Far-reaching conjectures connecting number theory and geometry
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In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and consequential conjectures about connections between number theory and geometry. 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. 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."[1]
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.
^"Math Quartet Joins Forces on Unified Theory". Quanta. December 8, 2015.
In representation theory and algebraic number theory, the Langlandsprogram is a web of far-reaching and consequential conjectures about connections between...
Phelan Langlands, CC FRS FRSC (/ˈlæŋləndz/; born October 6, 1936) is a Canadian mathematician. He is best known as the founder of the Langlandsprogram, a...
In mathematics, the geometric Langlands correspondence is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing...
Langlands dual Langlands group LanglandsprogramLanglands, Queensland, a locality in the Western Downs Region, Queensland, Australia Langlands Park, a rugby...
Gaitsgory and K. Vilonen: On the geometric Langlands conjecture, 2000. Recent Advances in the LanglandsProgram. 2003. arXiv:math/0303074. Bibcode:2003math...
equivalence between motivic and automorphic L-functions postulated in the Langlandsprogram can be tested. Automorphic forms realized in the cohomology of a Shimura...
mathematics, the local Langlands conjectures, introduced by Robert Langlands (1967, 1970), are part of the Langlandsprogram. They describe a correspondence...
absent in the Langlands correspondence. There are several other nonabelian theories, local and global, which provide alternatives to the Langlands correspondence...
algebraic groups over the p-adic numbers, with connections to the Langlandsprogram. She is a professor at the University of Bonn. Fintzen competed for...
Bonn. Her research interests include algebraic number theory and the Langlandsprogram. She was born in Bucharest and studied at Mihai Viteazul High School...
In mathematics, the Langlands group is a conjectural group LF attached to each local or global field F, that satisfies properties similar to those of...
Theorem, but pushed the whole of mathematics as a field towards the Langlandsprogram of unifying number theory. Wiles was born on 11 April 1953 in Cambridge...
outstanding contributions to Langlands' program in the fields of number theory and analysis, and in particular proved the Langlands conjectures for the automorphism...
ISBN 978-0-691-18443-2. On the geometric Langlands conjecture On a vanishing conjecture appearing in the geometric Langlands correspondence Dennis Gaitsgory at...
primarily with geometric representation theory and in particular the Langlandsprogram, tying number theory to algebraic geometry and quantum physics. Zhu...
mathematician who is active in algebraic geometry, especially in the Langlandsprogram, and a CNRS "Directeur de Recherches" at the Institute Fourier in...
2023 Ana Caraiani – "For diverse transformative contributions to the Langlandsprogram, and in particular for work with Peter Scholze on the Hodge-Tate period...
context of the Langlandsprogram, especially the theory of theta correspondence, the Gan–Gross–Prasad conjecture and the Langlandsprogram for Brylinski–Deligne...
invariant theory and the Erlangen program, has an impact in number theory via automorphic forms and the Langlandsprogram. There are diverse approaches to...
In number theory, the Shimura correspondence is a correspondence between modular forms F of half integral weight k+1/2, and modular forms f of even weight...
vast generalization of quadratic reciprocity. Robert Langlands formulated the Langlandsprogram, which gives a conjectural vast generalization of class...
that the first conjecture is true and the second one is false. The Langlandsprogram is a far-reaching web of these ideas of 'unifying conjectures' that...
In mathematics, the theta correspondence or Howe correspondence is a mathematical relation between representations of two groups of a reductive dual pair...
area of research in algebraic number theory is Iwasawa theory. The Langlandsprogram, one of the main current large-scale research plans in mathematics...
historical and recent, include Felix Klein's Erlangen program, Hilbert's problems, Langlandsprogram, and the Millennium Prize Problems. In the Mathematics...
is a special case of more general conjectures due to Robert Langlands. The Langlandsprogram seeks to attach an automorphic form or automorphic representation...
and the IAS maintains the key repository for the papers of Langlands and the Langlandsprogram. The IAS is a main center of research for homotopy type theory...
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