Rozansky homology and q,t Catalan numbers
I will discuss a recent proof of the Gorsky-Oblomkov-Rasmussen-Shende conjecture for (n,nm+1) torus knots, which generally expresses the Khovanov-Rozansky 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.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Piotr Kucharski
Knots, (extremal) A-polynomials, and BPS invariants
Co-author: Piotr Sulkowski (University of Warsaw, Caltech)

In this talk I will introduce a new class of algebraic curves called extremal A-polynomials 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 A-polynomials. 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.
14:30 to 15:30 Ramadevi Pichai
Arborescent knots, mutants - current status on their invariants
Computation of colored HOMFLY-PT 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 Chern-Simons 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
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, as well as a study of their deformations and stability properties. I will finish by discussing some implications for colored, triply-graded HOMFLY-PT homologies, including an exponential growth property conjectured by Gorsky, Gukov and Stosic.
Wednesday 12th April 2017
10:00 to 11:00 Pavel Putrov
Integrality in analytically continued Chern-Simons theory
Physics predicts existence of homological invariants of closed oriented 3-manifolds similar to Khovanov-Rozansky homology of knots in a 3-sphere. The decategorified version of such invariants are q-series with integer coefficients. In my talk I will discuss properties of such invariants, how they are related to Chern-Simons 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.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Tobias Ekholm
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. 
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
Thursday 13th April 2017
10:00 to 11:00 Mohammed Abouzaid
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 Gaiotto-Witten 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.
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 Labastida-Marino-Ooguri-Vafa 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.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Alexey Sleptsov
Colored knot invariants from Reshetikhin-Turaev approach
I will discuss Reshetikhin-Turaev approach for construction of colored quantum link invariants, which are colored HOMFLY polynomials in the case of sl(N). This approach involves quantum R-matrices and inclusive quantum Racah coefficients also known as 6-j symbols and provides a systematic way for calculation of colored invariants. I will present our recent results for three-strand knots and relation with an alternative approach coming from WZW CFT.
14:30 to 15:30 Amer Iqbal
BPS states of M5-brane on T^3
We will discuss a subclass of BPS states in the M5-brane 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.
15:30 to 16:00 Afternoon Tea
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons