Geometric Methods in Representation Theory Seminar
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: |
George Lusztig
Z/m graded Lie algebras and intersection cohomology |
Jiuzu Hong
Conformal blocks, Verlinde formula and diagram automorphisms |
You Qi
On the center of small quantum groups |
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 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 |
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 |