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."
The mathematical term perverse sheaves refers to the objects of certain abelian categories associated to topological spaces, which may be a real or complex manifold, or more general topologically stratified spaces, possibly singular.
In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker (1903) to make the formulas involving the solutions more symmetric. More generally, Jacquet (1966, 1967) introduced Whittaker functions of reductive groups over local fields, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL2(R).
Dennis Gaitsgory is an Israeli-American mathematician. He is a professor of mathematics at Harvard University and is known for his research on the geometric Langlands program.
In representation theory, a branch of mathematics, the Langlands dualLG of a reductive algebraic group G is a group that controls the representation theory of G. If G is defined over a field k, then LG is an extension of the absolute Galois group of k by a complex Lie group. There is also a variation called the Weil form of the L-group, where the Galois group is replaced by a Weil group. Here, the letter L in the name also indicates the connection with the theory of L-functions, particularly the automorphic L-functions. The Langlands dual was introduced by Langlands (1967) in a letter to A. Weil.
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.
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.
Victor Ginzburg is a Russian American mathematician who works in representation theory and in noncommutative geometry. He is known for his contributions to geometric representation theory, especially, for his works on representations of quantum groups and Hecke algebras, and on the geometric Langlands program. He is currently a Professor of Mathematics at the University of Chicago.
The mathematician Shmuel Aaron Weinberger is an American topologist. He completed a PhD in mathematics in 1982 at New York University under the direction of Sylvain Cappell. Weinberger was, from 1994 to 1996, the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, and he is currently the Andrew MacLeish Professor of Mathematics and chair of the Mathematics department at the University of Chicago.
Daqing Wan is a Chinese mathematician working in the United States. He received his Ph.D. from the University of Washington in Seattle in 1991, under the direction of Neal Koblitz. Since 1997, he has been on the faculty of mathematics at the University of California at Irvine; he has also held visiting positions at the Institute for Advanced Study in Princeton, New Jersey, Pennsylvania State University, the University of Rennes, the Mathematical Sciences Research Institute in Berkeley, California, and the Chinese Academy of Sciences in Beijing.
Edward Vladimirovich Frenkel is a Russian-American mathematician working in representation theory, algebraic geometry, and mathematical physics. He is a professor of mathematics at University of California Berkeley, a member of the American Academy of Arts and Sciences, and author of the bestselling book Love and Math.
In mathematics, the Satake isomorphism, introduced by Ichirō Satake, identifies the Hecke algebra of a reductive group over a local field with a ring of invariants of the Weyl group. The geometric Satake equivalence is a geometric version of the Satake isomorphism, proved by Ivan Mirković and Kari Vilonen.
Alexander Nikolaevich Varchenko is a Soviet and Russian mathematician working in geometry, topology, combinatorics and mathematical physics.
Bernd Siebert is a German mathematician who researches in algebraic geometry.
Zhiwei Yun is a Chinese-born American mathematician and writer. He is a Professor of Mathematics at MIT specializing in number theory, algebraic geometry and representation theory, with a particular focus on the Langlands program.
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.
Sug Woo Shin is a professor of mathematics at the University of California, Berkeley working in number theory, automorphic forms, and the Langlands program.
Francesco Damien "Frank" Calegari is a professor of mathematics at the University of Chicago working in number theory and the Langlands program.
David Erie Nadler is an American mathematician who specializes in geometric representation theory and symplectic geometry. He is currently a professor at the University of California, Berkeley.
