This is a research seminar, intended for mathematicians and physicists. For the speaker to successfully reach the audience in both fields, it is important to explain, as clearly as possible: the motivations for the work, questions addressed, key ideas. The audience may fail to appreciate the glory of the result, otherwise.

Gukov, Putrov and Vafa postulated the existence of some 3-manifold invariants, obtained by counting BPS states in the 3d N=2 theory T[M_3]. The GPV invariants take the form of power series converging in the unit disk, and whose radial limits at the roots of unity give the Witten-Reshetikhin-Turaev invariants. Furthermore, these power series have integer coefficients, and should admit a categorification. An explicit formula for the power series exists for negative definite plumbings. In this talk I will explain what should be the analogue of the GPV invariants for manifolds with torus boundary (such as knot complements), and propose a Dehn surgery formula for these invariants. The formula is conjectural, but it can be made explicit in the case of knots given by negative definite plumbings with an unframed vertex. This is joint work (in progress) with Sergei Gukov.

We explain how quantum affine algebras can be used to systematically construct "exotic" t-structures. One of the application is to obtain exotic t-structures on certain convolution varieties defined using affine Grassmannians (these varieties play an important role in the geometric Langlands program, knot homology constructions, the coherent Satake category etc.) As a special case we also recover the exotic t-structures of Bezrukavnikov-Mirkovic on Springer resolutions in type A. This is joint work with Clemens Koppensteiner.

Several deep mathematical and physical results such as Kontsevich's deformation-quantization, Drinfeld's associators, and the Deligne hypothesis are controlled by the vanishing of certain obstruction classes in the theory of differential graded operads. I will talk about a way to obtain such vanishing results, as well as higher-genus analogues, using a weight theory implied by a new motivic point of view on the conformal operad.

A conjecture of Dunfield-Gukov-Rasmussen predicts a family of differentials on reduced HOMFLYPT homology, indexed by the integers, that give rise to a corresponding family of reduced link homologies. We'll discuss a variant of this conjecture, constructing an unreduced link homology theory categorifying the quantum gl_n link invariant for all non-zero values of n (including negative values!). To do so, we employ the technique of annular evaluation, which uses categorical traces to define and characterize type A link homology theories in terms of simple data assigned to the unknot. Of particular interest is the case of negative n, which gives a categorification of the "symmetric webs" presentation of the type A Reshetikhin-Turaev invariant, and which produces novel categorifications thereof (i.e. distinct from the Khovanov-Rozansky theory).

A conjecture of Gorsky-Negut-Rasmussen asserts the existence of a pair of adjoint functors relating the Hecke category for symmetric groups and the Hilbert scheme of points in the plane. One topological consequence of this conjecture is the prediction of a deformation of the triply graded Khovanov-Rozansky link homology which restores the missing q->tq^{-1} symmetry of KR homology for links. In this talk I will discuss a candidate for such a deformation, constructed in joint work with Eugene Gorsky, which indeed facilitates connections with Hilbert schemes. For instance our main result explicitly computes the homologies (both deformed and undeformed) of the (n,nk) torus links, summed over all n\geq 0, as a graded algebra. Combining with work of Haiman this gives a functor from the Hecke category to sheaves on the relevant Hilbert scheme.

I will describe a construction which, for a given 4D N=2 Argyres-Douglas SCFT, seems to produce a three-dimensional TQFT, whose underlying modular tensor category coincides with that of a 2d chiral algebra of the parent 4d N=2 theory.

We revisit Donaldson-Witten theory, that is the N=2 topologically twisted super Yang-Mills theory with gauge group SU(2) or SO(3) on compact 4-manifolds. We study the effective action in the Coulomb branch of the theory and by considering a specific Q-exact deformation to the theory we find interesting connections to mock modular forms. A specific operator of this theory computes the famous Donaldson invariants and our analysis makes their computation more accessible than previously. We also extend these ideas to the case of ramified Donaldson-Witten theory, that is the theory in the presence of embedded surfaces. Our results make calculations of correlation functions of Coulomb branch operators more trackable and we hope that they can help in the search of new 4-manifold invariants. Based on collaborations with Jan Manschot, Greg Moore and Iurii Nidaiev.

I will present the rank N magnificent four theory, which is the supersymmetric localization of U(N) super-Yang-Mills theory with matter on a Calabi-Yau fourfold, and conjecture an explicit formula for the partition function Z: it has a free-field representation, and surprisingly it depends on Coulomb and mass parameters in a simple way. Based on joint work with N.Nekrasov.

In my talk I will consider a quantum integrable Hamiltonian system with two generic complex parameters q,t whose classical phase space is the moduli space of flat SL(2,C) connections on a genus two surface. This system and its eigenfunctions provide genus two generalization of the trigonometric Ruijsenaars-Schneider model and Macdonald polynomials, respectively. I will show that the Mapping Class Group of a genus two surface acts by automorphisms of the algebra of operators of this system. Therefore this algebra can be viewed as a genus two generalization of A_1 spherical Double Affine Hecke Algebra. Based on joint work with Sh. Shakirov.

One of the great surprises to emerge from string theory is the prediction of supersymmetric QFTs with interacting UV superconformal fixed points in 5d and 6d. Although 6d superconformal fixed points are believed classified, the classification of 5d superconformal fixed points remains an open problem. In this talk, I discuss recent progress towards a classification of 5d fixed points in terms of singular Calabi-Yau 3-folds. I will also explain how families of RG flows between different 5d fixed points are encoded in geometry, how these RG flows establish a direct connection with 6d fixed points, and how this connection can be made into a systematic classification program.