.jpg) | |
| Born | (1936-10-06)October 6, 1936 (age 89) New Westminster, British Columbia, Canada |
| Nationality | Canadian/American |
| Education | University of British Columbia (BSc,MSc) Yale University (PhD) |
| 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) Order of Canada (2019) |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Princeton University Middle East Technical University University of California, Berkeley Yale University Institute for Advanced Study |
| Thesis | Semi-Groups and Representations of Lie Groups (1960) |
| Doctoral advisor | Cassius Ionescu-Tulcea |
| Doctoral students | James Arthur Thomas Callister Hales Diana Shelstad |
Robert Phelan Langlands,CC FRS FRSC (/ˈlæŋləndz/; born October 6, 1936) is a Canadian mathematician.[1][2] He is best known as the founder of theLanglands program, a vast web of conjectures and results connectingrepresentation theory andautomorphic forms to the study ofGalois groups innumber theory,[3][4] for which he received the 2018Abel Prize. He isemeritus professor and occupiedAlbert Einstein's office at theInstitute for Advanced Study in Princeton, until 2020 when he retired.[5]
Langlands was born in New Westminster, British Columbia, Canada, in 1936 to Robert Langlands and Kathleen J Phelan. He has two younger sisters (Mary b. 1938; Sally b. 1941). In 1945, his family moved toWhite Rock, near the US border, where his parents had a building supply and construction business.[6][3][1]
He graduated fromSemiahmoo Secondary School and started enrolling at theUniversity of British Columbia at the age of 16, receiving his undergraduate degree in mathematics in 1957;[7] he continued at UBC to receive a M.Sc. in 1958. He then went toYale University, where he received a Ph.D. in 1960.[8]
His first academic position was atPrinceton University from 1960 to 1967, where he worked as an associate professor.[3] He spent a year in Turkey atMETU during 1967–68 in an office next toCahit Arf's.[9] He was a Miller Research Fellow at theUniversity of California, Berkeley, from 1964 to 1965, then was a professor atYale University from 1967 to 1972. He was appointed Hermann Weyl Professor at theInstitute for Advanced Study in 1972, and became professor emeritus in January 2007.[5]
Langlands' Ph.D. thesis was on the analytical theory ofLiesemigroups,[10] but he soon moved intorepresentation theory, adapting the methods ofHarish-Chandra to the theory ofautomorphic 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.[11][12]
He next constructed an analytical theory ofEisenstein series forreductive groups ofrank greater than one, thus extending work ofHans Maass, Walter Roelcke, andAtle Selberg from the early 1950s for rank one groups such as. This amounted to describing in general terms the continuousspectra 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 theWeil conjecture on Tamagawa numbers for the large class of arbitrary simply connectedChevalley 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.[13]
As a second application of this work, he was able to showmeromorphic continuation for a large class of-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 weakfunctional 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[14] making up what is often called theLanglands program. Very roughly speaking, they propose a huge generalization of previously known examples of reciprocity, including (a) classicalclass field theory, in which characters of local and arithmetic abelianGalois groups are identified with characters of localmultiplicative groups and the idele quotient group, respectively; (b) earlier results ofMartin Eichler andGoro Shimura in which theHasse–Weil zeta functions of arithmetic quotients of theupper half plane are identified with-functions occurring inHecke's theory ofholomorphic automorphic forms. These conjectures were first posed in relatively complete form in a famous letter to Weil,[14] written in January 1967. It was in this letter that he introduced what has since become known as the-group and along with it, the notion of functoriality.
The book byHervé Jacquet and Langlands on presented a theory of automorphic forms for thegeneral linear group, establishing among other things theJacquet–Langlands correspondence showing that functoriality was capable of explaining very precisely how automorphic forms for related to those forquaternion algebras. This book applied the adelictrace formula for 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 theHasse–Weil zeta functions of certainShimura varieties are among the-functions arising from automorphic forms.[15]
The functoriality conjecture is far from proven, but a special case (the octahedralArtin conjecture, proved by Langlands[16] and Tunnell[17]) was the starting point ofAndrew Wiles' attack on theTaniyama–Shimura conjecture andFermat's Last Theorem.
In the mid-1980s Langlands turned his attention[18] tophysics, particularly the problems ofpercolation and conformal invariance. In 1995, Langlands started a collaboration withBill Casselman at theUniversity 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-group. In recent years he has turned his attention back to automorphic forms, working in particular on a theme he calls "beyondendoscopy".[19]
Langlands has received the 1996Wolf Prize (which he shared withAndrew Wiles),[20] the 2005 AMSSteele Prize, the 1980Jeffery–Williams Prize, the 1988NAS Award in Mathematics from theNational Academy of Sciences,[21] the 2000 grande médaille de l'Académie des sciences de Paris, the 2006Nemmers Prize in Mathematics, the 2007Shaw Prize in Mathematical Sciences (withRichard Taylor) for his work on automorphic forms. In 2018, Langlands was awarded theAbel Prize for "his visionary program connecting representation theory to number theory".[22]
He was elected aFellow of the Royal Society of Canada in 1972 and aFellow of the Royal Society in 1981.[23][24] In 2012, he became a fellow of theAmerican Mathematical Society.[25] Langlands was elected as a member of theAmerican Academy of Arts and Sciences in 1990.[26] He was elected as a member of theNational Academy of Sciences in 1993[27] and a member of theAmerican Philosophical Society 2004.[28]
Among other honorary degrees, in 2003, Langlands received a doctoratehonoris causa fromUniversité Laval.[29]
In 2019, Langlands was appointed a Companion of theOrder of Canada.[30][31]
On January 10, 2020, Langlands was honoured atSemiahmoo Secondary, which installed a mural to celebrate his contributions to mathematics.
Langlands has been married to Charlotte Lorraine Cheverie (b 1935) since 1957. They have four children (2 daughters and 2 sons).[3] He holds Canadian and American citizenships.
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 English, French, Turkish and German, and reads (but does not speak) Russian.[32]
Robert Langlands, the mathematician who currently occupies Albert Einstein's office at the Institute for Advanced Study in Princeton
