Current Status
The Langlands conjectures for GL(1, K) follow from (and are essentially equivalent to) class field theory.
Langlands proved the Langlands conjectures for groups over the archimedean local fields R and C by giving the Langlands classification of their irreducible representations.
Lusztig's classification of the irreducible representations of groups of Lie type over finite fields can be considered an analogue of the Langlands conjectures for finite fields.
Andrew Wiles' proof of modularity of semi-stable elliptic curves over rationals can be viewed as an exercise in the Langlands conjectures. Unfortunately, his method cannot be extended to arbitrary number fields.
The Langlands conjecture for GL(2, Q) still remains unproved.
Laurent Lafforgue proved Lafforgue's theorem verifying the Langlands conjectures for the general linear group GL(n, K) for function fields K. This work continued earlier investigations by Vladimir Drinfel'd, who proved the case GL(2, K)
Read more about this topic: Langlands Program
Famous quotes containing the words current and/or status:
“The work of the political activist inevitably involves a certain tension between the requirement that positions be taken on current issues as they arise and the desire that one’s contributions will somehow survive the ravages of time.”
—Angela Davis (b. 1944)
“As a work of art it has the same status as a long conversation between two not very bright drunks.”
—Clive James (b. 1939)