Schedule 20192020
NOTE: Due to the current situation, all talks after March 16 will most likely be postponed or canceled. Please email the organizer to be placed on the mailing list, which will give you up to date information on which talks will be held, as well as time and location.
Location: MIT, 2449
Time: Wednesday 34 (alternate day Friday 34)
Contact: Andrei Neguț
Apr 15  Roman Gonin HSE Moscow 
TBD 
Apr 8  Lingfei Yi Caltech 
TBD 
Apr 1  Allen Knutson Cornell 
TBD 
Mar 25 
No Seminar 

Mar 18  José Simental Rodríguez UC Davis 
HarishChandra bimodules for type A rational Cherednik algebras HarishChandra bimodules first appeared in the representation theory of semisimple Lie algebras, and have since then been generalized to quantizations of symplectic singularities, where they categorify (part of) the top BorelMoore homology of the Steinberg variety. We consider the case of type A rational Cherednik algebras, that in a sense serves as a counterexample for many properties one might expect from the category of HC bimodules. For example, even the number of irreducible objects is highly dependent on the quantization parameter. We will give basic properties of this category, describe its irreducible objects and its relation to the category O and, time permitting, give a complete description via quivers with relations of the block containing the regular bimodule. 
Mar 11  Dragoș Oprea UC San Diego 
Virtual invariants of Quot schemes of surfaces Quot schemes of quotients of a trivial bundle of arbitrary rank on a nonsingular projective surface carry perfect obstruction theories and virtual fundamental classes whenever the quotient sheaf has at most 1dimensional support. The associated generating series of virtual Euler characteristics is conjectured to be a rational function. A similar conjecture can be made about more general descendent series with insertions obtained from the Chern characters of the tautological sheaf. I will discuss several results in this direction obtained jointly with Drew Johnson and Rahul Pandharipande. 
Mar 4  Valerio Toledano Laredo Northeastern 
The meromorphic Rmatrix of the Yangian Drinfeld proved that the Yangian Yg of a complex semisimple Lie algebra g gives rise to solutions of the quantum YangBaxter equations on irreducible, finitedimensional representations of Yg, which are rational in the spectral parameter. This result was recently extended by MaulikOkounkov for KacMoody algebras corresponding to quivers, and representations arising from geometry. Surprisingly perhaps, this rationality ceases to hold if one considers arbitrary finitedimensional representations of Yg, at least if one requires such solutions to be natural with respect to the representation and compatible with tensor products. I will explain how one can instead produce meromorphic solutions of the QYBE on all finitedimensional representations by resumming Drinfeld's universal Rmatrix R(s) of Yg. The construction hinges on resumming the abelian part of R(s), and on realizing its lower triangular part as a twist conjugating the standard coproduct of Yg to its deformed Drinfeld coproduct. This is joint work with Sachin Gautam and Curtis Wendlandt, and is based on arXiv:1907.03525 
Dec 11  Gurbir Dhillon Stanford/MIT 
The tamely ramified Fundamental Local Equivalence Let G be a reductive group with Langlands dual G'. Gaitsgory conjectured that affine Category O for G at a noncritical level should be equivalent to Whittaker Dmodules on the affine flag variety of G' at the dual level. We will provide motivation and background for this conjecture, which is some form of geometric Satake for quantum groups. We have proven this conjecture when the level is appropriately integral with Justin Campbell, and the general case is work in progress with Sam Raskin. Depending on time and interest, we will also indicate its relation to a BeilinsonBernstein style localization theorem for Walgebras, obtained jointly with Sam Raskin, and expected consequences thereof. 
Dec 4  MinhTam Trinh U Chicago 
Artin braids, annular invariants, and pointcounting Let beta be a positive braid on n strands. Using microlocal techniques, ShendeTreumannZaslow showed the following are proportional up to a qfactor: (1) the coefficient of the highest apower in the HOMFLY series of beta; (2) the coefficient of 1 when we write beta in the standard basis of the IwahoriHecke algebra of S_n. We give a new proof that moreover generalizes this result to any Weyl group W. To do this, we introduce a new class function on the Artin braid group of W that refines the Markov trace of JonesOcneanuGomi. Its output is a sum of qgraded total Springer representations, weighted by pointcounts on certain stacks over F_q. If time permits, we will show that special values of this class function are characters of virtual A_Wmodules, where A_W is the rational Cherednik algebra of W. 
Nov 20  Michael Groechenig Toronto 
Hypertoric Hitchin systems and padic integration A construction due to Hausel and Proudfoot associates to a graph a complexanalytic integrable system. In this talk I will describe a formalalgebraic analogue of their construction, and explain how one can compute the padic volumes of the generic Hitchin fibres in graphtheoretic terms. This is joint work with Michael McBreen. 
Nov 13  Thomas Lam Michigan/MIT 
Cohomology of cluster varieties I will talk about joint work with David Speyer where we study the singular cohomology of cluster varieties, and in particular about the "curious Lefschetz duality" that these cohomology rings satisfy. There are relations to the study of higher extension groups of Verma modules in the principal block of Category O, and to the study of rational q,tCatalan polynomials. 
Nov 6  Shuai Wang Columbia 
Nonrecursive formula for KazhdanLusztig polynomials for finite and affine G/P Brenti proves a nonrecursive formula for the KazhdanLusztig polynomials of Coxeter groups by combinatorial methods. In the case of the Weyl group of a split group over a finite field, a geometric interpretation of the formula is given by Sophie Morel via the "weighttruncation" theory of perverse sheaves developed by her. With suitable modifications of Morel's proof, we generalize the geometric interpretation to the case of finite and affine partial flag varieties. We also demonstrate the result for sl_3, sl_4 and affine sl_2 
Nov 1 NOTE: SPECIAL DAY  Andrey Smirnov UNC Chapel Hill 
3D mirror symmetry in representation theory
In this talk I outline recent developments in representation theory and enumerative geometry inspired by 3d mirror symmetry. 
Oct 23 
No Seminar 

Oct 16  Dmitrii Pirozhkov Columbia 
Semiorthogonal decompositions of $P^2$ and torsion objects For projective spaces we know many different semiorthogonal decompositions of their derived categories of coherent sheaves. It's generally expected, but not known aside from the case of P^1, that those standard decompositions are the only possible ones. For a projective plane I will show that a semiorthogonal decomposition D(P^2) = < A, B > is standard, i.e. it comes from the standard exceptional collection, under the additional assumption that either A or B contains a nonzero torsion object. I will also explain why this assumption is reasonable and what needs to be done to remove it. 
Oct 11 NOTE: SPECIAL DAY  Eugene Gorsky UC Davis 
Derived traces for Soergel categories It is well known that Hecke algebra is spanned by braids modulo skein relations, while the span of closed braids in the annulus modulo skein relations is isomorphic to the space of symmetric functions. I will describe a categorification of these results: the category of Soergel bimodules categories the Hecke algebra, while the annular closure corresponds to the formalism of "derived horizontal trace". Along the way, I will explicitly compute Hochschild homology of the category of Soergel bimodules. All notions will be explained in the talk. This is a joint work in progress with Matt Hogancamp and Paul Wedrich. 
Oct 2  Yu Zhao MIT 
Ktheoretic Hall algebra of a surface and FeiginOdesskii wheel conditions In this talk, we will discuss the Ktheoretic Hall algebra of a surface, which generalizes Grojnowski's and Nakajima's description of the cohomology of Hilbert schemes of points on surfaces. We will also discuss a wheel condition for the shuffle presentation of K theoretic Hall algebra, which originated from Feigin and Odesskii. 
Sep 25  Tudor Padurariu MIT 
Ktheoretic Hall algebras for quivers with potentials Given a quiver with potential, KontsevichSoibelman constructed a Hall algebra on the cohomology of the stack of representations of (Q,W). In particular cases, one recovers the Yangian of a quiver Q as defined by MaulikOkounkov. However, for general (Q,W), the Hall algebra has nice structure properties, for example DavisonMeinhardt proved a PBW theorem for it using the decomposition theorem. One can define a Ktheoretic version of this algebra using certain categories of singularities that depend on the stack of representations of (Q,W). In particular cases, these Hall algebras are quantum affine algebras. We show that some of the structure properties in cohomology, such as the PBW theorem, can be lifted to Ktheory, replacing the use of the decomposition theorem with semiorthogonal decompositions. 
Sep 18  Ivan Danilenko Columbia 
Slices of the Affine Grassmannian and Quantum Cohomology The Affine Grassmannian is an indscheme associated to a reductive group G. It has a cell structure similar to the one in the usual Grassmannian. Transversal slices to these cells give an interesting family of Poisson varieties. Some of them admit a smooth symplectic resolution and have an interesting geometry related to the representation theory of the Langlands dual group. We will focus on equivariant cohomology of such resolutions and will show how the trigonometric KnizhnikZamolodchikov equation arises as a quantum differential equation in this setting. 