Robert Langlands FRS
|Alma mater||University of British Columbia,|
|Known for||Langlands program|
|Awards||Jeffery–Williams Prize (1980)|
Cole Prize (1982)
Wolf Prize (1995–96)
Steele Prize (2005)
Nemmers Prize (2006)
Shaw Prize (2007)
Abel Prize (2018)
|Institutions||Princeton University, |
Institute for Advanced Study
|Doctoral advisor||Cassius Ionescu-Tulcea|
|Doctoral students||James Arthur|
Thomas Callister Hales
Robert Phelan Langlands FRS FRSC (//; born October 6, 1936) is an American-Canadian mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study of Galois groups in number theory, for which he received the 2018 Abel Prize. He is an emeritus professor and occupies Albert Einstein's office at the Institute for Advanced Study in Princeton.
He enrolled at the University of British Columbia at the age of 16, receiving his undergraduate degree in 1957; he 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 first academic position was at Princeton University from 1960 to 1967, where he worked as an Associate Professor. He was a Miller Research Fellow at the University of California Berkeley from 1964 to 1965 and between 1967 to 1972 he was at Yale University. He was appointed Hermann Weyl Professor at the Institute for Advanced Study in 1972, and became Professor Emeritus in January 2007.
Langlands' Ph.D. thesis was on the analytical theory of Lie semigroups, 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 forms, 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 Hans Maass, Walter Roelcke, and Atle 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 the Weil conjecture on Tamagawa numbers 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 Martin Eichler and Goro 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.
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 proven, 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 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. In recent years he has turned his attention back to automorphic forms, working in particular on a theme he calls `beyond endoscopy'.
Langlands has received the 1996 Wolf Prize (which he shared with Andrew Wiles), the 2005 AMS Steele Prize, the 1980 Jeffery–Williams Prize, the 1988 NAS Award in Mathematics from the National Academy of Sciences, the 2006 Nemmers Prize in Mathematics, the 2007 Shaw Prize in Mathematical Sciences (with Richard Taylor) for his work on automorphic forms. In 2018 Langlands was awarded the Abel Prize for "his visionary program connecting representation theory to number theory.".
Langlands spent a year in Turkey in 1967–68, where his office at the Middle East Technical University was next to that of Cahit Arf. In addition to his mathematical studies, Langlands likes to learn foreign languages, both for better understanding of foreign publications on his topic and just as a hobby. He speaks French, Turkish, German and Russian.
Langlands is married to Charlotte Lorraine Cheverie. They have bred four children.
Robert Langlands, the mathematician who currently occupies Albert Einstein's office at the Institute for Advanced Study in Princeton
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