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In mathematics, the Jacobi group, introduced by Eichler & Zagier (1985), is the semidirect product of the symplectic group Sp2n(R) and the Heisenberg group R1+2n. The concept is named after Carl Gustav Jacob Jacobi. Automorphic forms on the Jacobi group are called Jacobi forms.
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.
Raoul Bott was a Hungarian-American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem.
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand was a prominent Soviet mathematician. He made significant contributions to many branches of mathematics, including group theory, representation theory and functional analysis. The recipient of many awards, including the Order of Lenin and the first Wolf Prize, he was a Foreign Fellow of the Royal Society and professor at Moscow State University and, after immigrating to the United States shortly before his 76th birthday, at Rutgers University. Gelfand is also a 1994 MacArthur Fellow.
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups.
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by Hecke (1937), is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations.
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act.
In mathematics, the Selberg trace formula, introduced by Selberg (1956), is an expression for the character of the unitary representation of G on the space L2(G/Γ) of square-integrable functions, where G is a Lie group and Γ a cofinite discrete group. The character is given by the trace of certain functions on G.
Don Bernard Zagier is an American-German mathematician whose main area of work is number theory. He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany. He was a professor at the Collège de France in Paris, France from 2006 to 2014. Since October 2014, he is also a Distinguished Staff Associate at ICTP.
Martin Maximilian Emil Eichler was a German number theorist.
In number theory, the Stark conjectures, introduced by Stark and later expanded by Tate (1984), give conjectural information about the coefficient of the leading term in the Taylor expansion of an Artin L-function associated with a Galois extension K/k of algebraic number fields. The conjectures generalize the analytic class number formula expressing the leading coefficient of the Taylor series for the Dedekind zeta function of a number field as the product of a regulator related to S-units of the field and a rational number. When K/k is an abelian extension and the order of vanishing of the L-function at s = 0 is one, Stark gave a refinement of his conjecture, predicting the existence of certain S-units, called Stark units. Rubin (1996) and Cristian Dumitru Popescu gave extensions of this refined conjecture to higher orders of vanishing.
In mathematics, the Langlands classification is a description of the irreducible representations of a reductive Lie group G, suggested by Robert Langlands (1973). There are two slightly different versions of the Langlands classification. One of these describes the irreducible admissible (g,K)-modules, for g a Lie algebra of a reductive Lie group G, with maximal compact subgroup K, in terms of tempered representations of smaller groups. The tempered representations were in turn classified by Anthony Knapp and Gregg Zuckerman. The other version of the Langlands classification divides the irreducible representations into L-packets, and classifies the L-packets in terms of certain homomorphisms of the Weil group of R or C into the Langlands dual group.
In mathematics, the Arthur conjectures are some conjectures about automorphic representations of reductive groups over the adeles and unitary representations of reductive groups over local fields made by James Arthur (1989), motivated by the Arthur-Selberg trace formula.
In mathematics, a Jacobi form is an automorphic form on the Jacobi group, which is the semidirect product of the symplectic group Sp(n;R) and the Heisenberg group . The theory was first systematically studied by Eichler & Zagier (1985).
Jon Hal Folkman was an American mathematician, a student of John Milnor, and a researcher at the RAND Corporation.
In geometric group theory, the Rips machine is a method of studying the action of groups on R-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991.
Robert Wayne Thomason was an American mathematician who worked on algebraic K-theory. His results include a proof that all infinite loop space machines are in some sense equivalent, and progress on the Quillen–Lichtenbaum conjecture.
In mathematics, the Goncharov conjecture is a conjecture introduced by Goncharov (1995) suggesting that the cohomology of certain motivic complexes coincides with pieces of K-groups. It extends a conjecture due to Zagier (1991).
In mathematics, the Saito–Kurokawa lift takes elliptic modular forms to Siegel modular forms of degree 2. The existence of this lifting was conjectured in 1977 independently by Hiroshi Saito and Nobushige Kurokawa (1978). Its existence was almost proved by Maass (1979), and Andrianov (1979) and Zagier (1981) completed the proof.
YoungJu Choie is a South Korean mathematician who works as a professor of mathematics at the Pohang University of Science and Technology (POSTECH). Her research interests include number theory and modular forms.
In mathematics, a weak Maass form is a smooth function on the upper half plane, transforming like a modular form under the action of the modular group, being an eigenfunction of the corresponding hyperbolic Laplace operator, and having at most linear exponential growth at the cusps. If the eigenvalue of under the Laplacian is zero, then is called a harmonic weak Maass form, or briefly a harmonic Maass form.
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