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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.

Career

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 moved soon 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.

Functoriality, the L-group, the rigorous introduction of adele groups, and the consequent application of the representation theory of reductive groups over local fields changed drastically the way research in automorphic forms was carried out. Langlands' introduction of (or in cases where others had done previous work, emphasis on) these notions broke up large and to some extent intractable problems into smaller and more manageable pieces. For example, they made the infinite-dimensional representation theory of reductive groups into a major field of mathematical activity.

Functoriality is the conjecture that automorphic forms on different groups should be related in terms of their L-groups. As one example of this conjecture the letter to Weil raised the possibility of solving the well known conjecture of Emil Artin regarding the behaviour of Artin's L-functions, a hope partly realized in Langlands' later work on base change. In its application to Artin's conjecture, functoriality associated to every N-dimensional representation of a Galois group an automorphic representation of the adelic group of GL(N). In the theory of Shimura varieties it associates automorphic representations of other groups to certain l-adic Galois representations as well.

The book by Hervé Jacquet and Langlands on GL(2) presented a theory of automorphic forms for the general linear group GL(2), establishing among other things the Jacquet–Langlands correspondence showing that functoriality was capable of explaining very precisely how automorphic forms for GL(2) related to those for quaternion algebras. This book applied the adelic trace formula for GL(2) and quaternion algebras to do this. Subsequently James Arthur, a student of Langlands while he was at Yale, successfully developed the trace formula for groups of higher rank. This has become a major tool in attacking functoriality in general, and in particular has been applied to demonstrating that the Hasse-Weil zeta functions of certain Shimura varieties are among the L-functions arising from automorphic forms.

The functoriality conjecture is far from proved, but a special case (the octahedral Artin conjecture, proved by Langlands and Tunnell) was the starting point of Andrew Wiles' attack on the Taniyama-Shimura conjecture and Fermat's last theorem.

In the mid-1980s Langlands turned his attention to physics, particularly the problems of percolation and conformal invariance.

In recent years he has turned his attention back to automorphic forms, working in particular on a theme he calls `beyond endoscopy'.

In 1995 Langlands started a collaboration with Bill Casselman at the University of British Columbia with the aim of posting nearly all of his writings - including publications, preprints, as well as selected correspondence - on the Internet. The correspondence includes a copy of the original letter to Weil that introduced the L-group.

Langlands has received the 1996 Wolf Prize (which he shared with Andrew Wiles),[1] the 2005 AMS Steele Prize, the 1980 Jeffery-Williams Prize, the 2006 Nemmers Prize in Mathematics, and the 2007 Shaw Prize in Mathematical Sciences (with Richard Taylor) for his work on automorphic forms.

he was elected a Fellow of the Royal Society of London in 1981.

Publications

* Euler products, New Haven: Yale University Press, 1967, ISBN 0300013957
* On the Functional Equations Satisfied by Eisenstein Series, Berlin: Springer, 1976, ISBN 354007872X
* Base Change for GL(2), Princeton: Princeton University Press, 1980, ISBN 0691082723

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Robert Langlands