Zeta and L-functions | Langlands program | Representation theory of Lie groups | Automorphic forms | Conjectures

Langlands program

In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands , 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." The Langlands program consists of some very complicated theoretical abstractions, which can be difficult even for specialist mathematicians to grasp. To oversimplify, the fundamental lemma of the project posits a direct connection between the generalized fundamental representation of a finite field with its group extension to the automorphic forms under which it is invariant. This is accomplished through abstraction to higher dimensional integration, by an equivalence to a certain analytical group as an absolute extension of its algebra. Consequently, this allows an analytical functional construction of powerful invariance transformations for a number field to its own algebraic structure. The meaning of such a construction is nuanced, but its specific solutions and generalizations are very powerful. The consequence for proof of existence to such theoretical objects implies an analytical method in constructing the categoric mapping of fundamental structures for virtually any number field. As an analogue to the possible exact distribution of primes, the Langlands program allows a potential general tool for the resolution of invariance at the level of generalized algebraic structures. This in turn permits a somewhat unified analysis of arithmetic objects through their automorphic functions. Simply put, the Langlands philosophy allows a general analysis of structuring the abstractions of numbers. Naturally, this description is at once a reduction and over-generalization of the program's proper theorems, but these mathematical analogues provide the basis of its conceptualization. (Wikipedia).

Video thumbnail

The 2022 Summer School on the Langlands Program at IHES

The Langlands Program is a complex and far-reaching series of conjectures about the connections that can be built between very different areas of mathematics. The 2022 IHES Summer School will be fully dedicated to defining the state of the art of research on this fascinating subject and in

From playlist 2022 Summer School on the Langlands program

Video thumbnail

Brief introduction to the Langlands program

Popular presentation by Alex Bellos on the Langlands program, by Robert Langlands. This clip is a part of the Abel Prize Announcement 2018. You can see the full announcement here: https://youtu.be/6Zob3MeMIvc You can view Alex Bellos own YouTube channel here: https://www.youtube.com

From playlist Popular presentations

Video thumbnail

The Biggest Project in Modern Mathematics

In a 1967 letter to the number theorist André Weil, a 30-year-old mathematician named Robert Langlands outlined striking conjectures that predicted a correspondence between two objects from completely different fields of math. The Langlands program was born. Today, it's one of the most amb

From playlist Explainers

Video thumbnail

James Arthur: The Langlands program: arithmetic, geometry and analysis

Abstract: As the Abel Prize citation points out, the Langlands program represents a grand unified theory of mathematics. We shall try to explain in elementary terms what this means. We shall describe an age old question concerning the arithmetic prime numbers, together with a profound gene

From playlist Abel Lectures

Video thumbnail

Elementary Introduction to the Langlands Program, by Edward Frenkel (Part 1) [2015]

"Do we discover mathematics or do we invent it?" One of the most fascinating and important developments in mathematics in the last 50 years is the Langlands Program, a collection of ideas that provides a grand unification of many areas of mathematics. In September 2015, Edward Frenkel g

From playlist Number Theory

Video thumbnail

Eugen Hellman - 1/4 An Introduction to the Categorical p-adic Langlands (...)

An introduction to the ``categorical'' approach to the p-adic Langlands program, in both the ``Banach'' and ``analytic'' settings. Matthew Emerton (Chicago Univ.) Toby Gee (Imperial College) Eugen Hellman (Univ. Münster)

From playlist 2022 Summer School on the Langlands program

Video thumbnail

Toby Gee - 2/4 An Introduction to the Categorical p-adic Langlands (...)

An introduction to the ``categorical'' approach to the p-adic Langlands program, in both the ``Banach'' and ``analytic'' settings. Matthew Emerton (Chicago Univ.) Toby Gee (Imperial College) Eugen Hellman (Univ. Münster)

From playlist 2022 Summer School on the Langlands program

Video thumbnail

The Abel Prize announcement 2018 - Robert Langlands

0:52 Introduction by Alex Bellos, British writer, and science communicator 2:26 The Abel Prize announced by Ole Sejersted, President of The Norwegian Academy of Science and Letters 1:38 Citation by John Rognes, Chair of the Abel committee 7:18 Popular presentation of the prize winners work

From playlist The Abel Prize announcements

Video thumbnail

Abel Award Ceremony 2018 - Robert Langlands

0:02 Procession accompanied by the “Abel Fanfare” (Klaus Sandvik). Performed by musicians from The Staff Band of the Norwegian Armed Forces 0:52 His Majesty King Harald enters the University Aula 2:09 Dance of the Drums. Performed by Teodor Berg og Arild Torvik | Music: Gene Koshinski 7:19

From playlist Abel Prize Ceremonies

Video thumbnail

Kevin Buzzard (lecture 15/20) Automorphic Forms And The Langlands Program [2017]

Full course playlist: https://www.youtube.com/playlist?list=PLhsb6tmzSpiysoRR0bZozub-MM0k3mdFR http://wwwf.imperial.ac.uk/~buzzard/MSRI/ Summer Graduate School Automorphic Forms and the Langlands Program July 24, 2017 - August 04, 2017 Kevin Buzzard (Imperial College, London) https://w

From playlist MSRI Summer School: Automorphic Forms And The Langlands Program, by Kevin Buzzard [2017]

Video thumbnail

Lecture 1: Geometric Langlands and S-duality in N = 4 SYM by Sergei Gukov

Program: Quantum Fields, Geometry and Representation Theory ORGANIZERS : Aswin Balasubramanian, Saurav Bhaumik, Indranil Biswas, Abhijit Gadde, Rajesh Gopakumar and Mahan Mj DATE & TIME : 16 July 2018 to 27 July 2018 VENUE : Madhava Lecture Hall, ICTS, Bangalore The power of symmetries

From playlist Quantum Fields, Geometry and Representation Theory

Video thumbnail

The Geometric Langlands conjecture and non-abelian Hodge theory (Lecture 1) by Ron Donagi

Program: Quantum Fields, Geometry and Representation Theory ORGANIZERS : Aswin Balasubramanian, Saurav Bhaumik, Indranil Biswas, Abhijit Gadde, Rajesh Gopakumar and Mahan Mj DATE & TIME : 16 July 2018 to 27 July 2018 VENUE : Madhava Lecture Hall, ICTS, Bangalore The power of symmetries

From playlist Quantum Fields, Geometry and Representation Theory

Video thumbnail

Robbert Dijkgraaf tribute to Robert Langlands

This speech was held immediately after the announcement of the Abel Prize in March 2018. The speaker is Robbert Dijkgraaf, a theoretical physicist and string theorist, but also the director at Institute for Advanced Study in Princeton. Thumbnail photo: Andrea Kane

From playlist The Abel Prize announcements

Video thumbnail

Interview at Cirm: Michael Harris

Michael Harris is an American mathematician who deals with number theory and algebra. He made notable contributions to the Langlands program, for which he (alongside Richard Taylor) won the 2007 Clay Research Award. In particular, he (jointly with Taylor), proved the local Langlands conjec

From playlist English interviews - Interviews en anglais

Video thumbnail

Yiannis Sakellaridis - 1/2 Local and Global Questions “Beyond Endoscopy”

The near-completion of the program of endoscopy poses the question of what lies next. These talks will take a broad view of ideas beyond the program of endoscopy, highlighting the connections among them, and emphasizing the relationship between local and global aspects. Central among thos

From playlist 2022 Summer School on the Langlands program

Related pages

Jacquet–Langlands correspondence | Group representation | Algebraic variety | Conjecture | Levi decomposition | Invariant theory | Rational number | L-packet | General linear group | Quadratic reciprocity | Representation theory | Atle Selberg | Field extension | Global field | Real number | Artin reciprocity law | Birch and Swinnerton-Dyer conjecture | Holomorphic function | Prime number | Algebraic curve | Complex number | Restricted representation | P-adic number | Lafforgue's theorem | Lift (mathematics) | Semisimple Lie algebra | Abelian group | Hecke character | Fundamental lemma (Langlands program) | Finite field | Riemann hypothesis | Emil Artin | Étale fundamental group | Artin L-function | Analytic number theory | Algebraic number theory | Vector bundle | Morphism | Functor | Erich Hecke | Geometry | Siegel modular form | Langlands group | Lie group | Functional analysis | Local field | Class field theory | Homological algebra | Automorphism | Algebraic number field | Duality (mathematics) | Orbital integral | Algebraic geometry | Group theory | Hilbert modular form | Reciprocity law | Langlands dual group | Superstring theory | Algebraic group | Galois extension | Eisenstein series | Riemann zeta function | Monstrous moonshine | Langlands classification | Automorphic form | L-function | Spectrum (functional analysis) | Induced representation | Number theory | Reductive group | Galois group | Selberg trace formula | Local Langlands conjectures | Abstract algebra | Derived category | Adele ring | Automorphic L-function | Fundamental representation | Universal property | Hodge conjecture | Ideal class group | Spectral theory