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.

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

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

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.

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.

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.

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.

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

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

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.

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.

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.

* Most Points in Club History: 1, 554 ( 86 tries, 648 goals ) by Graeme Langlands from 1963 to 1976

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.

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 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.

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 ).

Located just south of East Kilbride, the reserve is accessed easiest from the A726, heading towards Langlands Golf Course & Auldhouse.

* Graeme Langlands ( c ) fullback from St. George

Barnes was born in Lochee, Dundee, the second of five sons of James Barnes, a skilled engineer and mill manager from Yorkshire, and his wife, Catherine Adam Langlands.

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.

St George's Captain-coach Graeme Langlands played with a misdirected pain killing injection that made his right leg go numb and prevented him from playing anywhere near his best.

* Most Goals in Club History: 648 by Graeme Langlands from 1963 to 1976

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.

* Edward Witten, Notes from the 2006 Bowen Lectures, an overview of Electric-Magnetic duality in gauge theory and its relation to the Langlands program

The pitch from the Langlands Park oval is currently the western-most pitch at the Gabba