Robert Phelan Langlands (born October 6, 1936, in New Westminster, British Columbia, Canada) is an emeritus professor at the Institute for Advanced Study. His work in automorphic forms and representation theory had a major effect on number theory.
Langlands received an undergraduate degree from the University of British Columbia in 1957, and continued on there to receive an M. Sc. in 1958. He then went to Yale University where he received a Ph.D. in 1960. His academic positions since then include the years 1960-67 at Princeton University, ending up as Associate Professor, and the years 1967-72 at Yale University. He was appointed Herman Weyl Professor at the Institute for Advanced Study in 1972, becoming Professor Emeritus in January 2007.
His Ph.D. thesis was on the analytical theory of semi-groups, but he soon moved into representation theory, adapting the methods of Harish-Chandra to the theory of automorphic forms. His first accomplishment in this field was a formula for the dimension of certain spaces of automorphic form, in which particular types of Harish-Chandra's discrete series appeared.
He next constructed an analytical theory of Eisenstein series for reductive groups of rank greater than one, thus extending work of Maass, Roelcke and Selberg from the early 1950s for rank one groups such as SL(2). This amounted to describing in general terms the continuous spectra of arithmetic quotients, and showing that all automorphic forms arise in terms of cusp forms and the residues of Eisenstein series induced from cusp forms on smaller subgroups. As a first application, he proved André Weil's conjecture about Tamagawa number for the large class of arbitrary simply connected Chevalley groups defined over the rational numbers. Previously this had been known only in a few isolated cases and for certain classical groups where it could be shown by induction.
As a second application of this work, he was able to show meromorphic continuation for a large class of L-functions arising in the theory of automorphic forms, not previously known to have them. These occurred in the constant terms of Eisenstein series, and meromorphicity as well as a weak functional equation were a consequence of functional equations for Eisenstein series. This work led in turn, in the winter of 1966/67, to the now well known conjectures making up what is often called the Langlands program. Very roughly speaking, they propose a huge generalization of previously known examples of reciprocity, including (a) classical class field theory, in which characters of local and arithmetic abelian Galois groups are identified with characters of local multiplicative groups and the idele quotient group, respectively; (b) earlier results of Eichler and Shimura in which the Hasse-Weil zeta functions of arithmetic quotients of the upper half plane are identified with L-functions occurring in Hecke's theory of holomorphic automorphic forms. These conjectures were first posed in relatively complete form in a famous letter to Weil, written in January 1967. It was in this letter that he introduced what has since become known as the L-group and along with it, the notion of functoriality.
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