To this day, ℓ-adic cohomology remains a fundamental tool for number theorists, with important applications to the Langlands program.

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.

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.

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