Recent Progress in Langlands Functoriality Conference, June 17-28, 2002, Luminy, France

Purdue University, West Lafayette IN

Investigators

Abstract

To recognize and build upon recent advances within the Langlands program, a conference with the above title has been approved by CIRM for June 17-28, 2002. The investigators, who are among the organizers, feel that such a conference will serve to advance the careers of any young researchers who are in attendance. We are proposing that ten junior people receive funds to support their participation in these proceedings. The amount requested will be sufficient for travel as well as local expenses. In administering these funds we shall make every effort to support those who have no other means of funding. The focus of this conference is to survey recent results within a branch of mathematics referred to as the Langlands program. The underlying philosophy of this program is that there should be deep, and in some sense, fundamental connections between three seemingly disparate mathematical fields: Algebraic Geometry, Number Theory, and Harmonic Analysis. The Shimura-Taniyama conjecture, which posited the equivalence of the theory of elliptic curves (geometric objects) and a certain classes of automorphic forms (analytic objects with number theoretic applications) is just a ``simple'' (that is simple within the context of the program) example of what the Langlands program is expected to produce. By proving this conjecture, Andrew Wiles delivered a long sought proof of Fermat's Last theorem, that for integers larger than 2, there is never a choice of two non-zero integers whose n-th powers sum to the n-th power of another non-zero integer. More general conjectures within the Langlands program can be stated with some more technical language, but have a similarity to the Fermat problem in that the proofs are much harder than the statements. The last several years have witnessed a series of remarkable breakthroughs in the Langlands program. Wiles's work is one of many examples of these exciting results. As outlined in the proposal, this is critical point in the history of the program, in that new ideas must be found to build upon this recent momentum. Thus, providing funding for relatively new researchers to attend a high level conference, such as the one being sponsored by CIRM, will serve the field, and the greater mathematical community greatly. It is through such development that mathematics continues to play a critical role in our society, both culturally and technologically.