Lecture 1
"Progress in p-adic representation theory of p-adic groups."
Friday, May 28, 3 PM, MS 6627

ABSTRACT : Continuous representations of p-adic Lie groups with p-adic coefficients arise naturally in a variety of situations in number theory. Attempts to develop a systematic theory of such representations have been hampered by the impossibility of defining Haar measure with p-adic values. Nevertheless there has been tremendous progress in recent years: Schneider and Teitelbaum have defined a category of p-adic representations to which differential techniques can be applied, Emerton has applied their work to p-adic modular forms, and Breuil has provided evidence for a p-adic local Langlands correspondence. The goal of this talk is to justify the speaker's conviction that this is the branch of representation theory that will see the most striking developments over the next decade.

Lecture 2
"Congruences of endoscopic and stable forms and applications to p-adic L-functions."
Tuesday, June 1, 4 PM, MS 6627

ABSTRACT: This is a report on joint work with J.-S. Li and C. Skinner. Inspired by ideas of Hida, we construct p-adic L-functions of holomorphic automorphic forms on Shimura varieties attached to unitary groups and show, using a p-adic variant of the Rallis inner product formula, that in certain cases they divide the characteristic ideals of Selmer groups.

Colloquium:
Thursday, June 3, 4 PM, MS 6627

ABSTRACT : The Langlands Conjectures comprise a far-reaching set of assertions, many of them proved, relating the Galois theory of number fields, function fields in one variable over finite fields, and related local fields, to objects arising in representation theory. In this talk I will concentrate on what the Langlands Conjectures predict about the structure of the absolute Galois group of the field of rational numbers and survey some joint work in progress with R. Taylor and with J.-S. Li and C. Skinner.

Background:

Number Theorist Michael Harris is a global leader in the arithmetic theory of automorphic representations. Over the last 25 years, Harris has made basic contributions to areas as diverse as special values of L-functions, p-adic L-functions, period relations, automorphic vector bundles, and most fundamentally, the Local Langlands Conjecture for GL(N), which he unexpectedly proved, with R. Taylor, in 1998. The last result is a "creative tour de force" (Milne) which has inspired rapid ongoing developments in the theory of Shimura varieties. Through his remarkable ability to appreciate immediately the meaning of new developments in the field, and his frequent collaborations with other mathematicians, he has been, and continues to be, a pioneer and architect of the extension of the theory of modular forms to higher dimensional settings.

Harris received his Ph. D in 1977 from Harvard University. He taught at Brandeis University until he moved to Paris VII in 1994. Since 2001 he has been a member of the Institut Universitaire de France, and is currently visiting Harvard University.

See http://www.math.jussieu.fr/~harris/ for more information about Michael Harris.