Langlands program
In mathematics, the Langlands program is a web of farreaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands (1967, 1970), it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as “a kind of grand unified theory of mathematics.”^{[1]}
Contents
Background
In a very broad context, the program built on existing ideas: the philosophy of cusp forms formulated a few years earlier by HarishChandra and Gelfand (1963), the work and approach of HarishChandra on semisimple Lie groups, and in technical terms the trace formula of Selberg and others.
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 (socalled functoriality).
For example, in the work of HarishChandra one finds the principle that what can be done for one semisimple (or reductive) Lie group, should be done for all. Therefore, once the role of some lowdimensional Lie groups such as GL(2) in the theory of modular forms had been recognised, and with hindsight GL(1) in class field theory, the way was open at least to speculation about GL(n) for general n > 2.
The cusp form idea came out of the cusps on modular curves but also had a meaning visible in spectral theory as 'discrete spectrum', contrasted with the 'continuous spectrum' from Eisenstein series. It becomes much more technical for bigger Lie groups, because the parabolic subgroups are more numerous.
In all these approaches there was no shortage of technical methods, often inductive in nature and based on Levi decompositions amongst other matters, but the field was and is very demanding.^{[2]}
And on the side of modular forms, there were examples such as Hilbert modular forms, Siegel modular forms, and thetaseries.
Objects
There are a number of related Langlands conjectures. There are many different groups over many different fields for which they can be stated, and for each field there are several different versions of the conjectures.^{[3]} Some versions^{[which?]} of the Langlands conjectures are vague, or depend on objects such as the Langlands groups, whose existence is unproven, or on the Lgroup 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:
 Representations of reductive groups over local fields (with different subcases corresponding to archimedean local fields, padic local fields, and completions of function fields)
 Automorphic forms on reductive groups over global fields (with subcases corresponding to number fields or function fields).
 Finite fields. Langlands did not originally consider this case, but his conjectures have analogues for it.
 More general fields, such as function fields over the complex numbers.
Conjectures
There are several different ways of stating the Langlands conjectures, which are closely related but not obviously equivalent.
Reciprocity
The starting point of the program may be seen as Emil Artin's reciprocity law, which generalizes quadratic reciprocity. The Artin reciprocity law applies to a Galois extension of an algebraic number field whose Galois group is abelian; it assigns Lfunctions to the onedimensional representations of this Galois group, and states that these Lfunctions are identical to certain Dirichlet Lseries or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters. The precise correspondence between these different kinds of Lfunctions constitutes Artin's reciprocity law.
For nonabelian Galois groups and higherdimensional representations of them, one can still define Lfunctions in a natural way: Artin Lfunctions.
The insight of Langlands was to find the proper generalization of Dirichlet Lfunctions, which would allow the formulation of Artin's statement in this more general setting. Hecke had earlier related Dirichlet Lfunctions with automorphic forms (holomorphic functions on the upper half plane of C that satisfy certain functional equations). 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. (This ring simultaneously keeps track of all the completions of Q, see padic numbers.)
Langlands attached automorphic Lfunctions to these automorphic representations, and conjectured that every Artin Lfunction arising from a finitedimensional representation of the Galois group of a number field is equal to one arising from an automorphic cuspidal representation. This is known as his "reciprocity conjecture".
Roughly speaking, the reciprocity conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from a Langlands group to an Lgroup. There are numerous variations of this, in part because the definitions of Langlands group and Lgroup are not fixed.
Over local fields this is expected to give a parameterization of Lpackets of admissible irreducible representations of a reductive group over the local field. For example, over the real numbers, this correspondence is the Langlands classification of representations of real reductive groups. Over global fields, it should give a parameterization of automorphic forms.
Functoriality
The functoriality conjecture states that a suitable homomorphism of Lgroups is expected to give a correspondence between automorphic forms (in the global case) or representations (in the local case). Roughly speaking, the Langlands reciprocity conjecture is the special case of the functoriality conjecture when one of the reductive groups is trivial.
Generalized functoriality
Langlands generalized the idea of functoriality: instead of using the general linear group GL(n), other connected reductive groups can be used. Furthermore, given such a group G, Langlands constructs the Langlands dual group ^{L}G, and then, for every automorphic cuspidal representation of G and every finitedimensional representation of ^{L}G, he defines an Lfunction. One of his conjectures states that these Lfunctions satisfy a certain functional equation generalizing those of other known Lfunctions.
He then goes on to formulate a very general "Functoriality Principle". Given two reductive groups and a (well behaved) morphism between their corresponding Lgroups, this conjecture relates their automorphic representations in a way that is compatible with their Lfunctions. This functoriality conjecture implies all the other conjectures presented so far. It is of the nature of an induced representation construction—what in the more traditional theory of automorphic forms had been called a 'lifting', known in special cases, and so is covariant (whereas a restricted representation is contravariant). Attempts to specify a direct construction have only produced some conditional results.
All these conjectures can be formulated for more general fields in place of Q: algebraic number fields (the original and most important case), local fields, and function fields (finite extensions of F_{p}(t) where p is a prime and F_{p}(t) is the field of rational functions over the finite field with p elements).
Geometric conjectures
The socalled geometric Langlands program, suggested by Gérard Laumon following ideas of Vladimir Drinfeld, arises from a geometric reformulation of the usual Langlands program that attempts to relate more than just irreducible representations. In simple cases, it relates ladic representations of the étale fundamental group of an algebraic curve to objects of the derived category of ladic sheaves on the moduli stack of vector bundles over the curve.
Current status
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 semistable elliptic curves over rationals can be viewed as an instance of the Langlands reciprocity conjecture, since the main idea is to relate the Galois representations arising from elliptic curves to modular forms. Although Wiles' results have been substantially generalized, in many different directions, the full Langlands conjecture for GL(2, Q) remains unproved.
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 Drinfeld, who proved the case GL(2, K).
Local Langlands conjectures
Philip Kutzko (1980) proved the local Langlands conjectures for the general linear group GL(2, K) over local fields.
Gérard Laumon, Michael Rapoport, and Ulrich Stuhler (1993) 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.
Richard Taylor and Michael Harris (2001) proved the local Langlands conjectures for the general linear group GL(n, K) for characteristic 0 local fields K. Guy Henniart (2000) gave another proof. Both proofs use a global argument. Peter Scholze (2013) gave another proof.
Fundamental lemma
In 2008, Ngô Bảo Châu proved the "fundamental lemma", which was originally conjectured by Langlands in 1983 and being required in the proof of some important conjectures in the Langlands program.^{[4]}^{[5]}
Notes
 ^ "Math Quartet Joins Forces on Unified Theory". Quanta. December 8, 2015.

^ Frenkel, Edward (2013). Love & Math. ISBN 9780465050741.
All this stuff, as my dad put it, is quite heavy: we've got Hitchin moduli spaces, mirror symmetry, Abranes, Bbranes, automorphic sheaves... One can get a headache just trying to keep track of them all. Believe me, even among specialists, very few people know the nuts and bolts of all elements of this construction.

^ Frenkel, Edward (2013), Love and Math: The Heart of Hidden Reality, Basic Books, p. 77, ISBN 9780465069958,
The Langlands Program is now a vast subject. There is a large community of people working on it in different fields: number theory, harmonic analysis, geometry, representation theory, mathematical physics. Although they work with very different objects, they are all observing similar phenomena.
 ^ Ham Chau (20090215). "Ngô Bao Châu, sommité mondiale des maths" (in French). Le Courrier du Vietnam.
 ^ Langlands, Robert P. (1983), Les débuts d'une formule des traces stable, Publications Mathématiques de l'Université Paris VII [Mathematical Publications of the University of Paris VII], 13, Paris: Université de Paris VII U.E.R. de Mathématiques, MR 0697567
References
 Arthur, James (2003), "The principle of functoriality", American Mathematical Society. Bulletin. New Series, 40 (1): 39–53, doi:10.1090/S0273097902009631, ISSN 00029904, MR 1943132
 Bernstein, J.; Gelbart, S. (2003), An Introduction to the Langlands Program, Boston: Birkhäuser, ISBN 3764332115
 Gelbart, Stephen (1984), "An elementary introduction to the Langlands program", American Mathematical Society. Bulletin. New Series, 10 (2): 177–219, doi:10.1090/S027309791984152376, ISSN 00029904, MR 0733692
 Frenkel, Edward (2005). "Lectures on the Langlands Program and Conformal Field Theory". arXiv:hepth/0512172.
 Gelfand, I. M. (1963), "Automorphic functions and the theory of representations", Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Djursholm: Inst. MittagLeffler, pp. 74–85, MR 0175997
 Harris, Michael; Taylor, Richard (2001), The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, 151, Princeton University Press, ISBN 9780691090900, MR 1876802
 Henniart, Guy (2000), "Une preuve simple des conjectures de Langlands pour GL(n) sur un corps padique", Inventiones Mathematicae, 139 (2): 439–455, Bibcode:2000InMat.139..439H, doi:10.1007/s002220050012, ISSN 00209910, MR 1738446
 Kutzko, Philip (1980), "The Langlands Conjecture for Gl_{2} of a Local Field", Annals of Mathematics, 112 (2): 381–412, doi:10.2307/1971151, JSTOR 1971151
 Langlands, Robert (1967), Letter to Prof. Weil
 Langlands, R. P. (1970), "Problems in the theory of automorphic forms", Lectures in modern analysis and applications, III, Lecture Notes in Math, 170, Berlin, New York: SpringerVerlag, pp. 18–61, doi:10.1007/BFb0079065, ISBN 9783540052845, MR 0302614
 Laumon, G.; Rapoport, M.; Stuhler, U. (1993), "Delliptic sheaves and the Langlands correspondence", Inventiones Mathematicae, 113 (2): 217–338, Bibcode:1993InMat.113..217L, doi:10.1007/BF01244308, ISSN 00209910, MR 1228127
 Scholze, Peter (2013), "The Local Langlands Correspondence for GL(n) over padic fields", Inventiones mathematicae, 192 (3): 663–715, arXiv:1010.1540, Bibcode:2013InMat.192..663S, doi:10.1007/s0022201204205
External links
 The work of Robert Langlands