Langlands program

Web of far-reaching and influential conjectures about connections between number theory and geometry

There are different types of objects for which the Langlands conjectures can be stated:

There are several different ways of stating the Langlands conjectures, which are closely related but not obviously equivalent.

The functoriality conjecture states that a suitable homomorphism of L-groups 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.

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.

To a lay reader or even nonspecialist mathematician, abstractions within the Langlands program can be somewhat impenetrable. However, there are some strong and clear implications for proof or disproof of the fundamental Langlands conjectures.

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.