April 26, 2015

Yale University

  Groups, Geometry, and Physics

                                                            In honor of Howard Garland

Schedule

Department of Mathematics / 12 Hillhouse Avenue / New Haven, CT 06511

Phone: +1 (203) 432 4180 Fax: +1 (203) 432-7316


LUNCH

Opening Remarks

Yongchang Zhu (Hong Kong)

Siegel-Weil formula and theta correspondence for loop groups

Doug Pickrell (Arizona)

Loops in Compact Lie Groups and Factorization

In this talk we discuss `root subgroup factorization', which is a refinement of the Riemann-Hilbert (or Birkhoff or triangular) factorization for a map from a circle into a compact Lie group, especially SU(2). Relative to this factorization, a number of interesting objects factor: Toeplitz determinants, a homogeneous Poisson structure on the loop group, and conjecturally an analogue of Haar measure for the loop group.


Stephen Miller (Rutgers) & Manish Patnaik (Alberta)

  Automorphic Forms on Loop Groups: On some recent work of Howard Garland

The study of discrete subgroups of Lie groups has been a fertile topic in modern mathematics.  This is particularly so of automorphic forms, which play a unifying role in number theory through the Langlands program.  In the 1970's Howard Garland began building a theory of automorphic forms on infinite-dimensional groups, which have the tantalizing possibility of providing important constructions that their finite-dimensional analogs cannot.


Our talk will give an overview of Garland's work on loop group Eisenstein series and how they relate to a potential generalization of the Langlands-Shahidi method for analyzing automorphic L-functions on finite dimenional Lie groups.  

Gregg Zuckerman (Yale)

Discrete subgroups, cohomology, and Laplacians 

In the 1960's and early 1970's, Howard Garland made many contributions to the theory of discrete subgroups of the rational points of real and p-adic algebraic groups.  A recurring theme in his work was the discovery of deep vanishing theorems for the cohomology of a discrete subgroup with coefficients in a finite dimensional complex representation.  Howard's methods of proof relied on the analysis of certain Hodge-Laplacians on the symmetric space of a real group or on the Bruhat-Tits building of a p-adic group. 


In the mid-1970's, Howard's vanishing theorems gained the serious attention of representation theorists who were studying the continuous cohomology of topological groups with coefficients in infinite dimensional topological representations.  Casselman, Wigner, Borel, Wallach, Vogan and the speaker all owe a serious debt to Howard's pioneering work.  Fast forwarding to 2015, I will sketch a comparison between currently known vanishing theorems for discrete subgroups of real groups and of p-adic groups.  These theorems reflect the radical differences between the classification of irreducible continuous representations, and the associated classification of irreducible unitary representations, for real algebraic groups and p-adic algebraic groups. 


BANQUET

10 AM

10:15 AM

11:30 AM

2:30 PM

4 PM

5:30 PM

LUNCH

12:30 PM

Talks will held in Dunham Lab 220

(Entrance from 10 Hillhoue Avenue, on 2nd Floor)

COFFEE BREAK (4th floor, Dunham Labs)

3:30 PM