FOM: Langlands Program

[Joe Shipman]

The following is taken from Ivars Peterson's "MathTrek" column of
11/22/99, available at www.maa.org :

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This fall, Brian Conrad and Richard Taylor of Harvard University, along
with Christophe Breuil of the Université Paris-Sud and Fred Diamond of
Rutgers University in New Brunswick, N.J., completed a proof of the
Taniyama-Shimura conjecture for all elliptic curves.

"The work was collaborative in nature," Conrad says. "Although we. .
.worked on different parts of the argument, there really was nontrivial
overlap among these parts, with questions and problems in one area
leading to questions and problems in other areas."

The conjecture "was widely believed to be unbreachable, until the summer
of 1993, when Wiles announced a proof that every semistable elliptic
curve is modular," Henri Darmon of McGill University in Montreal remarks
in the December Notices of the American Mathematical Society. "The
Shimura-Taniyama-Weil conjecture and its subsequent, just-completed
proof stand as a crowning achievement of number theory in the 20th
century."

Moreover, the Taniyama-Shimura conjecture fits into the so-called
Langlands program, formulated by Robert P. Langlands of the Institute
for Advanced Study in Princeton, N.J. This vast, visionary program
posits a bold, sweeping unification of important areas of mathematics.
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A "vast, visionary...bold, sweeping unification of important areas of
mathematics" sounds like something of foundational interest.  Can any
number theorists out there provide a concise description of the
Langlands Conjectures and the Langlands program?  (Is the "Langlands
Program" simply the effort to prove the Langlands Conjectures, or is
there more to it?)

-- Joe Shipman