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.

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

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

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.

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.

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.

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 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.

Roughly speaking, the reciprocity conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from a Langlands group to an L-group.

The Langlands program seeks to attach an automorphic form or automorphic representation ( a suitable generalization of a modular form ) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field.

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.

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.

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.

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.

In fact, Murty showed that Artin L-functions corresponding to irreducible representations of the Galois group of a solvable extension of the rationals are automorphic as predicted by the Langlands conjectures.

One of the aims of proposed non-abelian class field theory is to incorporate the complex-analytic nature of Artin L-functions into a larger framework, such as is provided by automorphic forms and Langlands ' philosophy.

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

pointed out that the Artin conjecture follows from strong enough results from the Langlands philosophy, relating to the L-functions associated to automorphic representations for GL ( n ) for all.

More precisely, the Langlands conjectures associate an automorphic representation of the adelic group GL < sub > n </ sub >( A < sub > Q </ sub >) to every n-dimensional irreducible representation of the Galois group, which is a cuspidal representation if the Galois representation is irreducible, such that the Artin L-function of the Galois representation is the same as the automorphic L-function of the automorphic representation.

Sometimes Hilbert's statements were not precise enough to specify a particular problem but were suggestive enough so that certain problems of more contemporary origin seem to apply, e. g. most modern number theorists would probably see the 9th problem as referring to the ( conjectural ) Langlands correspondence on representations of the absolute Galois group of a number field.

For example, over the real numbers, this is the Langlands classification of representations of real reductive groups.

The so-called geometric Langlands program, suggested by Gérard Laumon following ideas of Vladimir Drinfel'd, arises from a geometric reformulation of the usual Langlands program that attempts to relate more than just irreducible representations.

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.

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.

Another significant related point is that the Langlands approach stands apart from the whole development triggered by monstrous moonshine ( connections between elliptic modular functions as Fourier series, and the group representations of the Monster group and other sporadic groups ).

They play the central role in the Langlands correspondence which studies finite dimensional representations of the Galois group of the field and which is one of noncommutative extensions of class field theory.

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.

In 1731 he began his medical studies as an apprentice of George Langlands, a fellow of the Incorporation of Surgeons which preceded the Royal College of Surgeons of Edinburgh.

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 insight of Langlands was to find the proper generalization of Dirichlet L-functions, which would allow the formulation of Artin's statement in this more general setting.

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.

There are several other nonabelian theories, local and global, which provide alternative to the Langlands correspondence point of view.

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.

Later he worked for the Langlands Foundry Company Limited in Yarra Bank, Melbourne, which made locomotive boilers, wheels and gold mining equipment.

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.

The Eastern Suburbs Tigers are a rugby league club based at Langlands Park, which is in the suburb of Coorparoo in Brisbane, Australia.

Langlands Park is named for Langlands which was the name of the residence that previous existed on the property ( owned by the local Nicklin family ).

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.

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.

There are numerous variations of this, in part because the definitions of Langlands group and L-group are not fixed.

The Langlands conjectures for GL ( 1, K ) follow from ( and are essentially equivalent to ) class field theory.

These members are: Cllr Michael Holmes ( Labour ), Cllr Iain Langlands ( Conservative ) and Cllr Carol Puthucheary ( SNP ).

The associated reductive Lie groups are of significant interest: the Langlands program is based on the premise that what is done for one reductive Lie group should be done for all.

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 & 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.

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.

Artworks by Langlands & Bell are in the permanent collections of many prominent international art museums including the British Museum, Imperial War Museum, Tate and the V & A in London, MoMA, New York, the Carnegie Museum of Art, Pittsburgh, and the Yale Center for British Art, USA, and the State Hermitage Museum, St Petersburg, Russia.

Langlands Park regularly hosts training sessions for the Queensland and Australian Rugby League teams when they are playing in Brisbane.

His parents, Anastasia ( voiced by Thomasin Langlands ) and Percy ( voiced by Lawrence ) are also alcoholics, on welfare, fat, unattractive and extraordinarily stupid.