Representation Theory and Mathematical Physics Seminar

Spring 2019 Schedule:

 Jan. 29 Nicolai Reshetikhin Integrable systems of spin Calogero-Moser type related to symmetric spaces Feb. 5 Vera Serganova Temperley- Lieb algebra action on the category of representations of periplectic supergroup Feb. 12 No Seminar Feb. 19 Seminar Canceled Feb. 26 Nicolai Reshetikhin Integrable systems of spin Calogero-Moser type related to symmetric spaces, part II Mar. 5 Mar. 12 Mar. 19 Mar. 26 No Seminar Apr. 2 William Casper Matrix Orthogonal Polynomials and Representation Theory Apr. 9 Alexander Sherman Spherical Supervarieties Apr. 10* Yanpeng Lie New action-angle variables on coadjoint orbits Apr. 16 Anton Zeitlin Field Equations, Homotopy Gerstenhaber Algebras and Courant Algebroids Apr. 23 Carl Mautner Matroids, Schur algebras and category O Apr. 24* Nima Moshayedi Globalization of Split AKSZ Sigma Models in the BV-BFV Formalism Apr. 26* John Harnad Generating rationally weighted Hurwitz numbers with KP τ-functions Apr. 30 Natalia Rojkovskaia Vertex operators realizations of families of symmetric functions May 7 Daniil Rudenko May 15* Philippe Mathieu May 16* Ivan Contreras Poisson sigma models, abelianization and extended monodromy

* - special day/time

Seminar Archive

Jan. 29: Nicolai Reshetikhin (UC Berkeley)

Integrable systems of spin Calogero-Moser type related to symmetric spaces

The talk will start with a reminder of what is a Hamiltonian integrable system and what degenerate integrability, also know as superintegrability, means. Then examples of such systems on symplectic leaves of Poisson variety $K\subset {T}^{*}G/K$ will be constructed for a Lie group $G$ and a Lie subgroup $K\subset G$. If $G$ is a simple Lie group and $K$ is the subgroup of fixed points of the Chevalley automorphism of $G$, Hamiltonians of such integrable systems will be described explicitly.

Feb. 5: Vera Serganova (UC Berkeley)

Temperley- Lieb algebra action on the category of representations of periplectic supergroup

We define the action of infinitely generated Temperley-Lieb algebra on the category of representations of the supergroup P(n). The supergroup in question is an interesting super analogue of the orthogonal and symplectic groups. As an application of this construction we get algorithm computing characters of irreducible representation of P(n) and some other esults. As n tends to infinity, we obtain a new universal tensor category equipped with Temperley-Lieb algebra action. In this way we obtain representation of TL in the Fock space.

Apr. 2: William Casper (Louisiana State University)

Matrix Orthogonal Polynomials and Representation Theory

We present a solution of the matrix Bochner problem, a long-standing open problem in the theory of orthogonal polynomials, with applications to diverse areas of research including representation theory, random matrices, spectral theory, and integrable systems. Our solution is based on ideas applied by Krichever, Mumford, Wilson and others, wherein the algebraic structure of an algebra of differential operators influences the values of the operators within the algebra. By using a similar idea, we convert the matrix Bochner problem to one about noncommutative algebras of GK dimension 1 which are module finite over their centers. Then the problem is resolved using the representation theory of these algebras.

Apr. 9: Alexander Sherman (UC Berkeley)

Spherical Supervarieties

Spherical varieties are algebraic varieties with an action by a reductive group which admit an open Borel orbit. This extra condition on its symmetries connects their study to representation theory, makes tractable their classification, and yet is broad enough to have many rich examples.

We introduce a definition of a spherical supervariety, which is a simple generalization of the classical definition to the super world. Then, with a focus on the affine case, we look at certain properties of these spaces, highlighting some of the differences and similarities with the classical story.

Wednesday, Apr. 10, 4-5 PM in Evans 939: Yanpeng Lie (University of Geneva)

New action-angle variables on coadjoint orbits

The problem of constructing global action-angle variables on coadjoint orbits of compact Lie groups is one of the interesting questions in the theory of integrable systems. A fundamental contribution was made by Guillemin-Sternberg who constructed the Gelfand-Zeitlin integrable systems on coadjoint orbits of the groups SU(n) and SO(n). Recently, toric degeneration techniques allowed for the construction of global action-angle variables on rational coadjoint orbits of compact Lie groups of all types.

In this talk, I will present a new approach which aims at constructing global action-angle coordinates on all regular coadjoint orbits of compact Lie groups and on a large family of related Hamiltonian spaces. It combines the results of Ginzburg-Weinstein on the theory of Poisson-Lie groups and the theory of cluster algebras using the "partial tropicalization'' procedure.

The talk is based on joint works with A. Alekseev, A. Berenstein, B. Hoffman, and J. Lane.

Apr. 16: Anton M. Zeitlin (Louisiana State University)

Field Equations, Homotopy Gerstenhaber Algebras and Courant Algebroids

I will talk about the underlying homotopical structures within field equations, which emerge in string theory as conformal invariance conditions for sigma models. I will show how these, often hidden, structures emerge from the homotopy Gerstenhaber algebra associated to vertex and Courant algebroids, thus making all such equations the natural objects within vertex algebra theory.

Apr. 23: Carl Mautner (UC Riverside)

Matroids, Schur algebras and category O

The Schur algebra is a finite dimensional algebra that connects a number of interesting topics, including the modular representation theory of the symmetric and general linear groups and category O. I will discuss joint work with Tom Braden motivated the the theory of symplectic duality in which we introduce a similar algebra for any graph or, more generally, matroid. I will also discuss more recent work in progress with Jens Eberhardt, which relates these matroidal Schur algebras' to Braden-Licata-Proudfoot-Webster's hypertoric category O' via categorification.

Wednesday, Apr. 24, 3-4 PM in Evans 736: Nima Moshayedi (University of Zurich)

Globalization of Split AKSZ Sigma Models in the BV-BFV Formalism

I give an introduction to the BV-BFV formalism and discuss the setting of certain AKSZ theories. Moreover, I describe a globalization procedure using concepts of formal geometry, which extends the Quantum Master Equation for manifolds with boundary.

Friday, Apr. 26, 3-4 PM in Evans 740: John Harnad (CRM Montreal and Concordia University)

Generating rationally weighted Hurwitz numbers with KP $\tau$-functions

Hurwitz numbers enumerate branched coverings of the Riemann sphere with specified branching profiles. τ-functions of hypergeometric type for the KP and 2D-Toda integrable hierarchies serve as combinatorial generating functions for weighted sums over Hurwitz numbers, with weights chosen as symmetric functions of a set of auxiliary parameters determined by a weight generating function. This talk will explain how multicurrent correlators may be used to explicitly generate weighted Hurwitz numbers as weighted polyonomials in the Taylor coefficients of the weight generating function, without any knowledge required either of symmetric group characters or the Kostka matrices relating different bases of the ring of symmetric functions. The case of rational weight generating functions will be the main illustrative example.

Apr. 30: Natalia Rojkovskaia (Kansas State University)

Vertex operators realizations of families of symmetric functions

The algebra of charged free fermions participates in the construction of classical boson-fermion correspondence and provides vertex operator realization of Schur symmetric functions. We will show how vertex operator realizations of several other famous families of symmetric functions (Hall-Littlewood polynomials, shifted Schur functions, multiparameter Schur Q-functions) can be obtained by simple modifications of operators of charged free fermions and make some notes on the corresponding versions of boson-fermion correspondence.

Thursday, May 16, 4-5 PM in Evans 736: Ivan Contreras (Amherst College)

Poisson sigma models, abelianization and extended monodromy

The reduced phase space of the Poisson Sigma Model (PSM) comes equipped with a symplectic groupoid structure, when the worldsheet is a disk and the target Poisson structure is integrable. In this talk we describe an extension of this construction when we consider surfaces with arbitrary genus, obtaining the abelianization of the original groupoid. We will also describe the obstructions for smoothness of such abelianization, in terms of the extended monodromy groups. This can be seen as a generalization of the Hurewicz theorem to Lie groupoids and Lie algebroids. Joint work with Rui Fernandes.