Langlands generalized the idea of functoriality: instead of using the general linear group GL ( n ), other connected reductive groups can be used.

The Langlands decomposition is a slight refinement of the Levi decomposition for parabolic subgroups used in this context.

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

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 ).

What initially was very new in Langlands ' work, besides technical depth, was the proposed direct connection to number theory, together with the rich organisational structure hypothesised ( so-called functoriality ).

* The work of Robert Langlands

" As Langlands seems to suggest, Cordus was thus a man deeply misunderstood as a writer intending to vilify the royal family of the time, by his seemingly seditious work.

This development preceded the Langlands program by a few years, and can be regarded as complementary to it: Langlands ' work relates largely to Artin L-functions, which, like Hecke's L-functions, were defined several decades earlier, and to L-functions attached to general automorphic representations.

Taylor received the 2007 Shaw Prize in Mathematical Sciences for his work on the Langlands program with Robert Langlands.

In subsequent work, Taylor ( along with Michael Harris ) proved the local Langlands conjectures for GL ( n ) over a number field.

Drinfeld's work connected algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the geometric Langlands correspondence.

Langlands & Bell, are two fine artists who work collaboratively as a duo ; the two, Ben Langlands ( born London 1955 ) and Nikki Bell ( born London 1959 ), began collaborating in 1978, while studying Fine Art at Middlesex Polytechnic in North London, from 1977 to 1980.

In 2004 Langlands & Bell were also short-listed for the Turner Prize for the same work.

This was one of the major motivations for Langlands ' work.

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.

In fact the Langlands program ( or philosophy ) is much more like a web of unifying conjectures ; it really does postulate that the general theory of automorphic forms is regulated by the L-groups introduced by Robert Langlands.

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.

Robert Langlands showed how ( in generality, many particular cases being known ) the Riemann-Roch theorem could be applied to the calculation of dimensions of automorphic forms ; this is a kind of post hoc check on the validity of the notion.

The largest artworks to date by Langlands & Bell are, the 2004 Paddington Basin Bridge, designed in association with Atelier One ( structural engineers ), an 8 metre high x 45 metre long white metal and glass pedestrian bridge linking Paddington station and the new Paddington Basin Development, London, with a capacity of up to 20, 000 people per day ; Moving World ( Night & Day ) 2007, two 6 x 18 metre permanent outdoor sculptures of steel, glass, and digitally controlled neon at London Heathrow, Terminal 5 ; and China, Language of Places 2009, the 18 metre wall painting exhibited in English Lounge at Tang Contemporary Art, 798, Beijing in 2009.

Firstly Langlands and Deligne established a factorisation into Langlands – Deligne local constants ; this is significant in relation to conjectural relationships to automorphic representations.

Andrew Wiles ' proof of modularity of semi-stable elliptic curves over rationals can be viewed as an exercise in the Langlands conjectures.

Langlands then generalized these to automorphic cuspidal representations, which are certain infinite dimensional irreducible representations of the general linear group GL ( n ) over the adele ring of Q.

Langlands attached automorphic L-functions to these automorphic representations, and conjectured that every Artin L-function arising from a finite-dimensional representation of the Galois group of a number field is equal to one arising from an automorphic cuspidal representation.

" Automorphicity " of these modules and the Langlands correspondence could be then understood in terms of the action of Hecke operators.

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.

Ramanujan bounds for groups other than can be obtained as an application of known cases of Langlands functoriality.

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.

In 1974, at the age of twenty, Drinfeld announced a proof of the Langlands conjectures for GL < sub > 2 </ sub > over a global field of positive characteristic.

* 1941 – Graeme Langlands, Australian rugby player

* 1941 – Graeme Langlands, Australian rugby player

* Gérard Laumon La correspondance de Langlands sur les corps de fonctions ( d ' après Laurent Lafforgue ), Séminaire Bourbaki, 52e année, 1999 – 2000, no.

* Langlands – Deligne local constant

* There is a Deligne – Langlands conjecture of historical importance in relation with the development of the Langlands philosophy.

Langlands had problems with a groin injury and partially because he battled-on relying strongly on pain killers, St George were defeated 38 – 0.

* Graeme Langlands ( 1963 – 1976 )

* Graeme Langlands ( while playing ) 1972 – 76

In 1996-1997, a major survey exhibition Langlands & Bell Works 1986 – 1996 co-curated by the Serpentine Gallery, London, and Kunsthalle Bielefeld, Germany also toured to Cantieri Culturali alla Zisa, Palermo, Sicily, and Koldo Mitxelena, San Sebastián, Spain.

Robert Langlands argued in 1973 that the modern version of the should deal with Hasse – Weil zeta functions of Shimura varieties.