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Representation Theory Seminar
2013-2014

Fridays 2:00-3:00 in LCB 215

2012-2013: 2012-2013
2011-2012: 2011-2012
2010-2011: 2009-2010
2009-2010: 2009-2010
E. Cartan H. Weyl I. M. Gelfand Harish Chandra A. Borel R. Langlands

Date
 Speaker
Title
August 30
Moshe Adrian (University of Utah)
Jacquet's conjecture on the local converse problem for epipelagic supercuspidal representations
September 6
Baiying Liu (University of Utah)
Fourier coefficients of automorphic forms and Arthur classification, part 1
September 27
Baiying Liu (University of Utah)
Fourier coefficients of automorphic forms and Arthur classification, part 2
October 25
Veronika Ertl (University of Utah)
TBA
December 6
Wei Ho (Columbia University)
Families of lattice-polarized K3 surfaces
January 31
Matthew Housley (University of Utah)
Representation Theory and the Jones Polynomial
February 21
Thomas Nevins (University of Illinois at Urbana-Champaign)
Hamiltonian reduction in representation theory and algebraic geometry
February 28
Stephan Ramon Garcia (Pomona College)
Supercharacters on abelian groups
March 31
Luis Lomeli (University of Oklahoma)
On exterior and symmetric square \gamma-factors


August 30, 2013
Moshe Adrian
Title: Jacquet's conjecture on the local converse problem for epipelagic supercuspidal representations
Abstract
:   Let F be a non-archimedean local field of characteristic zero. For any irreducible admissible generic representation of GL(n,F), a family of twisted local gamma factors can be defined using Rankin-Selberg convolution or the Langlands-Shahidi method. Jacquet has formulated a conjecture on precisely which family of twisted local gamma factors can uniquely determine an irreducible admissible generic representation of GL(n,F). In joint work with Baiying Liu, we prove that Jacquet's conjecture is true for epipelagic supercuspidal representations of GL(n,F), supplementing recent results of Jiang, Nien, and Stevens.

September 6, 2013
Baiying Liu
Title: Fourier coefficients of automorphic forms and Arthur classification, part 1
Abstract
:   Fourier coefficients play an important role in the study of automorphic forms. For example, it is a well-known theorem, due to J. Shalika and I. Piatetski-Shapiro, independently, that any non-zero cuspidal automorphic form on GL_n(A) is generic, i.e. has a non-zero Whittaker-Fourier coefficient. On the other hand, a main theme in the theory of automorphic forms is to study the discrete spectrum. Recently, Arthur has classified the discrete spectrum of symplectic and special orthogonal groups up to automorphic L^2-packets parametrized by Arthur parameters. Towards understanding the structure of Fourier coefficients of representations in an automorphic L^2-packet, Jiang (2012) made a conjecture which gives the relation between the structure of Fourier coefficients and the structure of Arthur parameters. In this talk, focusing on symplectic groups, we introduce this conjecture and discuss some recent progress. This is a joint work with Prof. Dihua Jiang.

September 27, 2013
Baiying Liu
Title: Fourier coefficients of automorphic forms and Arthur classification, part 2
Abstract
:  

October 25, 2013
Veronika Ertl
Title: Overconvergent Chern classes
Abstract
:   For a proper smooth variety over a perfect field of characteristic p, crystalline cohomology is a good integral model for rigid cohomology and crystalline Chern classes are integral classes which are rationally compatible with the rigid ones. The overconvergent de Rham-Witt complex introduced by Davis, Langer and Zink provides an integral p-adic cohomology theory for smooth varieties designed to be compatible with rigid cohomology in the quasi-projective case. The goal of this talk is to describe the construction of integral Chern classes for smooth varieties rationally compatible with rigid Chern classes using the overconvergent complex.

December 6, 2013
Wei Ho
Title: Families of lattice-polarized K3 surfaces
Abstract
:   There are well-known explicit families of K3 surfaces equipped with a low degree polarization, e.g., quartic surfaces in P^3. What if one specifies multiple line bundles instead of a single one? We will discuss representation-theoretic constructions of such families, i.e., moduli spaces for K3 surfaces whose Neron-Severi groups contain specified lattices. These constructions, inspired by arithmetic considerations, also involve some explicit geometry and combinatorics. This is joint work with Manjul Bhargava and Abhinav Kumar.

January 31, 2014
Matthew Housley
Title: Representation Theory and the Jones Polynomial
Abstract
:   In this expository talk, I will introduce the Temperley-Lieb algebra, a remarkable quotient of the Hecke algebra for the symmetric group. Vaughn Jones exploited the natural multiplicative homomorphism from the braid group to the Temperley-Lieb algebra to construct the Jones polynomial. One may apply the same trick directly to the Hecke algebra to construct the HOMFLY polynomial, a finer invariant which subsumes both the Jones polynomial and the more classical Alexander polynomial. Conway and Kauffman showed how to recodify the underlying algebra diagrammatically to compute invariant polynomials via skein theory, leading eventually to diagrammatic encodings of representation categories for Lie algebras and quantum groups. This talk will not assume any knot theory background.

February 21, 2014
Thomas Nevins
Title: Hamiltonian reduction in representation theory and algebraic geometry
Abstract
:   Hamiltonian reduction arose as a mechanism for reducing complexity of systems in mechanics, but it also provides a tool for constructing complicated but interesting objects from simpler ones. I will illustrate how this works in representation theory and algebraic geometry via examples. I will explain a new structure theory, motivated by Hamiltonian reduction, for some categories (of D-modules) of interest to representation theorists, and, if there is time, indicate applications to the cohomology of (hyperkahler) manifolds. The talk will not assume that members of the audience know the meaning of any of the above-mentioned terms. The talk is based on joint work with K. McGerty.

February 28, 2014
Stephan Ramon Garcia
Title: Supercharacters on abelian groups
Abstract
:   The theory of supercharacters, which generalizes classical character theory, was recently developed in an axiomatic fashion by P. Diaconis and I.M. Isaacs, based upon earlier work of C. Andre. When this machinery is applied to abelian groups, a wide variety of applications emerge. In particular, we develop a broad generalization of the discrete Fourier transform along with several combinatorial tools. This perspective illuminates several classes of exponential sums (e.g., Gauss, Kloosterman, and Ramanujan sums) that are of interest in number theory. We also consider certain exponential sums that produce visually striking patterns of great complexity and subtlety. (Partially supported by NSF Grants DMS-1265973, DMS-1001614, and the Fletcher Jones Foundation)

March 31, 2014
Luis Lomeli
Title: On exterior and symmetric square \gamma-factors
Abstract
:   We make a study of an important case in the Langlands-Shahidi method, with emphasis on characteristic p. Namely, we consider smooth p-adic representations of GL(n) as representations of the Siegel Levi subgroup of a classical group. As a result, the Langlands-Shahidi local coefficient leads us to obtain exterior and symmetric square L-functions and root numbers. We explore several interconnections of the theory within the Langlands program. In particular, in joint work with G. Henniart, we establish compatibility with the local Langlands correspondence. Furthermore, in collaboration with R. Gunapathy, we explore the Deligne-Kazhdan transfer between close local fields of characteristic zero and characteristic p.

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