At present we are witnessing an important and natural expansion of
the scope of the classical Langlands program. This new mathematical
development makes use of the rich structure of congruences between
Fourier coefficients of modular forms and more generally of automorphic
representations, to tie together infinitely many otherwise disparate
automorphic representations into finite-dimensional parameter spaces.
By one count, there seems to be six independent essentially
simultaneous constructions currently underway, of parametrized
(p-adic) spaces of automorphic forms attached to algebraic groups
and their concomitant Galois representations. These parameter spaces
are called "eigenvarieties" or "Hecke varieties" and are being
constructed by different people in different but sometimes overlapping
contexts: for unitary groups of higher rank for symplectic groups
of high rank for general linear groups over number fields.
Eigenvarieties are a unifying force for classical and modern aspects
of number theory, algebraic geometry, analytic geometry (p-adic,
mainly) and the theory of group representations (both automorphic
representations and Galois representations). Some of this work has
already been used in important applications. The Eigenvarieties
program at Harvard University during the Spring semester 2006 is
intended to bring together many of the people working on these
constructions to provide intensive graduate courses on this material
and satellite seminars. 
The classical work of Ramanujan, that dealt with the arithmetic
properties of the Fourier coefficients of modular forms, unearthed
striking congruences that contain important number theoretic
information. These congruences suggest that a mysterious coherence
underlies a large assortment of basic arithmetic phenomena such as
the number of ways you can separate a collection of N objects into
subcollections or given a lattice in some Euclidean space the number
of lattice points closest to a given point or the number of solutions
of a system of polynomial equations modulo a prime number. An
extraordinary web of congruences acts as a virtual glue that binds
such problems together. In the intervening years the search for
congruences that have arithmetic applications that unify representation
theory, and the theory of modular forms has guided much number-theoretic
work. This search has been directly involved in many of the important
advances in number theory in the past few decades. For example it
played its role in the dramatic proof of modularity of elliptic
curves over the rational numbers, a few years ago. One is now on
the verge of a significant expansion of this enterprise. The hope
is that the Eigenvarieties program at Harvard University during the
Spring semester 2006 program will provide a milieu where further
progress can be made, where a coherent account of the current state
of knowledge will be established and where graduate students and
also post-docs and other interested mathematicians can gain mastery
of these new developments. Download this abstract in [PDF] form. |
