Representations and Lie Theory SeminarAutumn 2018
Time: Wednesday, 16:15  17:15


TIME  SPEAKER  TITLE 
August 29  Staff (après EtingofVarchenko)  Dynamical constructions in representation theory I. Fusion operator 
September 19  Staff (après EtingofVarchenko)  Dynamical constructions in representation theory II. Weyl group 
October 2 (Tuesday) 34PM Unusual time! Same place! 
Andrea Appel  The KZCasimir system and braided Coxeter categories 
October 10  No seminar  Autumn break 
October 24  Xiaomeng Xu  Stokes phenomenon, GelfandZeitlin systems and Frobenius manifolds 
October 31  
November 7  Petr Pushkar  Quantum K theory of the Grassmannian and the Baxter operator 
November 14  Alex Weekes  Yangians and KLR algebras 
November 21  No seminar  Thanksgiving break 
December 5  You Qi  On a tensor product categorification at prime roots of unity 
August 29: In this series of talks, we will go over the construction of a few dynamical operators which emerge naturally in the representation theory of Lie algebras. I will explain what the fusion operator is and why it satisfies the KnizhnikZamolodchikov (partial differential) equation. We will focus on the (dynamical) Weyl group symmetries of the fusion operator in a follow up talk.
September 19: In this part, I will present the definition and basic properties of a dynamical Weyl group action on a given representation of a simple Lie algebra (and time permitting, affine Lie algebra). I will not assume anything from the first talk.
October 2: It is wellknown that the monodromy data of the KZ equations of a semisimple Lie algebra are encoded by a nontrivial braided tensor structure on the category of its deformation modules. In this talk, I will introduce the equivariant Casimir connection of a symmetrizable KacMoody algebra and I will then show that the monodromy data of the associated KZCasimir joint system are described by the axiomatic of braided Coxeter categories, which are, informally, braided tensor categories carrying an action of a given generalized braid group on their objects. This is based on joint works with V. Toledano Laredo.
October 24: This talk will include a general introduction to a linear differential system with irregular singularities, and its relation with GelfandZeitlin systems, YangBaxter equations, and deformations of Frobenius manifolds.
November 7: In this talk I will define the quantum Ktheory of the Grassmannian and show its connection to representation theory of quantum groups and quantum integrable systems. In particular, the Baxter operator will be identified with operators of quantum multiplication by quantum tautological classes via Bethe equations. Based on joint works with P.Koroteev, A.Smirnov and A.Zeitlin.
November 14: In joint work with J. Kamnitzer, P. Tingley, B. Webster and O. Yacobi, we've studied the representation theory of algebras called truncated shifted Yangians. Recently, we proved that this representation theory is governed by some algebras closely related to KLR algebras. In this talk, I'll describe this connection through a few simple examples. I'll also describe some applications to some (possibly) more familiar problems in representation theory, like the study of GelfandTsetlin modules over U(gl(n)), and the calculation of characters of simple modules for Yangians.
December 5: Motivated by finding a categorical analogue of conformal blocks, we explain a formalism of extending a given categorical quantum group representation on a Weyl module to a certain tensor product representations. In particular, equipped with pdifferential graded structures, the machinery gives rise to a categorification of certain tensor product representations of Weyl modules at prime roots of unity. This is based on joint work and work in progress with M. Khovanov and J. Sussan.
Previous seminar (Spring 2017)
Previous seminar (Autumn 2017)