Lack of proof of this had been the main obstacle to completing many instances of the langlands program. Rational numbers and complex numbers were created to solve certain equations. Arithmetic applications of the langlands program springerlink. Robert langlands, mathematical visionary, wins the abel prize generations of researchers have pursued his langlands program, which seeks to create a grand unified theory of mathematics. Very loosely speaking, the langlands program can be viewed as an attempt to study to what extent lfunctions are really different from each other. Stephen gelbart, an elementary introduction to the langlands program, bull.
The langlands program envisions deep links between arithmetic. The categorical langlands duality is a conjectural equivalence of two categories associated to a reductive group and a smooth projective curve. The work of ngo bao chau, notices of the ams, 58, no. The conjectures of the langlands program have shaped research in number theory, representation theory, and other areas for many years, but.
This correspondence is relatively well understood when the connection has no singularities, i. Young mathematicians conference the ohio state university august 21, 2016. What is called the langlands correspondence in number theory. Workshop on motives, galois representations and cohomology around the langlands program. In these talks i will give an introduction to my geometrization conjecture of the local langlands correspondence.
Pdf eight dimensional physics and the langlands program a. The focus of the trimester will be on some manifestations of this program, including. In particular, we will establish a weak version of the jacquetlanglands correspondence in the torsion setting. The fundamental lemma was proved by gerard laumon and ngo b.
Stabilization of the trace formula, shimura varieties, and arithmetic applications ed. To begin with, langlands program is a synthesis of several important themes in classical number. The practice of mathematics ias school of mathematics. He is best known as the founder of the langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study of galois groups in number theory, for which he received the 2018 abel prize.
Abstractions blog robert langlands, mathematical visionary, wins the abel prize. Indam program on serre conjectures and the padic local langlands program wednesday, may 15th pascal boyer laga 11. Geometrization of the local langlands correspondence abstract. The langlands program an overview 211 introduction the langlands program predicts a correspondence between two types of objects. About iharas lemma in higher dimension in their work on satotate, clozelharristaylor proposed a generalization of the iharas lemma in higher dimension for some similitude groups. From quadratic reciprocity to langlands program a major task in mathematics is to solve equations. Langlands ars longa, vita brevis there are several central mathematical problems, or complexes of problems, that every mathematician who is eager to acquire some broad competence in the subject would like to understand, even if he has no ambition to attack them all. Therefore, we will give an introduction to the arithmetic setting and how. Ml ml the mathematical work of the 2010 fields medalists. This expository article is an introduction to the langlands functoriality conjectures and their applications to the arithmetic of representations of galois groups of number fields. Mod p langlands correspondences via arithmetic geometry notes from a minicourse at kias, august 2016 judith ludwig january 5, 2017 lecture 1. It was conjectured by robert langlands in the course of developing the langlands program. The 30yearold math sensation peter scholze is now one of the youngest fields medalists for the revolution that he launched in arithmetic geometry.
Langlands program, lie group, algebraic num ber theory, algebraic. Survey on the fundamental lemma university of chicago. It also is related to derived structures in the langlands program. Number systems and arithmetic jason mars thursday, january 24. Langlands in the context of endoscopy theory for automorphic representations in 26. Arithmetic applications of the langlands program michael harris universit. Whole numbers are the numbers used most often for counting and computation in everyday life. Geometry, representation theory, and the langlands program. Workshop on the arithmetic geometry of shimura varieties and rapoportzink spaces, kyoto university, japan, 74782011. In other words langlands program begins with visualizing how lfunctions are in the same time.
Hausdorff trimester program the arithmetic of the langlands program the langlands program aims to relate systems of polynomial equations with integer coefficients to automorphic forms, i. The global geometric langlands correspondence relates hecke eigensheaves on the moduli stack of gbundles on a smooth projective algebraic curve x and holomorphic gbundles with connection on x, where g is the langlands dual group of g. Born on october 6, 1936, in british columbia, robert langlands grew up in a small town where his father owned a building supply store. Yun thought that some of those ideas could be combined with techniques from algebraic geometry, but he told zhang he wasnt sure if. The geometric langlands correspondence is a geometric analogue of the arithmetic global langlands correspondence which conjectures a link between the absolute galois group galff of an algebraic number eld f and certain automorphic representations.
It is interesting because, at present, there seems to be no known algebraic mechanism that explains it. Prerequisites for the langlands program stony brook mathematics. The table below shows the specific wholenumber related objectives that are the. Basic arithmetic lesson 1 whole numbers 1 lesson 1 whole numbers introduction we will begin our study of basic arithmetic by learning about whole numbers. Edward frenkel is a professor of mathematics at the university of california, berkeley, member of the american academy of arts and sciences, and the winner of the hermann weyl prize in mathematical physics. The fundamental lemma and the related transfer conjecture were formulated by r. Geometric langlands david kazhdan notes by sam raskin. On the one side we have automorphic representations. Frank calegari, ana caraiani, laurent fargues, peter scholze the langlands program aims to relate systems of polynomial equations with integer coefficients to automorphic forms, i. Introduction to the langlands program stanford mathematics. Langlands program by discussing first lfunctions attached to rep resentations of the galois group of number fields.
Yun thought that some of those ideas could be combined with techniques from algebraic geometry, but he told zhang he wasnt sure if it was possible. The arithmetic of the langlands program hausdorff research. Survey on the fundamental lemma ngo bao ch au this is a survey on the recent proof of the fundamental lemma. Math quartet joins forces on unified theory quanta magazine. To begin with, langlands program is a synthesis of several important themes in classical number theory.
He is the author of a new york times bestseller love and math which is. The langlands program describes the representation theory of many important algebraic groups. Grojnowski indicates that, according to langlands, the representations of a reductive group are described in terms of a the langlands dual group and b some galois group. Modular arithmetic has been a major concern of mathematicians for at least 250 years, and is still a very active topic of current research. By kevin hartnett dan komodainstitute for advanced study robert langlands addresses a conference at the institute for advanced study in 2016. Dec 08, 2015 at that time, yun had been thinking about work zhang had done even earlier on a problem in the langlands program known as the arithmetic fundamental lemma. When i was a child i liked to add and subtract, says langlands. In this talk, we will introduce the main object of study of the local geometric langlands theory. Physicsmathematics summer institute phymsi, the cirm session on daha and langlands program, luminy, france, 620712011. In this article, i will explain what modular arithmetic is, illustrate why it is of importance for mathematicians, and discuss some recent breakthroughs. Based on his proof, which is concentrated around the moduli space of vector bundles on algebraic curves, drinfeld has expanded the class of.
Previous constructions have a purely arithmetic origin. Agnes algebraic geometry northeastern series, mit, 41432011. I will outline the materials and distribute the weekly lectures to volunteers. Hausdorff trimester program the arithmetic of the langlands. The langlands correspondence in arithmetic and geometry. The langlands program conjecturally relates homology of arithmetic groups to galois representations. Robert langlands, mathematical visionary, wins the abel prize. The langlands program envisions deep links between arithmeticandanalysis,andusesconstructionsinarithmetic to predict maps between spaces of functions on di. Shortly after formulating his program, however, langlands proposed to test it in two. Indam program on serre conjectures and the padic local. The langlands program aims to relate systems of polynomial equations with integer coefficients to automorphic forms, i. Thanks to the work of a great many people, the stable trace formula is now largely established in a version adequate for proving langlands functoriality in the. On the one hand the localization of the program to local fields leads to the conjecture of local langlands correspondences gelbart 84, page 34 210.
The purpose of this paper is to explain what the langlands program is about what new perspectives on number theory it affords, and what kinds of results it can be expected to prove. An overview of the geometric langlands program max planck. Introduction to mod p langlands correspondences and background on adic spaces 1. Workshop on motives, galois representations and cohomology. Proposed by robert langlands 1967, 1970, it seeks to relate galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Arithmetic applications of the langlands program michael harris.
At that time, yun had been thinking about work zhang had done even earlier on a problem in the langlands program known as the arithmetic fundamental lemma. On the other hand, the interpretation of the above story dually in arithmetic geometry in view of the function field analogy motivates the conjectural geometric langlands correspondence. The goal of this series of talks is to introduce the langlands program. Early sources, like the rhind papyrus, written around 1650 bce by ahmes, contains methods for solving linear equations. Introduction so far in the seminar there has been no discussion of the arithmetic aspects of the langlands program which originally motivated the development of a geometric program. For almost all its history, the study of modular arithmetic has been driven purely by. I will formulate this equivalence and explain what is currently known. The langlands program an overview ucsd mathematics. Lfunctions have their origins in number theory as a method for counting or manipulating various arithmetic objects, such as prime numbers.
In mathematics, the langlands program is a web of farreaching and influential conjectures about connections between number theory and geometry. Let p6 lbe a prime number and fq pbe a nite extension. Mod p langlands correspondences via arithmetic geometry. When rewritten in the case of other global elds that is function elds. Nov 22, 2011 lands program, initiated in kw, but also in the classical arithmetic langlands program in the recent proof of the fundamental lemma, see ngo 2006a, ngo 2006b. The fundamental lemma proved by ngo, following initial ideas of goreskykottwitzmacpherson and laumon. So the origin of the langlands program is in number theory.
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