The Langlands program is a far-reaching web of these ideas of ' unifying conjectures ' that link different subfields of mathematics, e. g. number theory and representation theory of Lie groups ; some of these conjectures have since been proved.

He has made outstanding contributions to Langlands ' program in the fields of number theory and analysis, and in particular proved the Langlands conjectures for the automorphism group of a function field.

The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles.

There are a number of related Langlands conjectures.

Some versions of the Langlands conjectures are vague, or depend on objects such as the Langlands groups, whose existence in unproven, or on the L-group that has several inequivalent definitions.

Moreover, the Langlands conjectures have evolved since Langlands first stated them in 1967.

There are different types of objects for which the Langlands conjectures can be stated:

Langlands did not originally consider this case, but his conjectures have analogues for it.

There are several different ways of stating Langlands conjectures, which are closely related but not obviously equivalent.

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.

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 )

proved the local Langlands conjectures for the general linear group GL ( 2, K ) over local fields.

proved the local Langlands conjectures for the general linear group GL ( n, K ) for positive characteristic local fields K. Their proof uses a global argument.

proved the local Langlands conjectures for the general linear group GL ( n, K ) for characteristic 0 local fields K. gave another proof.

The modularity theorem is a special case of more general conjectures due to Robert Langlands.