Ngô Bảo Châu proved an auxiliary but difficult statement, the so-called " Fundamental Lemma ", originally conjectured by Langlands.

Pupils were originally re-located to Langlands until the new school on this site was completed in early 2008.

Roughly speaking, the Langlands reciprocity conjecture is the special case of the functoriality conjecture when one of the reductive groups is trivial.

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 )

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

Another field, known as the Langlands program, likewise starts with apparently haphazard similarities ( in this case, between number-theoretical results and representations of certain groups ) and looks for constructions from which both sets of results would be corollaries.

However, the Langlands correspondence does not include as much arithmetical information about finite Galois extensions as class field theory does in the abelian case.

Contemporary successors of the theory are the Arthur-Selberg trace formula applying to the case of general semisimple G, and the many studies of the trace formula in the Langlands philosophy ( dealing with technical issues such as endoscopy ).

Langlands used the base change lifting to prove the tetrahedral case, and Tunnell extended his work to cover the octahedral case ; Wiles used these cases in his proof of the Taniyama – Shimura conjecture.

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:

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.

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.

* Local Langlands conjectures

According to Rebecca Langlands, Cordus's story "... is a tale which vividly demonstrates the possibility that a text might be received in a way which the author had not intended or anticipated, and be received in a way which might have dire consequences for author and text.

All irreducible unitary representations are admissible ( or rather their Harish-Chandra modules are ), and the admissible representations are given by the Langlands classification, and it is easy to tell which of them have a non-trivial invariant sesquilinear form.

Many generalisations have been sought of Kronecker's ideas ; they do however lie somewhat obliquely to the main thrust of the Langlands philosophy, and there is no definitive statement currently known.

Many famous footballers, such as Bob Fulton, Graeme Langlands, Mick Cronin, Rod Wishart, Paul McGregor, Craig Fitzgibbon, Luke Bailey, Steve Roach, Garry Jack, Warren Ryan, and the Stewart brothers Brett and Glenn, have come from the Illawarra region.

Drinfeld has also collaborated with Alexander Beilinson to rebuild the theory of vertex algebras, which have become increasingly important to conformal field theory, string theory, and the geometric Langlands program.

Langlands & Bell have exhibited internationally throughout their career including in exhibitions at Tate Britain and Tate Modern, the Imperial War Museum, the Serpentine Gallery, and the Whitechapel Art Gallery in London, at IMMA, Dublin, Kunsthalle Bielefeld, Germany, MoMA, New York, the Central House of the Artist, Moscow, Venice Biennale, Seoul Biennale, and CCA Kitakyushu and TN Probe, Tokyo in Japan.

Vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric Langlands correspondence.

Busway stations have been built at Stones Corner and Langlands Park.

Catania have triumphed in Serie " A " seven times since Langlands joined in 2014.

Most children who lived in these areas would have attended Langlands Primary, St Joseph's Primary or Carbrain Primary, and later Cumbernauld High School, Greenfaulds High School or Our Lady's High School.

As of 2006, Cumbernauld has lost many of its older and smaller primary schools, such as Langlands, which have been combined into numerous larger primary schools.

To this day, ℓ-adic cohomology remains a fundamental tool for number theorists, with important applications to the Langlands program.

Furthermore, given such a group G, Langlands constructs the Langlands dual group < sup > L </ sup > G, and then, for every automorphic cuspidal representation of G and every finite-dimensional representation of < sup > L </ sup > G, he defines an L-function.

The Langlands conjecture for GL ( 2, Q ) still remains unproved.

The general Langlands philosophy includes a comparable explanation of the connection of polynomials of degree m, and the representation theory for GL < sub > m </ sub >.

He facilitated the now-celebrated visit of Robert Langlands to Turkey ( now famous for the Langlands program, among many other things ); during which Langlands worked out some arduous calculations on the epsilon factors of Artin L-functions.