"[1] As an explanation to a non-specialist: the program provides constructs for a generalised and somewhat unified framework, to characterise the structures that underpin numbers and their abstractions; thus the invariants which base them through analytical methods.
This is accomplished through abstraction to higher dimensional integration, by an equivalence to a certain analytical group as an absolute extension of its algebra.
This allows an analytical functional construction of powerful invariance transformations for a number field to its own algebraic structure.
The consequence for proof of existence to such theoretical objects, implies an analytical method for constructing the categoric mapping of fundamental structures for virtually any number field.
This description is at once a reduction and over-generalization of the program's proper theorems – although these mathematical concepts illustrate its basic ideas.
The Langlands program is built on existing ideas: the philosophy of cusp forms formulated a few years earlier by Harish-Chandra and Gelfand (1963), the work and Harish-Chandra's approach on semisimple Lie groups, and in technical terms the trace formula of Selberg and others.
What was new in Langlands' work, besides technical depth, was the proposed connection to number theory, together with its rich organisational structure hypothesised (so-called functoriality).
In all these approaches technical methods were available, often inductive in nature and based on Levi decompositions amongst other matters, but the field remained demanding.
are vague, or depend on objects such as Langlands groups, whose existence is unproven, or on the L-group that has several non-equivalent definitions.
This offers numerous variations, in part because the definitions of Langlands group and L-group are not fixed.
Given two reductive groups and a (well behaved) morphism between their corresponding L-groups, this conjecture relates their automorphic representations in a way that is compatible with their L-functions.
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.
In 1998, 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) in the 1980s.
As the program posits a powerful connection between analytic number theory and generalizations of algebraic geometry, the idea of 'Functoriality' between abstract algebraic representations of number fields and their analytical prime constructions results in powerful functional tools allowing an exact quantification of prime distributions.
Furthermore, if the reciprocity of such generalized algebras for the posited objects exists, and if their analytical functions can be shown to be well-defined, some very deep results in mathematics could be within reach of proof.
Examples include: rational solutions of elliptic curves, topological construction of algebraic varieties, and the famous Riemann hypothesis.
[13] Such proofs would be expected to utilize abstract solutions in objects of generalized analytical series, each of which relates to the invariance within structures of number fields.
Simply put, the Langlands project implies a deep and powerful framework of solutions, which touches the most fundamental areas of mathematics, through high-order generalizations in exact solutions of algebraic equations, with analytical functions, as embedded in geometric forms.
It allows a unification of many distant mathematical fields into a formalism of powerful analytical methods.