| Speakers | Registration | Schedule | Abstracts | Local information |
Pramod Achar (Louisiana State University): \(A_{\infty}\) complexes of parity sheaves or Soergel bimodules
Notes by Jose Simental Rodriguez
Abstract: The homotopy category of chain complexes of parity sheaves or of Soergel bimodules is by now a well-known and important tool in geometric representation theory and categorification. But for some purposes, ordinary chain complexes are too "strict." In this talk, I will define "\(A_{\infty}\) complexes of parity sheaves (or Soergel bimodules)," a notion that allows more flexibility in certain homological constructions. As an application, I will explain how to use \(A_{\infty}\) complexes to construct a "nearby cycles functor" for parity sheaves. As explained in Rider's talk, this functor is a step towards redoing Gaitsgory's construction of central sheaves entirely in the language of parity sheaves and the Elias--Williamson calculus. (It is also related to independent work of Elias.) If time permits, I will discuss some other applications of
\(A_{\infty}\) complexes.
Rina Anno (Kansas State University): Skein relations in triangulated representations of diagrammatic categories
Notes by Jose Simental Rodriguez
Abstract: To construct a weak categorical representation of the category of tangles
one assigns a functor to every generator and checks that these functors
satisfy a list of relations (up to isomorphism). However when the representation
in question is triangulated, we can make an additional assumption that
some generators are shifted adjoints of others, and impose additional
relations, called the skein relations. Then the list of tangle relations to check
shrinks dramatically. Unlike tangle relations, skein relations are triangulated
in nature and can not be stated in the category of tangles itself. In this talk
I am going to discuss skein relations for a certain diagrammatic category related to the
category of \(\mathfrak{sl}_2\) webs (with crossings, without caps and cups).
Erik Carlsson (UC Davis): Geometry behind some new nabla formulas
Notes by Jose Simental Rodriguez
Abstract: I'll present some new formulas for powers of the nabla operator defined in the modified complete symmetric function basis, which is recent joint work with Anton Mellit. These formulas have several interesting geometric interpretations, including to the \((n,kn)\) affine Springer fiber, and its quotient by the lattice action studied by Goresky, Kottwitz, and Macpherson and others. I will explain our motivations and expected applications.
Sabin Cautis (University of British Columbia): Cluster theory of the coherent Satake category
Notes by Jose Simental Rodriguez
Abstract: Historically one studies the category of constructible perverse sheaves on
the affine Grassmannian. This leads to the constructible Satake category
and the well known (geometric) Satake equivalence. However, one can also
study the category of perverse coherent sheaves (the coherent Satake
category). Motivated by certain ideas in mathematical physics this
category is conjecturally governed by a cluster algebra structure.
We will illustrate the geometry of the affine Grassmannian and discuss
what we mean by a cluster algebra structure. We then describe a solution
to this conjecture in the case of general linear groups. This is joint
work with Harold Williams.
Ben Elias (University of Oregon): Gaitsgory's central complexes and the Gorsky-Negut-Rasmussen conjecture
Notes by Jose Simental Rodriguez
Abstract: When studying the representation theory of the Hecke algebra of the symmetric group, there is a large commutative subalgebra, the Jucys-Murphy subalgebra, which plays a crucial role. Analogous to the Cartan subalgebra of a semisimple lie algebra, simultaneously diagonalizing the Jucys-Murphy subalgebra is an immensely powerful tool. There is a categorification of this story, where the Hecke algebra is replaced by the triangulated Hecke category (whose objects are complexes), and the Jucys-Murphy subalgebra by a triangulated subcategory. The simultaneous diagonalization of the Jucys-Murphy complexes was recently achieved in joint work with Hogancamp.
An astounding recent conjecture of Gorsky-Negut-Rasmussen states that the Jucys-Murphy subcategory is equivalent to coherent sheaves on (roughly) the flag Hilbert scheme of points on the plane, and the center of the Hecke category is equivalent to coherent sheaves on (roughly) the ordinary Hilbert scheme. They reduce this conjecture to finding, inside the Hecke category, the central complex \(\mathcal{E}_1\) which corresponds to the first elementary symmetric polynomial in the Jucys-Murphy operators, and studying its properties.
Meanwhile, in the 90s, Gaitsgory constructed a monoidal functor, similar in spirit to geometric Satake, which (after rephrasing) goes from representations of \(\mathfrak{gl}_n\) to the center of the extended affine Hecke category. Just as a representation has a splitting into weight spaces, Gaitsgory's central complexes have a filtration by Wakimoto complexes. Recently, I have given an explicit construction of the image \(\mathcal{V}\) of the standard representation. (There is an independent construction of \(\mathcal{V}\) in another context due to Achar-Rider). I also conjecture that there is a flattening functor from the extended affine Hecke category to the finite Hecke category, which sends Wakimoto complexes to Jucys-Murphy complexes, and sends \(\mathcal{V}\) to \(\mathcal{E}_1\).
In this talk I try to give an overview of the many moving parts in this picture, and I will try to describe the explicit construction of \(\mathcal{V}\).
Jim Haglund (University of Pennsylvania): Three Faces of the Delta Conjecture
Notes by Jose Simental Rodriguez
Abstract: The Delta Conjecture says that a certain symmetric function, expressed in terms of Macdonald polynomial operators, equals a weighted sum over Dyck lattice paths. It contains the well-known Shuffle Theorem of Carlsson and Mellit as a special case. There is also a third side to the problem that has emerged, centered around the goal of showing the symmetric function side also has a representation-theoretic interpretation. We will overview some of this work, including a recent conjecture of Mike Zabrocki which says the two sides of the Delta Conjecture equal the bigraded character of a generalization of the diagonal coinvariant ring. We will also discuss work of D'Adderio, Iraci, and Wyngaerd, who have used plethystic calculus to prove special cases of the Delta Conjecture.
Matt Hogancamp (University of Southern California): Derived traces of the type A Hecke category
Notes by Jose Simental Rodriguez
Abstract: In this talk I will discuss joint work (in progress) with Eugene Gorsky and Paul Wedrich concerning derived versions of the horizontal and vertical trace of the Hecke category in type A. Such notions have been studied by other authors (often under the heading of "character sheaves"), but our approach will be quite a bit more concrete than one often finds in the literature. I will begin by defining explicitly what we mean by derived (or dg) horizontal and vertical trace and how they are related. We will then consider Soergel bimodules in type A, whose derived vertical trace we compute explicitly (extending a result of Elias-Lauda) and whose horizontal trace we conjecturally describe (extending results of Queffelec-Rose). Unlike its underived counterpart, the derived horizontal trace of SBim is expected to inherit categorical analogues of various skein theoretic operations, including the insertion of a full twist. This provided the main motivation for this work.
Anthony Licata (Australian National University): Braids, triangulated autoequivalences, and geometric group theory.
Notes by Jose Simental Rodriguez ,
notes by Minh-Tam Trinh
Abstract: Many dynamical and group theoretic questions about mapping class groups of a surface are studied by considering the action of the mapping class group on the Thurston compactification of the Teichmuller space of the surface. The goal of this talk will be to explain a (largely conjectural) parallel story where the mapping class group is replaced by the autoequivalence group of a triangulated category; we'll use the most basic categorical braid group action as a running example. This is joint work with Asilata Bapat and Anand Deopurkar.
Ivan Loseu (University of Toronto): Representations of quantized Gieseker varieties and
higher rank Catalan numbers
Notes by Jose Simental Rodriguez ,
notes by Minh-Tam Trinh
Abstract: A quantized Gieseker variety is an associative algebra quantizing the global
functions on a Gieseker moduli space. This algebra arises as a quantum Hamiltonian
reduction of the algebra of differential operators on a suitable space. It depends on
one complex parameter and has interesting and beautiful representation theory.
For example, when it has finite dimensional representations, there is a unique
simple one and all finite dimensional representations are completely reducible.
In fact, this is a part of an ongoing project with Pavel Etingof and Vasily Krylov,
one can explicitly construct the irreducible finite dimensional representation using
a cuspidal equivariant D-module on \(\mathfrak{sl}_n\) and get an explicit dimension (and character)
formula. This formula gives a "higher rank" version of rational Catalan numbers.
I'll introduce all necessary definitions, describe the results mentioned above and,
time permitting talk about open problems.
Anton Mellit (University of Vienna): Nabla operator: explicit formulas and the Springer object.
Abstract: I will report on my ongoing project with Erik Carlsson motivated by the nabla positivity conjecture. I will explain how the \(q,t\) polynomials \((\nabla s_\mu, s_\lambda)\) can be interpreted in two settings: one is coming from the Hilbert scheme, and another is expected to come from Rouquier complexes, where the construction involves the Springer object. Then I'll present explicit formulas we have obtained so far.
Brendon Rhoades (UC San Diego): Spanning configurations
Notes by Jose Simental Rodriguez
Abstract: A sequence \((W_1, W_2, \dots, W_r)\) of subspaces of a fixed finite-dimensional complex vector space \(V\) is a
Laura Rider (University of Georgia): Nearby cycles for parity sheaves and the affine Hecke category
Notes by Jose Simental Rodriguez
Abstract: Achar recently introduced a "nearby cycles formalism" in the framework of chain complexes of parity sheaves. In joint work with Achar, we compute the output of this functor in two related settings. The first is affine space, stratified by the action of a torus, and the second is the global Schubert variety associated to the first fundamental coweight of the group \(PGL_n\). The latter is a parity-sheaf analogue of Gaitsgory's central sheaf construction. See also talks by Elias and Achar.
Jose Simental Rodriguez (UC Davis): Harish-Chandra bimodules for quantized Hilbert schemes
Notes
Abstract: We consider categories of Harish-Chandra bimodules over a quantization of the Hilbert scheme of points in \(\mathbb{C}^2\), aka a spherical rational Cherednik algebra, that categorify the top Borel-Moore homology of the Steinberg variety. While many of the properties are similar to those of the corresponding category \(\mathcal{O}\) (e.g. the existence of exact restriction functors) some others are more complicated (even the number of simples is highly dependent on the period of the quantization, for example). We will give basic properties of this category, as well as the construction of restriction, induction and coinduction functors and a duality functor that comes, essentially, from interchanging the x- and y-axis in \(\mathbb{C}^2\) and explains the mysterious "double wall-crossing" bimodule of Bezrukavnikov-Losev.
Paul Wedrich (Australian National University): Categorical invariants of annular links
Notes by Jose Simental Rodriguez
Abstract: A classical result of Turaev identifies the positive HOMFLYPT skein algebra of the annulus with the algebra of symmetric functions. Queffelec and Rose categorified this using annular webs and foams. I will recall their construction and compute explicit symmetric functions and their categorical analogues for some links. This is joint work with Eugene Gorsky.
Elijah Bodish (University of Oregon): Clasp Formulas for \(B_2\) Webs
Notes by Jose Simental Rodriguez
Abstract: In 1997 Kuperberg gave a generators and relations description for the tensor category generated by the fundamental representations of the quantum group associated to a rank two root system. In an attempt to better understand Kuperberg`s category in type \(B_2\) we construct an algorithm to find a so called "light-leaves" basis. This basis naturally gives rise to recursive formulas for projectors in this category. The formulas for the coefficients provide further evidence for Elias`s "clasp conjecture".
Jay Hathaway (University of Oregon): The Endomorphism X of the Gaitsgory's Central Complex for the Standard Representation
Abstract: The tautological bundle on the flag Hilbert scheme of points in the plane carries an endomorphism X arising from the coordinate function x on the plane. This endomorphism is upper triangular with respect to the flag filtration of the tautological bundle. There is a complex of affine Soergel bimodules with a filtration, analogous to the filtration of the tautological bundle, which categorifies the Jucys-Murphy elements in the affine Hecke algebra. I'll explain how an upper triangular endomorphism of this complex of Soergel bimodules, analogous to X, can be computed explicitly by lifting an endomorphism of the associated graded.
Graham Hawkes (UC Davis): Generalized \(q,t\)-Catalan numbers from a combinatorial perspective
Abstract: Recent work of Gorsky, Negut and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov-Rozansky knot homology produces a family of polynomials in \(q\) and \(t\) labeled by integer sequences. The rational \(q,t\)-Catalan numbers are special cases of this construction. We give a purely combinatorial treatment of these polynomials and show that in many cases they have nonnegative integer coefficients.
For sequences of length at most 4, we prove that these coefficients enumerate subdiagrams in a certain fixed Young diagram and give an explicit symmetric chain decomposition of the set of such diagrams. This strengthens results of Lee, Li and Loehr for \((4,n)\) rational \(q,t\)-Catalan numbers. The talk is based on the joint paper arXiv:1905.10973 with E. Gorsky and A. Schilling.
Dmytro Matvieievskyi (Northeastern University): On \(G\)-equivariant quantizations of nilpotent coadjoint orbits.
Notes by Jose Simental Rodriguez
Abstract: Let \(\mathfrak{g}\) be a simple Lie algebra over
\(\mathbb{C}\), \(G\) be the corresponding simply connected algebraic
group and \(\mathbb{O}\subset \mathfrak{g}^*\) be a nilpotent coadjoint
orbit. In this talk I will prove that the set of \(G\)-equivariant
formal graded quantizations of \(\mathbb{O}\) is an affine space.
The key part of the proof is to construct a bijection between the sets
of \(G\)-equivariant formal graded quantizations of \(\mathbb{O}\) and its
affinization \(Spec(\mathbb{C}[\mathbb{O}])\). The latter set is an
affine space due to a result of Losev. This talk is based on
arXiv:1810.11531
Jianping Pan (UC Davis): Virtualization of root systems and Littelmann Path Model
Slides
Abstract: I will talk about basics about root systems, representation theoretic motivation for studying crystals, virtualization of the root system and Littelmann path model, with pictures and examples. Our main result is showing the natural embedding of weight lattices from a diagram folding is a virtualization map for the Littelmann path model, which recovers a result of Kashiwara. This talk is based on this paper .
Wencin Poh (UC Davis): Characterization of queer supercrystals
Slides
Abstract: We provide a characterization of the crystal bases for the quantum queer
superalgebra recently introduced by Grantcharov et al. This characterization is a
combination of local queer axioms generalizing Stembridge's local axioms for crystal
bases for simply-laced root systems, which were recently introduced by Assaf and
Oguz, with further axioms and a new graph \(G\) characterizing the relations of
the type \(A\) components of the queer crystal. We provide a counterexample to
Assaf's and Oguz' conjecture that the local queer axioms uniquely characterize
the queer supercrystal. We obtain a combinatorial description of the graph \(G\) on
the type \(A\) components by providing explicit combinatorial rules for the odd
queer operators on certain highest weight elements. This is joint work with Maria Gillespie, Graham Hawkes and Anne Schilling based on arXiv:1809.04647 .
Minh-Tam Trinh (University of Chicago): Markov Traces and Rational Cherednik Algebras
Notes
Abstract: Gorsky, Oblomkov, Rasmussen, and Shende observed an identity relating the HOMFLY polynomial of the \((m, n)\)-torus knot and the simple spherical module of the rational DAHA of type \(A_{n-1}\) of parameter m/n. They also conjectured a refinement that would relate the Khovanov-Rozansky homology of the former and an interesting \(t\)-grading on the latter. We first generalize their identity beyond type A. In doing so, we introduce a bigraded virtual module of the graded affine Hecke algebra that we can use to sharpen their homological conjecture, and more broadly, (a generalization of) the Oblomkov-Rasmussen-Shende conjecture on locally-planar algebraic curves.
Yue Zhao (UC Davis): A combinatorial description of some representations of dAHA and
dDAHA of type \(BC\)
Notes by Jose Simental Rodriguez
Abstract: For each \(GL_N\)-module \(M\), Etingof , Freund and Ma defined a Schur Weyl like functor \(F_{n}^{\lambda}\) which takes \(M\) to a representation of dAHA and dDAHA of type \(BC\). In the Let \(\mathcal{R}(G/K)\) be the algebra of \(\lambda\)-twisted functions on \(G/K\). This is also a module for \(\mathcal{D}(G/K)\) which is the algebra of differential operators and hence a module of \(GL_N\). In this talk, we will describe some properties of the representation \(F_{n}^{\lambda}(\mathcal{R}(G/K))\). There are also quantum versions of the above: \(\mathcal{D}_q(G/K)\), \(\mathcal{R}_q(G/K)\) and \(\mathcal{F}_n(\mathcal{R}_q(G/K))\). Our ultimate goal is to give a similar description of the representation \(\mathcal{F}_{n}(\mathcal{R}_q(G/K))\).