Abstracts
September 17: Gus Schrader.
Title: Dehn twists and Whittaker functions
Abstract:
The quantum higher Teichmuller theory developed by Fock and Goncharov associates to a hyperbolic marked surface S and a simple Lie group G an infinite dimensional unitary representation of the mapping class group of S. In this talk, I'll describe the solution of the spectral problem for the operators representing Dehn twists in quantum higher Teichmuller theory for G=SL(n) and explain how the Dehn twist eigenfunctions are given by with Whittaker functions for (the modular double of) the quantum group U_q(sl(n)). Joint work with Alexander Shapiro.
September 24: Yakov Kononov
Title: Capped equivariant vertex with 2 legs
Abstract:
We derive an explicit formula for the capped vertex with 2 legs twisted by the bundles O ⊕ O(−1). Using this result we obtain a formula for the operator corresponding to horizontal legs of resolved conifold, and in particular prove its polynomiality in the Kahler variable.
October 1: Ivan Loseu.
Title: Harish-Chandra bimodules over quantized symplectic singularities
Abstract:
We study a certain category of bimodules over a filtered algebra quantizing the algebra of functions on a conical symplectic singularity. The bimodules we care about are so called Harish-Chandra bimodules. This notion first appeared in the case of universal enveloping algebras of semisimple Lie algebras in the work of Harish-Chandra on representations of the corresponding complex Lie groups. Since then it was generalized to filtered quantizations of algebras of functions on affine Poisson varieties. The goal of this talk is to explain a classification of the simple Harish-Chandra bimodules with full support over quantizations of conical symplectic singularities that have no slices of type E_8. We will see that these irreducible bimodules are in one-to-one correspondence with the irreducible representations of a suitable finite group. The group in question arises as the quotient of the algebraic fundamental group of the open leaf by a normal subgroup depending on the quantization parameter in a way that will be explained in the talk. I will not assume any preliminary knowledge of conical symplectic singularities, their quantizations etc.
October 8: Ben Davison
Title: Yangians from Nakajima quiver varieties
Abstract:
One can associate to any finite graph Q the skew-symmetic Kac-Moody Lie algebra g_Q. While this algebra is always infinite, unless Q is a Dynkin diagram of type ADE, g_Q shares a lot of the nice features of a semisimple Lie algebra. In particular, the cohomology of Nakajima quiver varieties associated to Q gives a geometric representations of g_Q. Encouraged by this story, one could hope to define the Yangian of g_Q, for general Q, as a subalgebra of the algebra of endomorphisms of cohomology of quiver varieties. In fact there are two approaches to doing this: firstly via the stable envelope construction of Maulik and Okounkov, secondly via the preprojective Hall algebra of Schiffmann and Vasserot. Via another Hall algebra construction, due to Kontsevich and Soibelman, and work of myself and Meinhardt on BPS Lie algebras, these approaches turn out to be more or less the same. In showing this we show that the correct Lie algebra of endomorphisms associated to Q is cohomologically graded, with zeroeth piece the old Kac-Moody Lie algebra: the best way to build a (Borcherds) Lie algebra out of Q is directly from geometric representation theory.
October 15: Konstantin Matveev
Title: Macdonald positivity, Kerov conjecture, vertex models.
Abstract:
I will talk about the recent proof of the Kerov conjecture (1992) classifying the homomorphisms from the algebra of symmetric functions to reals with non-negative values on all the Macdonald functions. I will discuss the history of the problem, connections with totally non-negative matrices, asymptotic representation theory, Gibbs measures on branching graphs. I will explain ideas behind the proof and talk about a new direction: Hall-Littlewood positive homomorphisms into matrices (instead of reals) and relations with vertex models.
October 22: Alex Weekes
Title: Affine Grassmannian slices and their categories O.
Abstract:
One of the most important classes of spaces in geometric representation theory are the affine Grassmannians, whose geometry is intimately connected to the representations of Lie groups. In this talk I will discuss a manifestation of this connection: recent joint work studying categories of modules over deformation quantizations of affine Grassmannian slices, and connections to work of Khovanov-Lauda, Rouquier, and Webster on categorification of representations. This progress has been aided by Braverman-Finkelberg-Nakajima's realization of affine Grassmannian slices as Coulomb branches. This all works only in symmetric type; time permitting I will describe on-going work aiming to extend to symmetrizable types.
Joint with J. Kamnitzer, P. Tingley, B. Webster and O. Yacobi.
October 29 : Alexei Oblomkov
Title: Matrix factorizations and knot homology
Abstract:
Recently, several groups of researchers proposed relations between the HOMFLY-PT homology and sheaves on the Hilbert schemes of points in the plane, see the work of Aganagic, Cherednik, Gorsky, Negut, Hogancamp, Oblomkov, Rasmussen, Rozansky, Shende. In my talk, I will discuss an approach due to Oblomkov and Rozansky. For a braid $b$, we construct a two-periodic complex of coherent sheaves $S_b$ such that $H^*(S_b)$ is the HOMFLYPT homology of the closure of $b$.
In the heart of our construction is a realization of the braid group inside some specific category of matrix factorizations. The main goal of the talk is to give an introduction to the theory of matrix factorization and to construct the above mentioned braid group action. As an application we will compute the knot homology of the torus knots.
November 5 : Changjian Su
Title: K-theoretic stable basis, Motivic Chern classes, and Iwahori invariants of principal series
Abstract:
In this talk, we will relate the following objects: Maulik--Okounkov's K-theoretic stable basis for the Springer resolution, Brasselet–Schurmann–Yokura’s motivic Chern classes for the Schubert cells, and Iwahori-invariants of an unramified principal series representation of the Langlands dual p-adic group. With this, we can prove a conjecture of Bump–Nakasuji–Naruse about Casselman basis in the unramified principal series. Based on joint work with P. Aluffi, L. Mihalcea and J. Schurmann.
November 12 : Andrei Neguț
Title: (parabolic) W-algebras and moduli of (parabolic) sheaves
Abstract:
There is a general picture that physicists call AGT, but geometers would interpret as the connection between the cohomology/K-theory of the moduli space of rank r sheaves on a surface and q-W-algebras for gl_r. I'll present a survey of this connection, plus a tentative generalization to moduli space of parabolic (i.e. endowed with a flag structure on a divisor) sheaves. The algebraic object is expected to match the, yet undefined, q-W-algebra associated to gl_r and arbitrary nilpotent element.
November 19 : Andrei Smirnov
Title: Elliptic stable envelopes and 3D mirror symmetry
Abstract:
In this talk we consider a simplest Nakajima variety X = T*Gr(k,n) (the cotangent bundle of Grassmannian of k-planes in n-space ) and its "3D mirror" X'. We consider explicit formulas for the elliptic stable envelopes for both X and X' and discuss relations among them.
November 26: Sylvain Carpentier
Title: The role of PreHamiltonian differential and difference operators in (classical) integrable systems.
Abstract:
We discuss a relatively new algebraic structure in the theory of integrable systems (of PDEs and differential-difference equations): the class of differential, or difference, operators such that their image is a sub Lie algebra of the algebra of evolutionary vector fields. These operators, called PreHamiltonian, encode most attributes of integrability for a given system. We will explain how they provide a natural non skew-symmetric generalization of the Hamiltonian (local and non-local) formalism, and discuss what is their geometric nature.
December 3: Sasha Tsymbaliuk
Title: Coulomb branches, shifted quantum algebras and modified q-Toda systems.
Abstract:
In the recent series of papers by Braverman-Finkelberg-Nakajima a mathematical construction of the Coulomb branches of 3d N=4 quiver gauge theories was proposed (they are symplectic dual to the corresponding well-understood Higgs branches). They can be also realized as slices in the affine Grassmannian and therefore admit a multiplication.
In this talk, we shall discuss the quantizations of these Coulomb branches and their K-theoretic analogues, and the (conjectural) down-to-earth realization of these quantizations via shifted Yangians and shifted quantum affine algebras. Those admit a coproduct quantizing the aforementioned multiplication of slices. In type A, they also act on equivariant cohomology/K-theory of parabolic Laumon spaces.
As another interesting application, the shifted quantum affine algebras in the simplest case of sl(2) give rise to a new family of 3^{n-2} q-Toda systems of sl(n), generalizing the well-known one due to Etingof and Sevostyanov. If time permits, we shall explain how to obtain 3^{rk(g)-1} modified q-Toda systems for any simple Lie algebra g.
This talk is based on the joint works with M. Finkelberg and R. Gonin.
December 6: Michael McBreen
Title: C*-equivariant Homological Mirror Symmetry for Hypertoric Varieties
Abstract:
Hypertoric varieties are basic examples of symplectic resolutions, a class of algebraic symplectic varieties which plays a key role in contemporary geometric representation theory. I will discuss joint work with Ben Webster, which constructs a derived equivalence between C*-equivariant coherent sheaves on a hypertoric variety (the 'B-model') and a category of microlocal sheaves on a multiplicative hypertoric variety (the 'A-model'), and explain how this relates to the usual formulation of mirror symmetry. I will give special attention to the notion of microlocal Hodge structure on the A-model, which reproduces the C*-action on the B-model. All of this will be served as a bite-sized nugget, by focusing on the case of T*P^1.
January 28: Dawei Chen
Title: Volumes and intersection theory on moduli spaces of Abelian differentials
Abstract:
Computing volumes of moduli spaces has significance in many fields. For instance, the celebrated Witten’s conjecture regarding intersection numbers on the moduli space of curves has a fascinating connection to the Weil-Petersson volume, which motivated Mirzakhani to give a proof via Teichmueller theory, hyperbolic geometry, and symplectic geometry. The initial two other proofs of Witten’s conjecture by Kontsevich and by Okounkov-Pandharipande also used various ideas in ribbon graphs, Gromov-Witten theory, and Hurwitz theory. In this talk I will introduce an analogue of Witten’s intersection numbers on moduli spaces of Abelian differentials that can be used to compute the Masur-Veech volumes. This is joint work with Martin Moeller, Adrien Sauvaget, and Don Zagier (arXiv:1901.01785).
February 4: Alexander Varchenko
Title: Equivariant quantum differential equation, Stokes bases, and K-theory for a projective space
Abstract:
I will consider the equivariant quantum differential equation for the projective space P^{n−1} and describe the Stokes bases of the differential equation at its irregular singular point in terms of the exceptional bases of the equivariant K-theory algebra of P^{n−1} and a suitable braid group action on the set of exceptional bases.
February 11: Iza Stuhl
Title: Hard-core disks on 2D lattices
Abstract:
It is well-known that in a plane, the maximum-density configuration of hard-core (non-overlapping) disks of diameter D is given by a triangular/hexagonal arrangement (Fejes Tóth, Hsiang). If the disk centers are placed at sites of a lattice, say, a unit triangular lattice or a unit square lattice, then we get a discrete analog of this problem, with the Euclidean exclusion distance. (An alternative is to use the graph distance.)
I will discuss high-density Gibbs/DLR measures for the hard-core model of these lattices, for a large value of fugacity z. According to the Pirogov-Sinai theory, the extreme Gibbs measures are obtained via a polymer expansion from dominating periodic ground states. For the hard-core model the ground states are associated with maximally dense configurations, and dominance is determined by counting defects in local excitations.
On a triangular lattice we have a complete description of the extreme Gibbs measures for a large z and any D; a convenient tool here is the Eisenstein integer ring. For a square lattice, the situation is made more complicated by various (related) phenomena: sliding, non-tessellation etc. Here, some results are available; conjectures of various generality and precision can also be proposed.
A number of our results are computer-assisted (and require a large memory, which is a restriction).
This is a joint work with A. Mazel and Y. Suhov.
February 19: Alexander Shapiro
Title: Postive Peter-Weyl theorem.
Abstract: The classical Peter-Weyl theorem asserts that the regular representation of a compact Lie group on the space of square-integrable functions decomposes as the direct sum of all irreducible unitary representations of . In the talk, I will use positive representations of cluster varieties, to obtain a "non-compact" quantum analogue of the Peter-Weyl theorem. This is a joint work with Ivan Ip and Gus Schrader.
March 11 : Amol Aggarwal
Title: Large Genus Asymptotics in Strata of Abelian Differentials
Abstract: Masur-Veech volumes and Siegel-Veech constants are geometric quantities associated with strata of the moduli space of Abelian differentials. In this talk, I will outline how a combinatorial analysis of explicit but intricate formulas (due to Eskin-Okounkov and Eskin-Masur-Zorich) for these quantities can be used to analyze their behavior in the large genus limit.
March 25 : Mark Shoemaker
Title: Matrix Factorizations in Gromov—Witten theory
Abstract: Originally introduced by Eisenbud in the context of commutative algebra, matrix factorizations have since earned a prominent role in mathematical physics. In this talk I will describe recent work using matrix factorizations to define a cohomological field theory given the input data of a gauged linear sigma model. This construction gives a new description of the virtual class in Gromov-Witten theory, FJRW theory, and so-called hybrid models. This is joint work with Ciocan-Fontanine, Favero, Guere, and Kim.
April 1 : Petr Pushkar
Title: Transfer matrix and quasimaps with defects
Abstract:
Abstract: I will talk about a (at least partial) realization of the transfer matrix in terms of the enumerative geometry of Grassmannians. To do this I will use the moduli space of quasimaps with defects and will build on techniques used to obtain the Baxter operator as an operator of quantum multiplication by tautological classes.
April 8: Will Sawin
Title: The polar multiplicities of the nilpotent cone in the module space of Higgs bundles
Abstract: The polar multiplicities are some intersection-theoretic invariants associated to a conical cycle in the cotangent bundle of a variety and a point on that variety. Motivated by a problem in number theory, we aim to compute these for the nilpotent cone in the moduli space of Higgs bundles. We begin to answer this by handling the first nontrivial case (rank 2, genus 0, with level structure).
April 22 : Yuri Tschinkel
Title: Equivariant birational geometry and modular symbols
Abstract:
We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups (joint with M. Kontsevich and V. Pestun).
April 29 : Martina Lanini
Title: The Steinberg-Lusztig tensor product theorem for abstract Fock space
Abstract:
The abstract Fock space, constructed in joint work with A.Ram and P.Sobaje, is a combinatorial gadget which generalizes Lecler-Thibon's realization of the classical Fock Space to any Lie type, and encodes decomposition numbers for quantum groups at roots of unity. In this talk I will discuss joint work with Arun Ram, in which we establish the analogue of the Steinberg-Lusztig Tensor product Theorem for abstract Fock space. This result has several consequences, such as a new proof of the Casselman-Shalika formula, and a character formula for a certain class of modules for affine Kac-Moody algebras at a negative level.