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.

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.

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.

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.​

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.

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.

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.

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. 

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.

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.

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.