Abstract: Defined by Haiman, wreath Macdonald polynomials are generalizations of Macdonald polynomials to wreath products of symmetric groups with a fixed cyclic group. Using a wreath analogue of the Frobenius characteristic, they can be viewed as partially-symmetric functions. Relatively little is known about them. In this talk, we present novel difference operators that are diagonalized on the wreath Macdonald polynomials. Their formulas are quite complicated, but they give strong evidence that bispectral duality holds in the wreath case. This is joint work with Daniel Orr and Mark Shimozono.

Slides 

Abstract: The theory of Hall algebras has known many spectacular developments and applications since the discovery by Ringel of their connection with quantum groups. One important object arising naturally in the study of Hall algebras is the integration map defined by Reineke, which allows to produce certain celebrated wall-crossing identities. In this talk I will first focus on the Dynkin case and show how the integration map can be interpreted in a natural way via the representation theory of quantum affine algebras. I will then explain how this opens perspectives towards an analogous interpretation for more general quivers, relying on the framework of quantum cluster algebras. This is ongoing joint work with Lang Mou.

Slides 

Abstract: The last two decades have seen great success in studying representation theoretic objects through geometric techniques. One small part of this story involves Nakajima quiver varieties, curve counting, and a mysterious string-theoretic duality. More specifically, curve counting in Nakajima varieties turns out to be governed by certain q-difference equations that, after a nontrivial amount of work, can be seen to coincide with the some well-known equations from representation theory. Moreover, these curve counts are expected to possess deep nontrivial symmetries that have only been understood in very specific examples. I will provide an overview of the main concepts and results related to these ideas, discuss my own contributions, and mention some future directions.

Video Slides 

Abstract: We construct a quantum loop group associated to an arbitrary symmetric generalized Cartan matrix by defining appropriate versions of the Drinfeld-Serre relations. Explaining the meaning of the word "appropriate" and specifying the relations will be the main purpose of the talk.

Video

Abstract: The Oblomkov–Rasmussen–Shende conjecture relates the homologies of the Hilbert schemes of a plane curve singularity to the triply-graded Khovanov–Rozansky (i.e., HOMFLYPT) homology of its link, via an identity in variables a, q, t. Two major cases are known: (1) the t = -1 limit, settled a decade ago by Maulik; (2) the lowest-a-degree, q = 1 limit of the "torus link" case, settled jointly by Elias–Hogancamp, Mellit, and Gorsky–Mazin, using (q, t)-Catalan combinatorics as an essential bridge. An unpublished research statement of Shende speculated that the ORS conjecture could be proved in a third, totally different way, via a wild analogue of the P = W phenomenon in nonabelian Hodge theory. He and his coauthors carried out most of this approach for the "torus-knot" subcase of case (1). We extend their work, and also refine it enough to handle the (more difficult) torus-knot subcase of case (2). The key is our new geometric model for Khovanov–Rozansky homology, which realizes the t variable as cohomological degree. If there is time, we will explain how this flavor of nonabelian Hodge theory is related to the noncrossing-nonnesting dichotomy in Catalan combinatorics.

Video

Abstract: The P=W conjecture, first proposed by de Cataldo-Hausel-Migliorini in 2010, gives a link between the topology of the moduli space of Higgs bundles on a curve and the Hodge theory of the corresponding character variety, using non-abelian Hodge theory.  In this talk, I will explain this circle of ideas and discuss a recent proof of the conjecture for GLn (joint with Junliang Shen).

Abstract: Braid varieties are smooth affine varieties associated to any positive braid. Special cases of braid varieties include Richardson varieties, double Bruhat cells, and double Bott-Samelson cells. Cluster algebras are a class of commutative rings with a rich combinatorial structure, introduced by Fomin and Zelevinsky. I'll discuss joint work with P. Galashin, T. Lam and D. Speyer in which we show the coordinate rings of braid varieties are cluster algebras, proving and generalizing a conjecture of Leclerc in the case of Richardson varieties. Seeds for these cluster algebras come from "3D plabic graphs", which are bicolored graphs embedded in a 3-dimensional ball that generalize Postnikov's plabic graphs for positroid varieties.

Video Slides 

Abstract: The vertex functions are generating functions counting rational curves in a quiver variety. They also give a basis of solutions to quantum differential equation associated with the quiver variety. In my talk I discuss a construction of certain polynomial solutions of quantum differential equation modulo a prime p. I also describe a number of conjectures relating the p-adic limit of these solutions to the vertex functions. The talk is based on a joint investigation in progress with A. Varchenko. 

Abstract: We study, using the extended isomonodromy deformation, the WKB approximation of Stokes matrices of a class of meromorphic linear ODE systems of Poincare rank 1 on the projective line that appear in various contexts of geometry. We show that, via the degenerate Riemann-Hilbert map, the WKB approximation of Stokes matrices recovers the Gelfand-Tsetlin integrable systems whose action variables match with period on spectral curves. If time permits, we will also briefly discuss the potential ramifications to cluster theory, spectral networks  and gl(n)-crystals (in the quantum setting). The talk is based on joint work with Anton Alekseev and Xiaomeng Xu and ongoing discussions with Andrew Neitzke.

Abstract: We construct an algebraic vector bundle over the deformation to the normal cone for an embedding of manifolds through a rescaling of a vector bundle over the ambient space. This method generalizes the construction of the spinor rescaled bundle over the tangent groupoid by Nigel Higson and Zelin Yi. Applications of this construction include local index formula, equivariant index formula, Kirillov formula and Witten and Novikov deformation of de Rham operator.

Video

Abstract: Nichols algebras are fundamental invariants of large classes of Hopf algebras.  I will survey from scratch those arising from abelian groups, focusing on the methods of computation.

Video Slides

Abstract: We construct a non-finite type four-dimensional Weinstein domain M_{univ} and describe a HMS correspondence between distinguished birational transformations of the projective plane preserving a standard holomorphic volume form and symplectomorphisms of M_{univ}. The space M_{univ} is universal in the sense that it contains every Liouville manifold mirror to a log Calabi-Yau surface as a Weinstein subdomain; after restricting to these subdomains, we recover a mirror correspondence between the automorphism group of any open log Calabi-Yau surface and the symplectomorphism group of its mirror. This is joint work with Ailsa Keating. 

Video Slides

Abstract: In joint work with Tom Braden we give a purely algebraic description of the category of perverse sheaves (with coefficients in any field) on S^n(C^2), the n-fold symmetric product of the plane.  In particular, using the geometry of the Hilbert scheme of points, we relate this category to the symmetric group and its representation ring.  Our work is motivated by analogous structure appearing in the Springer resolution and Hilbert-Chow morphism.

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