2016 |

University of North Carolina at Chapel Hill

Fridays 4pm, PH-367 or PH-385

The aim of this seminar is to bring speakers from this area and outside to speak on topics related to Representation Theory (specially geometric and topological methods employed in Representation Theory). The speakers are expected to give their talks at a level suitable for graduate students. The seminar is organized by Shrawan Kumar.

2016 Fall

Date	Speaker	Affiliation	Title
Nov 18	Mike Schuster	University of Georgia	Sub-cones of the additive eigencone
Oct 14	George Lusztig	MIT	Z/m graded Lie algebras and intersection cohomology
Sep 30	Jiuzu Hong	UNC Chapel Hill	Conformal blocks, Verlinde formula and diagram automorphisms
Sep 16	You Qi	Yale University	On the center of small quantum groups

Mike Schuster

Sub-cones of the additive eigencone

Abstract:
The additive eigenvalue problem for SL(n) is the problem of finding all possible sets of eigenvalues of three or more Hermitian matrices adding to the zero matrix. The set of solutions of this problem forms a convex polyhedral cone called the additive eigencone. Its interest lies in its close relationships with the representation theory of algebraic groups and cohomology of flag varieties. In this talk I will discuss special sub-cones of eigencones – called sub-eigencones – which satisfy a strong functoriality property. In particular, I will discuss a new family of examples of sub-eigencones arising from subgroups fixed by inner automorphisms.

George Lusztig

Z/m graded Lie algebras and intersection cohomology
Abstract: Let g be a semisimple Lie algebra with a given grading (g_i) where i runs over Z/m, a finite cyclic group. The variety of nilpotent elements in g_i (for fixed i) decomposes in finitely many orbits under the action of a certain algebraic group. The aim of the talk is the study of the intersection cohomology of the closure of any one of these orbits with coefficients in certain local systems. This is a joint work with Zhiwei Yun.

Jiuzu Hong

Conformal blocks, Verlinde formula and diagram automorphisms
Abstract: The Verlinde formula computes the dimension of conformal blocks associated to simple Lie algebras and Riemann surfaces. If the simple Lie algebra admits a nontrivial diagram automorphism, then this automorphism acts on the space of conformal blocks naturally. I will report a recent result on the analogue of Verlinde formula for the trace of this automorphism on conformal blocks.
Motivated by this formula and Jantzen’s twining formula, I will also give a conjecture on the Verlinde formula for the dimension of conformal blocks associated to twisted affine Lie algebras.

You Qi

On the center of small quantum groups
Abstract: We will report some recent progress on the problem of finding the center of small quantum groups. This will be based on joint work with Anna Lachowska.

2016 Spring

Date	Speaker	Affiliation	Title
April 15	Syu Kato	Kyoto University	An algebraic study of extension algebras
April 8	John Duncan	Emory University	K3 Surfaces and Modular Forms
March 25	Ivan Loseu	Northeastern University	Deformations of symplectic singularities and the orbit method
Feb 26	Jiuzu Hong	UNC Chapel Hill	Affine Grassmannian and basis theory
Feb 19	Linhui Shen	Northwestern University	Donaldson-Thomas transformations for moduli spaces of local systems on surfaces

Syu Kato

An algebraic study of extension algebras
Abstract: The BGG category $\mathcal O$ and affine Hecke algebras share a common feature that the algebra we concern is realized by the $\mathrm{Ext}$ of the direct sum of all the possible intersection cohomology complexes relevant to the analysis.

We formalize such a geometric situation by the finiteness of orbits of a linear algebraic group action and some purity conditions on intersection cohomology complexes so that the $\mathrm{Ext}$-algebras satisfies an analogous structure like a highest weight category.

This formulation allows us to describe the minimal projective resolution of standard modules of $\mathrm{Ext}$-algebras, and give some non-trivial criterion of the purity of intersection cohomology complexes (in turn). This have some applications to the module category of affine Hecke algebras, a categorification of Kostka polynomials, and a proof of the positivity of PBW bases of quantum groups.

John Duncan

K3 Surfaces and Modular Forms
Abstract: Mukai demonstrated a curious connection between K3 surfaces and sporadic simple groups in 1988, by showing that the finite groups of symplectic automorphisms of a complex K3 surface are the subgroups of the largest Mathieu group that have five orbits (including a fixed point) in their natural action on 24 points. More recently, Huybrechts established a derived generalization of this, relating groups of symplectic autoequivalences of K3 surfaces to subgroups of the Conway group. We will present a construction which attaches modular forms to these autoequivalences. These forms are conjectured to encode equivariant versions of the enumerative K3 invariants.

Ivan Loseu
Deformations of symplectic singularities and the orbit method
Abstract: Symplectic singularities were introduced by Beauville in 2000. These are especially nice singular Poisson algebraic varieties that include symplectic quotient singularities and the normalizations of orbit closures in semisimple Lie algebras. Poisson deformations of conical symplectic singularities were studied by Namikawa who proved that they are classified by a points of a vector space. Recently I have proved that quantizations of a conical symplectic singularities are still classified by the points of the same vector spaces. I will explain these results and then apply them to establish a version of Kirillov’s orbit method for semisimple Lie algebras.

Jiuzu Hong
Affine Grassmannian and basis theory
Abstract: The geometry of affine Grassmannian is now getting more and more important in algebraic geometry, representation theory, number theory and even in categorification and link homology. In this talk, I will only restrict myself to the connection with representation theory. I will give an introduction to geometric Satake correspondence and explain how certain bases of representations arise from it. These bases are closely related to canonical bases and can be studied via tropical geometry by the works of Kamnitzer and Goncharov-Shen. In a sequel of this talk, I will talk about some applications of this beautiful theory in representation theory.

Linhui Shen

Donaldson-Thomas transformations for moduli spaces of local systems on surfaces
Abstract: Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in single formal automorphism of the cluster variety, called the DT-transformation.An oriented surface S with punctures, and a finite number of special points on the boundary give rise to a moduli space, closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure. We determine the DT-transformation of this space.This is a joint work with Alexander Goncharov.