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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, 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."
In mathematics, the local Langlands conjectures, introduced by Robert Langlands, are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group G over a local field F, and representations of the Langlands group of F into the L-group of G. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois groups.
Danny Matthew Cornelius Calegari is a mathematician and, as of 2023, a professor of mathematics at the University of Chicago. His research interests include geometry, dynamical systems, low-dimensional topology, and geometric group theory.
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.
Michael Rapoport is an Austrian mathematician.
In mathematics, the Fontaine–Mazur conjectures are some conjectures introduced by Fontaine and Mazur (1995) about when p-adic representations of Galois groups of number fields can be constructed from representations on étale cohomology groups of a varieties. Some cases of this conjecture in dimension 2 were already proved by Dieulefait (2004).
Kari Kaleva Vilonen is a Finnish mathematician, specializing in geometric representation theory. He is currently a professor at the University of Melbourne.
Mihnea Popa is a Romanian-American mathematician at Harvard University, specializing in algebraic geometry. He is known for his work on complex birational geometry, Hodge theory, abelian varieties, and vector bundles.
Jun-Muk Hwang is a South Korean mathematician, specializing in algebraic geometry and complex differential geometry.
Gan Wee Teck is a Malaysian mathematician. He is a Distinguished Professor of Mathematics at the National University of Singapore (NUS). He is known for his work on automorphic forms and representation theory in the context of the Langlands program, especially the theory of theta correspondence, the Gan–Gross–Prasad conjecture and the Langlands program for Brylinski–Deligne covering groups.
Oscar Randal-Williams is a British mathematician and professor at the University of Cambridge, working in topology.
Anton Yurevich Alekseev is a Russian mathematician.
Shen Weixiao is a Chinese mathematician, specializing in dynamical systems.
Eric Jean-Paul Urban is a professor of mathematics at Columbia University working in number theory and automorphic forms, particularly Iwasawa theory.
Jacques Tilouine is a professor of mathematics at Université Sorbonne Paris Nord working in number theory and automorphic forms, particularly Iwasawa theory.
Sug Woo Shin is a professor of mathematics at the University of California, Berkeley working in number theory, automorphic forms, and the Langlands program.
Philipp Habegger is a Swiss mathematician and a professor of mathematics at the University of Basel who works in Diophantine geometry.
Bhargav Bhatt is a mathematician who is the Fernholz Joint Professor at the Institute for Advanced Study and Princeton University and works in arithmetic geometry and commutative algebra.
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.
Tsachik Gelander is an Israeli mathematician working in the fields of Lie groups, topological groups, symmetric spaces, lattices and discrete subgroups. He is a professor in Northwestern University.
References
- ↑ "Frank Calegari". International Mathematical Olympiad . Retrieved 6 March 2020.
- ↑ Frank Calegari at the Mathematics Genealogy Project
- ↑ "Francesco Damien Calegari". Institute for Advanced Study. 9 December 2019. Retrieved 6 March 2020.
- 1 2 "Frank Calegari". University of Chicago . Retrieved 6 March 2020.
- ↑ "Mathematische Zeitschrift | Editors". Springer Science+Business Media . Retrieved 6 March 2020.
- ↑ "Editorial Board". Annals of Mathematics. Retrieved 6 March 2020.
- ↑ "Frank Calegari" . Retrieved 6 March 2020.
- ↑ "Family, Collaborators, Students" . Retrieved 6 March 2020.
