From a recent attempt to vulgarize the Langlands program:
| “Galois’ work is a great example of the power of a mathematical insight….
And then, 150 years later, Langlands took these ideas much farther. In 1967, he came up with revolutionary insights tying together the theory of Galois groups and another area of mathematics called harmonic analysis. These two areas, which seem light years apart, turned out to be closely related.” — Frenkel, Edward (2013-10-01). (Links to related Wikipedia articles have been added.) |
| Wikipedia on the Langlands program
The starting point of the program may be seen as Emil Artin’s reciprocity law [1924-1930], which generalizes quadratic reciprocity. The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions to the one-dimensional representations of this Galois group; and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters. The precise correspondence between these different kinds of L-functions constitutes Artin’s reciprocity law. For non-abelian Galois groups and higher-dimensional representations of them, one can still define L-functions in a natural way: Artin L-functions. The insight of Langlands was to find the proper generalization of Dirichlet L-functions, which would allow the formulation of Artin’s statement in this more general setting. |
| From “An Elementary Introduction to the Langlands Program,” by Stephen Gelbart (Bulletin of the American Mathematical Society, New Series , Vol. 10, No. 2, April 1984, pp. 177-219)
On page 194: “The use of group representations in systematizing and resolving diverse mathematical problems is certainly not new, and the subject has been ably surveyed in several recent articles, notably [ Gross and Mackey ]. The reader is strongly urged to consult these articles, especially for their reformulation of harmonic analysis as a chapter in the theory of group representations. In harmonic analysis, as well as in the theory of automorphic forms, the fundamental example of a (unitary) representation is the so-called ‘right regular’ representation of G…. Our interest here is in the role representation theory has played in the theory of automorphic forms.* We focus on two separate developments, both of which are eventually synthesized in the Langlands program, and both of which derive from the original contributions of Hecke already described.” [ Gross ] K. I. Gross, On the evolution of non-commutative harmonic analysis . Amer. Math. Monthly 85 (1978), 525-548. [ Mackey ] G. Mackey, Harmonic analysis as the exploitation of symmetry—a historical survey . Bull. Amer. Math. Soc. (N.S.) 3 (1980), 543-698. * A link to a related Math Overflow article has been added. |
In 2011, Frenkel published a commentary in the A.M.S. Bulletin
on Gelbart’s Langlands article. The commentary, written for
a mathematically sophisticated audience, lacks the bold
(and misleading) “light years apart” rhetoric from his new book
quoted above.
In the year the Gelbart article was published, Frenkel was
a senior in high school. The year was 1984.
For some remarks of my own that mention
that year, see a search for 1984 in this journal.