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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.
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Langlands and program
To this day, ℓ-adic cohomology remains a fundamental tool for number theorists, with important applications to the Langlands program.
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.
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.
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.
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.
* Langlands program
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.
# REDIRECT Langlands program
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.
Taylor received the 2007 Shaw Prize in Mathematical Sciences for his work on the Langlands program with Robert Langlands.
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.
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 program
Langlands and is
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.
For example, over the real numbers, this is the Langlands classification of representations of real reductive groups.
Roughly speaking, the Langlands reciprocity conjecture is the special case of the functoriality conjecture when one of the reductive groups is trivial.
The modularity theorem is a special case of more general conjectures due to Robert Langlands.
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 ).
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.
Often, the Langlands correspondence is viewed as a nonabelian class field theory and indeed when fully established it will contain a very rich theory of nonabelian Galois extensions of global fields.
It also does not include an analog of the existence theorem in class field theory, i. e. the concept of class fields is absent in the Langlands correspondence.
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.
The reserve is owned by South Lanarkshire Council and maintained by The Friends of Langlands Moss L. N. R.
Located just south of East Kilbride, the reserve is accessed easiest from the A726, heading towards Langlands Golf Course & Auldhouse.
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.
The Chief Executive of HEFCE is Sir Alan Langlands ( since 1 April 2009 ), previously Vice-Chancellor of the University of Dundee and former chief executive of the NHS.
* There is a Deligne – Langlands conjecture of historical importance in relation with the development of the Langlands philosophy.
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.
It is this concept that is basic to the formulation of the Langlands philosophy.
Langlands and these
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.
" 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.
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.
Langlands and conjectures
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.
* Local Langlands conjectures
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