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."
Richard Lawrence Taylor is a British mathematician working in the field of number theory. He is currently the Barbara Kimball Browning Professor in Humanities and Sciences at Stanford University.
In mathematics, a totally disconnected group is a topological group that is totally disconnected. Such topological groups are necessarily Hausdorff.
In mathematics, a Drinfeld module is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka is a sort of generalization of a Drinfeld module, consisting roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it.
Alexander A. Beilinson is the David and Mary Winton Green University professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1999, Beilinson was awarded the Ostrowski Prize with Helmut Hofer. In 2017, he was elected to the National Academy of Sciences. In 2018, he received the Wolf Prize in Mathematics and in 2020 the Shaw Prize in Mathematics.
Li Jianshu, also known as Jian-Shu Li, is a Chinese mathematician working in representation theory and automorphic forms. He is the founding director of the Institute for Advanced Study in Mathematics at Zhejiang University and Professor Emeritus at the Hong Kong University of Science and Technology.
In mathematics, the Jacquet–Langlands correspondence is a correspondence between automorphic forms on GL2 and its twisted forms, proved by Jacquet and Langlands (1970, section 16) in their book Automorphic Forms on GL(2) using the Selberg trace formula. It was one of the first examples of the Langlands philosophy that maps between L-groups should induce maps between automorphic representations. There are generalized versions of the Jacquet–Langlands correspondence relating automorphic representations of GLr(D) and GLdr(F), where D is a division algebra of degree d2 over the local or global field F.
In the mathematical theory of automorphic forms, the fundamental lemma relates orbital integrals on a reductive group over a local field to stable orbital integrals on its endoscopic groups. It was conjectured by Robert Langlands (1983) in the course of developing the Langlands program. The fundamental lemma was proved by Gérard Laumon and Ngô Bảo Châu in the case of unitary groups and then by Ngô (2010) for general reductive groups, building on a series of important reductions made by Jean-Loup Waldspurger to the case of Lie algebras. Time magazine placed Ngô's proof on the list of the "Top 10 scientific discoveries of 2009". In 2010, Ngô was awarded the Fields Medal for this proof.
Hervé Jacquet is a French American mathematician, working in automorphic forms. He is considered one of the founders of the theory of automorphic representations and their associated L-functions, and his results play a central role in modern number theory.
Diana Frost Shelstad is a mathematician known for her work in automorphic forms. She is a professor at Rutgers University–Newark. She earned her doctorate at Yale University in 1974 studying real reductive algebraic groups.
Marie-France Vignéras is a French mathematician. She is a Professor Emeritus of the Institut de Mathématiques de Jussieu in Paris. She is known for her proof published in 1980 of the existence of isospectral non-isometric Riemann surfaces. Such surfaces show that one cannot hear the shape of a hyperbolic drum. Another highlight of her work is the establishment of the mod-l local Langlands correspondence for GL(n) in 2000. Her current work concerns the p-adic Langlands program.
Michael Howard Harris is an American mathematician known for his work in number theory. He is a professor of mathematics at Columbia University and professor emeritus of mathematics at Université Paris Cité.
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 the Weil group. It was given that name by Robert Kottwitz. In Kottwitz's formulation, the Langlands group should be an extension of the Weil group by a compact group. When F is local archimedean, LF is the Weil group of F, when F is local non-archimedean, LF is the product of the Weil group of F with SU(2). When F is global, the existence of LF is still conjectural, though James Arthur gives a conjectural description of it. The Langlands correspondence for F is a "natural" correspondence between the irreducible n-dimensional complex representations of LF and, in the local case, the cuspidal automorphic representations of GLn(AF), where AF denotes the adeles of F.
In mathematics, Lafforgue's theorem, due to Laurent Lafforgue, completes the Langlands program for general linear groups over algebraic function fields, by giving a correspondence between automorphic forms on these groups and representations of Galois groups.
David Ginzburg is a professor of mathematics at Tel Aviv University working in number theory and automorphic forms.
David Soudry is a professor of mathematics at Tel Aviv University working in number theory and automorphic forms.
Jacques Tilouine is a professor of mathematics at Université Sorbonne Paris Nord working in number theory and automorphic forms, particularly Iwasawa theory.
Adrian Ioviță is a Romanian-Canadian mathematician, specializing in arithmetic algebraic geometry and p-adic cohomology theories.
Thomas Hermann Geisser is a German mathematician working at Rikkyo University. He works in the field of arithmetic geometry, motivic cohomology and algebraic K-theory.
Sebastiaan Johan Edixhoven was a Dutch mathematician who worked in arithmetic geometry. He was a professor at University of Rennes 1 and Leiden University.
