1999-2000

Schedule, Fall 1999

September 1:

Organizational Meeting

September 8:

Speaker: Rebecca Herb, UMCP

Title: Orbital integrals on p-adic Lie algebras

September 15:

Speaker: David Kazhdan, Harvard University

Title: Character sums associated to prehomogeneous vector spaces

September 22:

Speaker: Susana Salamanca-Riba, New Mexico State Univ. and UMCP

Title: Strictly small representations and a reduction theorem for the unitary dual, I

September 29:

Speaker: Susana Salamanca-Riba, New Mexico State Univ. and UMCP

Title: Strictly small representations and a reduction theorem for the unitary dual, II

October 6: no meeting

October 13: no meeting

October 20:

Speaker: Jeffrey Adams, UMCP

Title: An introduction to quantum groups, I

Note: This will be the first lecture in a series of "learning seminars" on quantum groups and their connection with representation theory. The series will start from scratch and is especially recommended for graduate students. We will start with the definition and basic properties of U_q(sl₂).

October 27:

Speaker: Freydoon Shahidi (Purdue)

Title: Another approach to L-functions with applications: A survey

October 27:

Extra Meeting at 3PM, joint with Algebra Seminar

Speaker: V. Ponomarev

Title: Graphs and Representations, I

November 3:

Speaker: Alain Valette (Neuchâtel) (joint with Algebra Seminar)

Title: Elementary construction of regular graphs with large girth

Abstract: The girth g(X) of a graph X is the length of its shortest circuit. If X is a finite, connected, k-regular graph, it is easy to see that g(X) <= 2 + 2 log_k-1|X|. If X_m is a family of such graphs, we say that this family has large girth if there exists C > 0 such that g(X_m) >= C log_k-1|X_m|. By non-constructive methods, Erdös and Sachs produced in 1963 families with large girth and C = 1. In 1987, Lubotzky-Phillips-Sarnak and independently Margulis used adelic groups to construct families of (p+1)-regular graphs (p an odd prime) with large girth and C = 4/3 (the constant C = 4/3 holds the world record). We give an elementary construction of these graphs and proof of the girth estimate, using Jacobi's four-squares theorem, the arithmetic of integral quaternions, and properties of the finite groups SL(2, p). This is joint work with Giuliana Davidoff and Peter Sarnak.

November 3:

Extra Meeting at 3PM, joint with Algebra Seminar

Speaker: V. Ponomarev

Title: Graphs and Representations, II

November 4:

Related Talk, not officially part of the Seminar

Speaker: Adam Koranyi (Lehman College, CUNY)

Title: Higher codimension CR structures on nilpotent Lie groups

November 10:

Speaker: Scott Wolpert, UMCP

Title: Semi-classical limits for the hyperbolic plane, I

November 17:

Speaker: Scott Wolpert, UMCP

Title: Semi-classical limits for the hyperbolic plane, II

November 24: no meeting

December 1:

Speaker: David Vogan (MIT)

Title: TBA

December 8:

Speaker: Jeffrey Adams, UMCP

Title: An introduction to quantum groups, II

Monday, December 13, 3 PM, room 3206:

Speaker: Peter Trapa (Inst. for Advanced Study)

Title: Unitary representation of reductive Lie groups

Abstract: The purpose of this talk is to highlight recent progress in three approaches to the classical (and still open) problem of describing all continuous, irreducible, length-preserving actions of a real Lie group on a Hilbert space. The first approach is external: one instead considers a larger, more tractable set of representations --- the admissible ones --- and classifies these with the hope that the unitary representations will be easy to recognize in an appropriate classification. The second approach is internal: one tries to build all unitary representations without recourse to nonunitary ones. This is the theory of geometric quantization and the orbit method. The final approach is pragmatic: for particular applications, there are unitary representations which are simply not needed, so one should really refine the original problem and concentrate on those unitary representations relevant for applications. I hope to conclude with some evidence for a miraculous conjecture describing all unitary representations relevant for arithmetic applications.

December 15, room 3206:

Speaker: Jiu-Kang Yu (Princeton Univ.)

Title: Supercuspidal representations and Hecke algebra isomorphisms

Abstract: I will give a general survey of the program of understanding representations of p-adic groups using the compact open subgroups, including the theory of types, minimal K-types, and Hecke algebra isomorphisms. In particular, I will introduce a general construction of supercuspidal representations via induction from compact open subgroups, which conjecturally exhausts all supercuspidal representations in the tame case.

Tuesday, December 21, 9-11 AM, room 1310:

Speaker: Jeffrey Adams (UMCP)

Title: Quantum Groups III

Abstract: I'm going to try to define and state the main properties of Kashiwara's crystal bases.

Wednesday, December 22, 2 PM, room 3206:

Speaker: Alexander Braverman (MIT)

Title: Gamma-functions of representations and Langlands lifting

Schedule, Spring 2000

Wednesday-Thursday, January 26-27. (No lecture Tuesday because of the snowstorm.)

Special Lecture Series

Speaker: David Kazhdan (Harvard)

Titles:

1. Special bases and Kashiwara's crystals (Wed. at 2:30)
2. Geometric and unipotent crystals (Thu. at 12:30)
3. Examples and appications (Thu. at 1:30)

Wednesday, February 23

Speaker: David Goldberg (Purdue)

Title: K-types for admissible representations of

    SL(n) over a p-adic field

Tuesday, February 29, 11 AM (MTH 3206 - note special day, room, and time)

Speaker: Kari Vilonen (Brandeis)

Title: Langlands dual and the affine Grassmannian

Wednesday, March 15

Speaker: David Renard (Poitiers, France)

Title: Irreducible genuine characters of the metaplectic

    group: KL-algorithm and Vogan's duality

Wednesday, April 12:

Speaker: Wulf Rossmann (Ottawa)

Title: Abelian systems and characters

Note: This is the second talk in a two-lecture series.

    The first talk on "action-angle variables on flag manifolds"

    is in the Geometry/Topology Seminar on Monday, April 10.

Wednesday, April 19,

Speaker: Ramin Takloo-Bighash (Johns Hopkins)

Title: Local L-factors for the symplectic group order 4

Thursday, April 20, 4 PM:

Speaker: Victor Kac (MIT)

Title: Infinite-dimensional algebras of symmetries and

    the Standard Model

Wednesday, April 26:Cancelled

Speaker: Nils Andersen

Title:

Wednesday, May 3:

Speaker: Jim Cogdell (Oklahoma State)

Title: On lifting from classical groups to GL(n)

Wednesday, May 10:

Speaker: Steve Devlin (UMCP)

Title: TBA

Schedule, Spring 2000