Langlands program

"Arithmetic automorphic representation theory is one of the most active areas in current mathematical research, centering around what is known as the Langlands program."

"The Langlands Program was initiated by Robert Langlands in the late 1960s in order to connect number theory and harmonic analysis [L1]. In the last 40 years a lot of progress has been made in proving the Langlands conjectures, but much more remains to be done. We still do not know the underlying reasons for the deep and mysterious connections suggested by these conjectures. But in the meantime, these ideas have propagated to other areas of mathematics, such as geometry and representation theory of infinite-dimensional Lie algebras, and even to quantum physics, bringing a host of new ideas and insights. There is hope that expanding the scope of the Langlands Program will eventually help us get the answers to the big questions about the Langlands duality. ... [L1] R. Langlands, Problems in the theory of automorphic forms, in Lect. Notes in Math. 170, pp. 18–61, Springer Verlag, 1970."

"At the heart of Langlands' program is the general notion of an "automorphic representation" π and its L-function L(s, π ) ... both defined via group theory and the theory of harmonic analysis on so-called adele groups ... The conjectures of Langlands ... amount (roughly) to the assertion that the other zeta-functions arising in number theory are but special realizations of these L(s, π ). Herein lies the agony as well as the ecstasy of Langlands' program. To merely state the conjectures correctly requires much of the machinery of class field theory, the structure theory of algebraic groups, the representation theory of real and p-adic groups, and (at least) the language of algebraic geometry. In other words, though the promised rewards are great, the initiation process is forbidding."

"The functoriality conjecture is at the heart of the Langlands program and will undoubtedly remain as a challenge to number theorists for many decades to come. Shortly after formulating his program, however, Langlands proposed to test it in two interdependent settings. The first was the framework of Shimura varieties, already understood by Shimura as a natural setting for a non-abelian generalization of the Shimura-Taniyama theory of complex multiplication. The second was the phenomenon of endoscopy, which can be seen alternatively as a classification of the obstacles to the stabilization of the trace formula or as an opportunity to prove the functoriality conjecture in some of the most interesting cases. After three decades of research, much of it by Langlands and his associates, these two closely related experiments are coming to a successful close, at least for classical groups, thanks in large part to the recent proof of the so-called Fundamental Lemma by Waldspurger, Laumon, and especially Ngô."

"... it was Langlands who, in 1966 at the age of 30, amalgamated and vastly generalized the essential ideas from his contemporaries as well as the recent past: • the ideas of Selberg, Harish-Chandra, and Gelfand which are rooted in Eisenstein series, harmonic analysis, and representation theory of certain classes of noncompact groups; • the usage of adelic structure on groups championed by Godement as well as Tamagawa and Satake; and • Artin’s legacy on class field theory and his quest for a nonabelian class field theory. Among what Langlands had created was a series of interlocking conjectures which later became the foundation of the Langlands program. Those conjectures seek to connect deep arithmetic questions with the highly structured theory of infinite-dimensional representations of Lie groups. The latter is part of harmonic analysis. Those visionary conjectures have exposed, quite unexpectedly, the deeply entwined nature of several seemingly unrelated branches of mathematics."