Department of mathematics

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.

2017 Spring

Date Speaker Affiliation Title
April 21 Reuven Hodges Northeastern TBA
April 7 Curtis Porter NCSU TBA
Mar 31 4:40pm Gabor Pataki UNC Combinatorial characterizations in semidefinite programming duality
Mar 24 Xuhua He Maryland Cocenters and representations of p-adic groups
Mar 10 Olivier Debarre University Paris 7 and EMS UNEXPECTED ISOMORPHISMS BETWEEN HYPERKÄHLER FOURFOLDS
Feb 10 Goncalo Oliveira Duke Gerbes on G2-manifolds

 Gabor Pataki Combinatorial characterizations in semidefinite programming duality Abstract: I will discuss optimization problems over affine slices of positive definite symmetric matrices. These problems, called semidefinite programs (SDPs), have numerous applications. Several basic properties of SDP’s are rooted in the fact that the linear image of the cone of symmetric positive semidefinite matrices may not be closed. Examples of these properties include the non-triviality of the infeasibility problem (when the set of positive semidefinite matrices has an empty intersection with an affine subspace) and pathological properties of dual SDP’s. In this talk I survey recent, somewhat surprising combinatorial type characterizations for several fundamental problems in SDP duality. The main tool is very simple: we use elementary row operations – inherited from Gaussian elimination – to bring an SDP to a format in which properties – such as infeasibility – are trivial to recognize. Part of this work is joint with Minghui Liu. Xuhua He Cocenters and representations of p-adic groups Abstract: It is known that the number of conjugacy classes of a finite group equals the number of irreducible representations (over complex numbers). The conjugacy classes of a finite group give a natural basis of the cocenter of its group algebra. Thus the above equality can be reformulated as a duality between the cocenter (i.e. the group algebra modulo its commutator) and the finite dimensional representations.Now let us move from the finite groups to the $p$-adic groups. In this case, one needs to replace the group algebra by the Hecke algebra. The work of Bernstein, Deligne and Kazhdan in the 80’s establish the duality between the cocenter of the Hecke algebra and the complex representations. It is an interesting, yet challenging problem to fully understand the structure of the cocenter of the Hecke algebra.In this talk, I will discuss a new discovery on the structure of the cocenter and then some applications to the complex and modular representations of $p$-adic groups, including: a generalization of Howe’s conjecture on twisted invariant distributions, trace Paley-Wiener theorem for smooth admissible representations, and the abstract Selberg Principle for projective representations. It is partially joint with D. Ciubotaru. Olivier Debarre UNEXPECTED ISOMORPHISMS BETWEEN HYPERKÄHLER FOURFOLDS Abstract: In 1985, Beauville and Donagi showed by an explicit geometric construction that the variety of lines contained in a Pfaffian cubic hypersurface in $P^5$ is isomorphic to a canonical desingularization of the symmetric self-product of a K3 surface (called its Hilbert square). Both of these projective fourfolds are hyperkähler (or symplectic): they carry a symplectic 2-form. In 1998, Hassett showed by a deformation argument that this phenomenon occurs for countably many families of cubic hypersurfaces in $P^5$. Using the Verbitsky-Markman Torelli theorem and results of Bayer-Macri, we show these unexpected isomorphisms (or automorphisms) occur for many other families of hyperkähler fourfolds. This involves playing around with Pell-type diophantine equations. This is joint work with Emanuele Macrí. Goncalo Oliveira Gerbes on G2-manifolds Abstract: On a projective complex manifold, the Abelian group of Divisors (Div) maps onto that of holomorphic line bundles (the Picard group). I shall explain a similar construction for G2-manifolds. This uses coassociative submanifolds to define an analogue of Div, and a gauge theoretical equation for a connection on a gerbe to define an analogue of the Picard group.

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.