Representation Theory and Mathematical Physics Seminar

The seminar has moved to Fridays 9:30-10:30am Los Angeles time.
Please email to get zoom address and password.

Organized by Semeon Artamonov, Nicolai Reshetikhin, Vera Serganova, and Alexander Shapiro.

Spring 2020 Schedule:

Feb 4Anne DranowskiCalculating Duistermaat-Heckman measures of a perfect basis
Feb 18Kostya TolmachovSymmetric braiding on the derived category of character sheaves on GL(n)
Feb 25Sam GunninghamThe finiteness conjecture for Skein modules
Mar 3Catharina StroppelSemi-infinite highest weight categories
Mar 10Alexey ElaginDerived categories of coherent sheaves on affine schemes and their thick subcategories
Apr 17Victor OstrikIncompressible tensor categories
Apr 24Paul WedrichInvariants of 4-manifolds from Khovanov-Rozansky link homology
May 1Daniil RudenkoRational Elliptic Surfaces and Trigonometry of Non-Euclidean TetrahedraVideo
May 8Sergei GukovEquivariant Verlinde formula for Coulomb branchesSlidesVideo
May 15Siddhartha SahiMetaplectic Representations of Hecke algebrasSlidesVideo
May 22Andrey SmirnovElliptic stable envelopes and symplectic dualitySlidesVideo
May 29Anton MellitAq,t algebra and shuffle conjecturesSlidesVideo
Jun 5Leonid RybnikovBethe subalgebras in Yangians and the wonderful compactificationVideo
Jun 12Mikhail BershteinTwisted representations of toroidal gl_1SlidesVideo
Jun 26Robert WestonBaxter's Q-operator for Open Quantum Spin ChainsSlidesVideo

* - special day/time

Seminar Archive

Fall 2019, Spring 2019, Fall 2018, Fall 2017

Feb 4: Anne Dranowski (University of Toronto)

Calculating Duistermaat-Heckman measures of a perfect basis

By work of Lusztig, generic modules for the preprojective algebra give bases for representations of ADE groups. By work of Mirkovic and Vilonen, MV cycles give bases for these representations as well. While both bases were known to be crystal bases practically since their inception, only recently did we see polytope descriptions of their crystal structures. The crystal structure on MV cycles was modelled on their moment polytopes by Kamnitzer. The crystal structure on generic modules was modelled on their Harder-Narasimhan polytopes by Baumann, Kamnitzer and Tingley. Miraculously, these descriptions turned out to coincide (HN polytopes agreeing with MV polytopes). Polytopes being inherently geometric, the combinatorial coincidence called for an upgrade, and in recent work by Baumann, Kamnitzer and Knutson, the authors asked whether equal polytopes have equal equivariant volumes. By identifying open subsets in MV cycles with generalized orbital varieties, we will show that the answer to this question in type A is no.

Feb 18: Kostya Tolmachov (Perimeter Institute and the University of Toronto)

Symmetric braiding on the derived category of character sheaves on GL(n)

In view of recent developments in knot homologies, it is sometimes desirable to be able to compute Schur functors in certain triangulated categories of adjoint-equivariant sheaves on GL(n). However, canonical braided structures in these categories are not symmetric, so aforementioned functors are not well-defined. I will report on a joint work with Roman Bezrukavnikov, where this deficiency is fixed for the derived category of character sheaves.

Feb 25: Sam Gunningham (King's College London)

The finiteness conjecture for Skein modules

The Kauffman bracket skein module of an oriented 3-manifold M is a vector space (depending on a parameter q) which is generated by framed links in M modulo certain skein relations. The goal for the talk is the explain our recent proof (joint with David Jordan and Pavel Safronov) that the skein module of a closed 3 manifold is finite dimensional for generic q, confirming a conjecture of Witten. The proof involves understanding skein modules in terms of deformation quantizations of SL(2,C)-character varieties.

Mar 3: Catharina Stroppel (University of Bonn)

Semi-infinite highest weight categories

Highest weight categories are ubiquitous in representation theory and can be viewed as an axiomatisation of classical highest weight theory of for instance Lie algebras. Motivated by categorification and diagram algebras we like to generalise this notion to certain infinite dimensional settings which also covers more naturally the classical cases. In the talk I will review shortly the notion of highest weight categories in a more modern language and then generalize to the infinite setting. The main result will be a Ringel duality theorem relating categories of comodules and categories of modules and as an application a new definition of based quasi-hereditary algebras. A list of concrete examples for this setup will be given. This is based on joint work with Jon Brundan.

Mar 10: Alexey Elagin (Institute for Information Transmission Problems, HSE University)

Derived categories of coherent sheaves on affine schemes and their thick subcategories

I will tell about derived categories of coherent sheaves on affine schemes and their thick subcategories. How many such subcategories are there? For regular schemes X the classification of thick subcategories in Db(coh X) is known while for most non-regular ones it seems to be a wild problem. I will recall about some non-regular affine schemes where the classification of thick subcategories is possible and argue that for the second infinitesimal neighborhood of a point on affine space the classification is hardly possible. If time permits I will explain that thick subcategories in derived categories of coherent sheaves on affine schemes usually has no strong generators. This is a joint work with Valery Lunts.

Apr 17: Victor Ostrik (University of Oregon)

Incompressible tensor categories

This talk is based on joint work with Benson and Etingof. We say that a symmetric tensor category is incompressible if there is no symmetric tensor functor from this category to a smaller tensor category. Our main result is a construction of new examples of incompressible tensor categories in positive characteristic.

Apr 24: Paul Wedrich (MSRI)

Invariants of 4-manifolds from Khovanov-Rozansky link homology

Ribbon categories are 3-dimensional algebraic structures that control quantum link polynomials and that give rise to 3-manifold invariants known as skein modules. I will describe how to use Khovanov-Rozansky link homology, a categorification of the gl(N) quantum link polynomial, to obtain a 4-dimensional algebraic structure that gives rise to vector space-valued invariants of smooth 4-manifolds. The technical heart of this construction is the newly established functoriality of Khovanov-Rozansky homology in the 3-sphere. Based on joint work with Scott Morrison and Kevin Walker.

May 1: Daniil Rudenko (University of Chicago)

Rational Elliptic Surfaces and Trigonometry of Non-Euclidean Tetrahedra

I will explain how to construct a rational elliptic surface out of every non-Euclidean tetrahedra. This surface "remembers" the trigonometry of the tetrahedron: the length of edges, dihedral angles and the volume can be naturally computed in terms of the surface. The main property of this construction is self-duality: the surfaces obtained from the tetrahedron and its dual coincide. This leads to some unexpected relations between angles and edges of the tetrahedron. For instance, the cross-ratio of the exponents of the spherical angles coincides with the cross-ratio of the exponents of the perimeters of its faces. The construction is based on relating mixed Hodge structures, associated to the tetrahedron and the corresponding surface.

May 8: Sergei Gukov (California Institute of Technology)

Equivariant Verlinde formula for Coulomb branches

This will be more of an "algebra and geometry" seminar, as we will be discussing geometric constructions of various algebraic structures: Galois actions on modular categories, module categories of quantum groups and logarithmic vertex algebras. Although I hope to convey the "big picture" as well, the focus will be on simple, easy-to-compute aspects of this relation. Much of the talk will be based on prior work, but if time permits I will try to incorporate elements of the ongoing work with Po-Shen Hsin, Hiraku Nakajima, Sunghyuk Park, Du Pei, and Nikita Sopenko.

May 15: Siddhartha Sahi (Rutgers University)

Metaplectic Representations of Hecke algebras

Cherednik's double affine Hecke algebra (DAHA) admits an important representation, called the polynomial representation, which plays a key role in the theory of Macdonald polynomials. We describe a "metaplectic" generalization of this representation, which is motivated by the theory of Weyl group multiple Dirichlet series. More precisely for each integer n we describe a certain representation of the DAHA, which depends on [n/2] auxiliary parameters g_i, and which for n=1 reduces to the polynomial representation. As a direct consequence we obtain a family of "metaplectic" polynomials, which generalizes Macdonald polynomials.

When the parameters g_i are specialized to certain Gauss sums, then the resulting polynomials are closely related to the p-parts of certain Weyl group multiple Dirichlet series, which have been studied by Kazhdan-Patterson for GL(n), and by Bump-Brubaker-Chinta-Friedberg-Gunnells for other groups.

This is joint work with Jasper Stokman and Vidya Venkateswaran.

May 22: Andrey Smirnov (University of North Carolina, Chapel Hill)

Elliptic stable envelopes and symplectic duality

In this talk I'll explain the following idea: "the elliptic stable envelopes of symplectic dual varieties coincide." I'll describe a simplest example of T*P1 in details and discuss other cases in which the statement is proven.

May 29: Anton Mellit (University of Vienna)

Aq,t algebra and shuffle conjectures

Aq,t algebra was introduced by Erik Carlsson and myself for the purpose of proving the shuffle conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov. It turned out to lie somewhat between the elliptic Hall algebra (EHA) and the double affine Hecke algebra (DAHA), more flexible than the former, but not as big as the latter. The shuffle conjecture can be formulated as an identity in EHA, but for proving it we needed a recursion whose intermediate steps lied in the bigger algebra Aq,t. Topological realization of Aq,t turned out to be useful for proving the rational shuffle conjecture and for computing the triply graded homology of torus knots. There is also realization on the K-theory of Hilbert schemes which helps to compute certain natural operators there. I will try to explain the main ideas, the big picture and discuss new developments.

Jun 5: Leonid Rybnikov (HSE University)

Bethe subalgebras in Yangians and the wonderful compactification

Bethe subalgebras in the Yangian Y(g) are maximal commutative subalgebras depending on a parameter from the corresponding adjoint Lie group G. For g=sl2, these subalgebras are generated by the integrals of the XXX Heisenberg magnet chain. We extend the family of Bethe subalgebras to the De Concini--Procesi wonderful compactification G⊃G and describe the subalgebras corresponding to generic points of any stratum in G as Bethe subalgebras in the Yangian of the corresponding Levi subalgebra in g. We expect that monodromy of solutions of Bethe ansatz equations along some particular paths in this extended parameter space gives rise to a group of piecewise linear transformations generalizing the cactus group action on Kashiwara crystals (due to Henriques and Kamnitzer).

This is a joint work with Aleksei Ilin,

Jun 12: Mikhail Bershtein (HSE University, Landau Institute for Theoretical Physics)

Twisted representations of toroidal gl_1

Fock module is a basic representation of quantum toroidal gl(1); it can be identified with equivariant K-theory of Hilbert scheme of points on C2. We study a twisted Fock module which is the same vector space with an action toroidal algebra twisted by SL(2,Z) automorphism. Surprisingly, certain explicit formulas for this module lead to an appearance of an auxiliary quantum affine gl(n)-action on (twisted) Fock space. Conjecturally, this construction has geometric application (conjecture of Gorsky and Negut on K-theoretic stable bases). If time permits I will also talk about finitization of this story and DAHA.

Based on joint work with R. Gonin.

Jun 26: Robert Weston (Heriot-Watt University)

Baxter's Q-operator for Open Quantum Spin Chains

Baxter introduced the Q-operator in 1972 as an ingenous way of obtaining Bethe Ansatz equations for closed quantum spin chains without a Bethe Ansatz. In the late 1990s this operator was finally understood in the language of the quantum inverse scattering method/quantum groups. The Q-operator is a purely representation theory object: it is a trace of the universal R-matrix over an infinite-dimensional representation of a Borel subalgebra of the underlying quantum group. In this seminar, which is based on joint work with Bart Vlaar, I will develop an analogous algebraic construction of a Q-operator for open quantum spin chains. The complication for open systems is that there are now four algebras in play: the quantum group, two Borel subalgebras and a coideal subalgebra of the quantum group. The coideal subalgebra is associated with the solution of the reflection equation that describes the boundary conditions of the open quantum spin chain. Our Q-operator construction involves representations and associated homomorphisms of all four algebras. In the simple diagonal-boundary XXZ example that we consider, the resulting Q-operator reproduces the known open Bethe Ansatz equations of Sklyanin.