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.

A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers - connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for SL(N). We introduce a difference equation version of opers called q-opers and prove a q-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted q-opers with regular singularities on the projective line. We show that the quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars-Schneider model may be viewed as a special case of the q-Langlands correspondence. We also describe an application of q-opers to the equivariant quantum K-theory of the cotangent bundles to partial flag varieties.

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.

A classical result of Turaev identifies the skein algebra of the annulus with the algebra of symmetric functions in infinitely many variables. Queffelec and Roze categorified this using annular webs and foams. I will recall their construction and compute explicit symmetric functions and their categorical analogues for some links. As an application, I will describe spectral sequences computing categorical invariants of generalized Hopf links. The talk is based on a joint work with Paul Wedrich.

Virasoro constraints are omnipresent in enumerative geometry. Recently, Kontsevich and Soibelman introduced a generalization of Virasoro constraints in the form of Airy structures. It can also be understood as an abstract framework underlying the topological recursion of Chekhov, Eynard and Orantin. In this talk I will explain how the triumvirate of Virasoro constraints, Airy structures and topological recursion can be generalized to W-algebra constraints, higher Airy structures and higher topological recursion. I will briefly discuss the enumerative geometric meaning of the resulting W-constraints in the context of open and closed intersection theory on the moduli spaces or curves with r-spin structure and its variants.

Calabi-Yau manifolds have played a central role in both string theory and mathematics for decades, but in spite of this no Ricci-flat metric on a compact non-toroidal Calabi-Yau manifold is known. I will discuss a new physically motivated approach toward the determination of such metrics for K3 surfaces. The key remaining step is the determination of a BPS spectrum of a heterotic little string theory on T^2. I will use string dualities to provide a number of mathematical reformulations of this problem, ranging from open string reduced Gromov-Witten theory for the mirror K3 surface (in accordance with the SYZ conjecture) to Donaldson-Thomas theory for auxiliary Calabi-Yau threefolds. Finally, I will discuss new approximations to K3 metrics near the semi-flat limit that require only a minimal knowledge of this BPS spectrum.

This is a joint work with A. Oblomkov exploring the relation between the HOMFLY-PT link homology and coherent sheaves over the Hilbert scheme of points on C^2.

We consider a special object in the 2-category related to the Hilbert scheme of n points on C^2. We define a homomorphism from the braid group on n strands to the monoidal category of endomorphisms of this object. We prove that the space of morphisms between the images of a braid and of the identity braid is the invariant of a link constructed by closing the braid. Conjecturally, this space is the triply-graded HOMFLY-PT homology.

From the TQFT point of view, we consider a B-twisted 3d N=4 SUSY YM with matter, whose Higgs branch is the Hilbert scheme.

Link homology appears as the Hilbert space of a 2-disk. Its boundary carries a flag variety-based sigma model, and the Kahler parameters of the flag variety braid as one goes around the disk.

From the IIA string theory point of view, the points on C^2 are BPS particles coming from a 2-disk shaped stack of n D2-branes located in one of the fibers at the North Pole of P^1, which forms the base of a resolved conifold. The D2-branes end on a stack of NS5 branes which form a closed braid in the other fiber.

I will describe joint work with Ciprian Manolescu on constructing an analogue of instanton Floer homology replacing the group SU(2) by SL(2,C). Having failed to do so using the standard Floer theoretic tools of gauge theory and symplectic topology, we turned to sheaf theory to produce an invariant. After describing our approach, I will discuss some features of this theory that are expected to be visible from a Floer-theoretic point of view, but that we cannot currently access.

Based on the representation theory of quantum toroidal algebras we propose a generalization of the refined topological vertex formalism incorporating additional "Higgsed" vertices and lines apparently corresponding to refined Lagrangian branes. We find rich algebraic structure associated to brane diagrams incorporating the new vertices and lines. In particular, we build the screening charges associated to W-algebras of types gl(n) and gl(n|m), and more generally to Y-algebras of Gaiotto and Rapcak. The resulting refined partition functions coincide with partition functions of certain interacting 5d-3d-1d systems of quiver gauge theories (including quivers associated with superalgebras). Our formalism also automatically incorporates Ruijsenaars-Schneider Hamiltonians and their supersymmetric generalizations which act on the refined partition functions.

3 dimensional N=4 supersymmetric quantum field theories have two distinguished topological twists, called Higgs and Coulomb (though we periodically get confused about which is which). These two twists manifest very interesting mathematical objects in Lie theory and algebraic geometry, which don't seem to obviously be related, except through this bridge in QFT. I'll do my best to explain what physicists know to mathematicians, what mathematicians know to physicists, and if I fail at both, hopefully there will be some comedy value in my attempt.

Both the Higgs bundle moduli space and the moduli space of flat connections have a natural stratification induced by a C* action. In both of these stratifications, each stratum is a holomorphic fibration over a connected component of complex variations of Hodge structure. While the nonabelian Hodge correspondence provides a homeomorphism between Higgs bundles and flat connections, this homeomorphism does not preserve the respective strata. The closed stratum on the Higgs bundle side is the image of the Hitchin section and the closed stratum in the space of flat connections is the space of opers. In this talk, we will show how many of the relationships between opers and the Hitchin section extend to general strata. In particular, we will show that the conformal limit identifies certain holomorphic Lagrangian subspaces of the stratifications.

It has been understood for some time now that many highlights of Lie theory, such as the representation-theoretic theory of special functions, or the Kazhdan-Lusztig theory, have a natural extension to a much broader setting, the boundaries of which are yet to be explored. In this extension, the focus is shifting from a group G to various classes of algebraic varieties that possess the key features of T*G/B. While there are some proposal about what should replace a Lie algebra, root systems, etc., it is less clear what should be the group, or multiplicative analog of these structures. Reflecting the nature of the field, the talk will combine a review of established partial results with unsubstantiated speculations.

Wilson loops are important observables in gauge theory. In this talk, we study half-BPS Wilson loops of a large class of five dimensional supersymmetric quiver gauge theories with 8 supercharges, in a nontrivial instanton background. The Wilson loops are codimension 4 defects of the quiver gauge theory, and their interaction with self-dual instantons is captured by a 1d ADHM quantum mechanics. We compute the partition function as its Witten index. It turns out that we can understand the 5d physics in 3d gauge theory terms. This comes about from so-called gauge/vortex duality; namely, we study the vortices on the Higgs branch of the 5d theory, and reinterpret its partition function from the point of view of the vortices. This perspective has an advantage: it has a dual description in terms of "deformed" Toda Theory on a cylinder, in the Coulomb gas formalism. We show that the gauge theory partition function is equal to a (chiral) correlator of the deformed Toda Theory, with stress tensor and higher spin operator insertions. We derive all the above results from type IIB string theory, compactified on a resolved ADE singularity X times a cylinder with punctures. The 5d quiver gauge theory arises as the low energy limit of a system of D5 branes wrapping various two-cycles of X, the Wilson loops are D1 branes, and the duality to Toda theory emerges after introducing additional D3 branes.