Robert Langlands

Robert Phelan Langlands (born October 6, 1936 in New Westminster, British Columbia, Canada) was one of the most influential
mathematicians of the 20th century,and remains influential in the 21st.His work in automorphic forms and representation theoryhad a major effect on number theory. He has alsodone original work in mathematically rigorous statistical physicswhere his work has had less effect.

Langlands received an undergraduate degreefrom the University of British Columbia in 1957,and continued on there to receive an M. Sc. there in 1958.He then went to Yale University where he receiveda Ph.D. in 1960. His academic positions since then includethe 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, becomingProfessor Emeritus in January 2007.

His Ph.D. thesis was onthe analytical theory of semi-groups, but he moved soon intorepresentation theory, adapting the methods of Harish-Chandrato the theory of automorphic forms. His first accomplishment in this field was a formula for the dimensionof certain spaces of automorphic form, in which particular types of Harish-Chandra's discrete series appeared.

He next constructed an analytical theory of Eisenstein seriesfor reductive groupsof 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 generalterms the continuous spectra of arithmetic quotients, and showingthat all automorphic forms arise in termsof cusp forms and the residues of Eisenstein seriesinduced from cusp forms on smaller subgroups.As a first application, he proved
André Weil's conjecture about Tamagawanumber 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 casesand for certain classical groups where it could be shown by induction.

As a second application of his work on Eisenstein series,he was able to show meromorphiccontinuation for a large class of $L$-functions arising in the theory of automorphic forms,not previously known to have them. These occurred inthe constant terms of Eisenstein series,and meromorphicity as wellas a weak functional equationwere a consequence of functionalequations for Eisenstein series. This work led in turn,in the winter of 1966/67, to the now wellknown conjectures making upwhat is often called the Langlands program.Very roughly speaking, they propose a huge generalization ofpreviously known examples of reciprocity, including (a) classical class field theory, in which characters of local and arithmetic abelian Galois groupsare identified with characters of local multiplicative groups and the idele quotient group,respectively; (b) earlier results of Eichler and Shimura in whichthe Hasse-Weil zeta functions of arithmetic quotients of the upper half plane are identified with $L$-functions occurringin Hecke's theory of holomorphic automorphic forms. These conjectures were first posed in relatively completeform in a famous letter to Weil, written in January 1967.It was in this letter thathe introduced what has since become known as the $L$-groupand along with it, the notion of functoriality.

Functoriality, the $L$-group,the rigorous introduction of adele groups, and the consequentapplication of the representation theory of reductivegroups over local fieldschanged drastically the way research in automorphic formswas carried out.In effect, Langlands' introduction of(or in cases where others had done previous work, emphasis on) these notionsbroke up large and to some extent intractableproblems into smaller and more manageable pieces.Among other things, they made the infinite-dimensional representation theory of reductive groupsinto a major field of mathematical activity.

Functoriality is the conjecturethat automorphic forms on different groups should be relatedin terms of their $L$-groups. As one exampleof this conjecture the letter to Weil raised the possibility of solvingthe well known conjecture of Emil Artin regardingthe 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 $N$-dimensionalrepresentation of a Galois group an automorphic representation ofthe 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 surprisingly complete theory of automorphic forms for the general linear group $GL(2)$, establishing among other things that functoriality was capable ofexplaining very precisely how automorphic forms for $GL(2)$related to those for quaternion algebras. This bookapplied 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, spent much work in successfullydeveloping the trace formula for groupsof higher rank. This has become a major toolin attacking functoriality in general,and in particular has beenapplied to demonstrating that the Hasse-Weil zeta functionsof 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 BillCasselman at the University of British Columbiawith the aim of posting nearly all of his writings - includingpublications, preprints, as well as selected correspondence - on the Internet.The correspondence includes a copy ofthe original letter to Weilthat introduced the $L$-group.

Langlands has received the 1996 Wolf Prize(which he shared with Andrew Wiles), 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.

External links

* [http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/intro.html The work of Robert Langlands (a nearly complete archive)]