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Arithmetic Group
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm_2(\Z). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Finally, these two topics join in the theory of automorphic forms which is fundamental in modern number theory. History One of the origins of the mathematical theory of arithmetic groups is algebraic number theory. The classical reduction theory of quadratic and Hermitian forms by Charles Hermite, Hermann Minkowski and others can be seen as computing fundamental domains for the action of certain arithmetic groups on the relevant symmetric spaces. The topic was related to Minkowski's geometry of numbers and the early development of the study of arithmetic invariant of number fields such as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacques Tits
Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Life and career Tits was born in Uccle to Léon Tits, a professor, and Lousia André. Jacques attended the Athénée of Uccle and the Free University of Brussels. His thesis advisor was Paul Libois, and Tits graduated with his doctorate in 1950 with the dissertation ''Généralisation des groupes projectifs basés sur la notion de transitivité''. His academic career includes professorships at the Free University of Brussels (now split into the Université Libre de Bruxelles and the Vrije Universiteit Brussel) (1962–1964), the University of Bonn (1964–1974) and the Collège de France in Paris, until becoming emeritus in 2000. He changed his citizenship to French in 1974 in order to teach at the Collège de France, which at that point required ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Hyperbolic 3-manifold
In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space \mathbb H^3 by an arithmetic Kleinian group. Definition and examples Quaternion algebras A quaternion algebra over a field F is a four-dimensional central simple F-algebra. A quaternion algebra has a basis 1, i, j, ij where i^2, j^2 \in F^\times and ij = -ji. A quaternion algebra is said to be split over F if it is isomorphic as an F-algebra to the algebra of matrices M_2(F); a quaternion algebra over an algebraically closed field is always split. If \sigma is an embedding of F into a field E we shall denote by A \otimes_\sigma E the algebra obtained by extending scalars from F to E where we view F as a subfield of E via \sigma. Arithmetic Kleinian groups A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Symmetric Space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis. In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (''M'', ''g'') is said to be symmetric if and only if, for each point ''p'' of ''M'', there exists an isometry of ''M'' fixing ''p'' and acting on the tangent space T_pM as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Arthur (mathematician)
James Greig Arthur (born May 18, 1944) is a Canadian mathematician working on automorphic forms, and former President of the American Mathematical Society. He is a Mossman Chair and University Professor at the University of Toronto Department of Mathematics. Education and career Born in Hamilton, Ontario, Arthur graduated from Upper Canada College in 1962, received a BSc from the University of Toronto in 1966, and a MSc from the same institution in 1967. He received his PhD from Yale University in 1970. He was a student of Robert Langlands; his dissertation was ''Analysis of Tempered Distributions on Semisimple Lie Groups of Real Rank One''. Arthur taught at Yale from 1970 until 1976. He joined the faculty of Duke University in 1976. He has been a professor at the University of Toronto since 1978. He was four times a visiting scholar at the Institute for Advanced Study between 1976 and 2002. Contributions Arthur is known for the Arthur–Selberg trace formula, generalizing th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Selberg's Trace Formula
In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of a Lie group on the space of square-integrable functions, where is a cofinite discrete group. The character is given by the trace of certain functions on . The simplest case is when is cocompact group action, cocompact, when the representation breaks up into discrete summands. Here the trace formula is an extension of the Induced representation#Alternate formulations, Frobenius formula for the character of an induced representation of finite groups. When is the cocompact subgroup of the real numbers , the Selberg trace formula is essentially the Poisson summation formula. The case when is not compact is harder, because there is a spectrum (functional analysis), continuous spectrum, described using Eisenstein series. Selberg worked out the non-compact case when is the group ; the extension to higher rank groups is the Arthur–Selberg trace formula. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Langlands
Robert Phelan Langlands, (; born October 6, 1936) is a Canadian mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study of Galois groups in number theory, for which he received the 2018 Abel Prize. He was an emeritus professor and occupied Albert Einstein's office at the Institute for Advanced Study in Princeton, until 2020 when he retired. Career Langlands was born in New Westminster, British Columbia, Canada, in 1936 to Robert Langlands and Kathleen J Phelan. He has two younger sisters (Mary b 1938; Sally b 1941). In 1945, his family moved to White Rock, near the US border, where his parents had a building supply and construction business. He graduated from Semiahmoo Secondary School and started enrolling at the University of British Columbia at the age of 16, receiving his undergraduate degree in Mathematics in 1957; he continued at UBC to receive an M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langlands Program
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 , 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 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 integrati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marina Ratner
Marina Evseevna Ratner (russian: Мари́на Евсе́евна Ра́тнер; October 30, 1938 – July 7, 2017) was a professor of mathematics at the University of California, Berkeley who worked in ergodic theory. Around 1990, she proved a group of major theorems concerning unipotent flow (mathematics), flows on homogeneous spaces, known as Ratner's theorems. Ratner was elected to the American Academy of Arts and Sciences in 1992, awarded the Ostrowski Prize in 1993 and elected to the United States National Academy of Sciences, National Academy of Sciences the same year. In 1994, she was awarded the John J. Carty Award for the Advancement of Science, John J. Carty Award from the National Academy of Sciences. Biographical information Ratner was born in Moscow, Russian SFSR to a Jewish family, where her father was a plant physiologist and her mother a chemist. Ratner's mother was fired from work in the 1940s for writing to her mother in Israel, then considered an enemy of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ratner's Theorems
In mathematics, Ratner's theorems are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The theorems grew out of Ratner's earlier work on horocycle flows. The study of the dynamics of unipotent flows played a decisive role in the proof of the Oppenheim conjecture by Grigory Margulis. Ratner's theorems have guided key advances in the understanding of the dynamics of unipotent flows. Their later generalizations provide ways to both sharpen the results and extend the theory to the setting of arbitrary semisimple algebraic groups over a local field. Short description The Ratner orbit closure theorem asserts that the closures of orbits of unipotent flows on the quotient of a Lie group by a lattice are nice, geometric subsets. The Ratner equidistribution theorem further asserts that each such orbit is equidistributed in its closure. The Ratner measure classification theorem is the weaker statement that e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oppenheim Conjecture
In Diophantine approximation, the Oppenheim conjecture concerns representations of numbers by real quadratic forms in several variables. It was formulated in 1929 by Alexander Oppenheim and later the conjectured property was further strengthened by Harold Davenport and Oppenheim. Initial research on this problem took the number ''n'' of variables to be large, and applied a version of the Hardy-Littlewood circle method. The definitive work of Grigory Margulis, Margulis, settling the conjecture in the affirmative, used methods arising from ergodic theory and the study of discrete subgroups of semisimple Lie groups. Short description Meyer's theorem states that an indefinite integral quadratic form ''Q'' in ''n'' variables, ''n'' ≥ 5, nontrivially represents zero, i.e. there exists a non-zero vector ''x'' with integer components such that ''Q''(''x'') = 0. The Oppenheim conjecture can be viewed as an analogue of this statement for forms ''Q'' that are not multi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
