# Robert Langlands

**Robert Langlands** (born 1936 in Canada) is one of the most significant mathematicians of the 20th century, with
profound insights in number theory and representation theory.

Langlands received his PhD from Yale University in 1960. During the early 1960s he developed the general theory of Eisenstein series for discrete groups, initiated by Atle Selberg; this, roughly speaking, is the continuous spectrum theory of automorphic forms, for arithmetic groups in semisimple groups. Though his work was strong, he was not offered tenure at Princeton University. He spent a year in Turkey, working in isolation, during which time he had profound insights. His subsequent work shook mathematics (in a famous anecdote, André Weil complained that a conversation with Langlands had induced a headache). He has been a permanent member of the Institute for Advanced Study since the early 1970s.

Langlands is the author of the Langlands program, a deep web of conjectures connecting number theory and representation theory.

Langlands understood that the theory of automorphic representation offers a generalization of class field theory, a central topic in algebraic number theory. Thus, in crude terms, to every representation of a Galois group there should be associated an automorphic form. Taken to its logical and organisational conclusion, this leads to his famous functoriality conjecture, which altered the understanding of key issues in number theory.

To give evidence for this idea, by working out special cases, Hervé Jacquet and Langlands developed an idea of the Russian mathematicians, that representation theory is the setting for the theory of automorphic forms. Using every tool at their disposal, they gave a surprisingly complete theory of automorphic forms on the general linear group GL(2), establishing important cases of functoriality.

Subsequently Langlands and James Arthur developed the Selberg trace formula as a method of attacking functoriality in general.

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 the proof of Fermat's last theorem.

Since the mid-1980s he has turned his attention to physics, where his contributions have been less influential than his earlier work.

In 1996, Langlands received the Wolf Prize for his work on the Langlands program.