Langlands's 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 \mathrm{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, 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's 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". ==Awards and honors==