Zeta and L-functions | Langlands program | Representation theory of Lie groups | Automorphic forms | Conjectures
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).
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
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
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
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
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
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
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
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
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
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]
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
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
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
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
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
Sam Raskin - 1/2 What does geometric Langlands mean to a number theorist?
Sam Raskin (Univ. Texas)
From playlist 2022 Summer School on the Langlands program