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Dixmier
Jacques Dixmier (born 24 May 1924) is a French mathematician. He worked on operator algebras, especially C*-algebras, and wrote several of the standard reference books on them, and introduced the Dixmier trace and the Dixmier mapping. Biography Dixmier received his Ph.D. in 1949 from the University of Paris, and his students include Alain Connes. In 1949 upon the initiative of Jean-Pierre Serre and Pierre Samuel, Dixmier became a member of Bourbaki, in which he made essential contributions to the Bourbaki volume on Lie algebras. After retiring as professor emeritus from the University of Paris VI, he spent five years at the Institut des Hautes Études Scientifiques. Often, there is made the erroneous claim that Dixmier originated the name ''von Neumann algebra'' for the operator algebras introduced by John von Neumann, but Dixmier said in an interview that the name originated from a proposal by Jean Dieudonné. Dixmier was an invited speaker at the International Congress o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dixmier Trace
In mathematics, the Dixmier trace, introduced by , is a non-normal trace on a space of linear operators on a Hilbert space larger than the space of trace class operators. Dixmier traces are examples of singular traces. Some applications of Dixmier traces to noncommutative geometry are described in . Definition If ''H'' is a Hilbert space, then ''L''1,∞(''H'') is the space of compact linear operators ''T'' on ''H'' such that the norm :\, T\, _ = \sup_N\frac is finite, where the numbers ''μ''''i''(''T'') are the eigenvalues of , ''T'', arranged in decreasing order. Let :a_N = \frac. The Dixmier trace Tr''ω''(''T'') of ''T'' is defined for positive operators ''T'' of ''L''1,∞(''H'') to be :\operatorname_\omega(T)= \lim_\omega a_N where lim''ω'' is a scale-invariant positive "extension" of the usual limit, to all bounded sequences. In other words, it has the following properties: *lim''ω''(''α''''n'') ≥ 0 if all ''α''''n'' ≥ 0 (positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dixmier Mapping
In mathematics, the Dixmier mapping describes the space Prim(''U''(''g'')) of primitive ideals of the universal enveloping algebra ''U''(''g'') of a finite-dimensional solvable Lie algebra ''g'' over an algebraically closed field of characteristic 0 in terms of coadjoint orbits. More precisely, it is a homeomorphism from the space of orbits ''g''*/''G'' of the dual ''g''* of ''g'' (with the Zariski topology) under the action of the adjoint group ''G'' to Prim(''U''(''g'')) (with the Jacobson topology). The Dixmier map is closely related to the orbit method, which relates the irreducible representations of a nilpotent Lie group to its coadjoint orbits. introduced the Dixmier map for nilpotent Lie algebras and then in extended it to solvable ones. describes the Dixmier mapping in detail. Construction Suppose that ''g'' is a completely solvable Lie algebra, and ''f'' is an element of the dual ''g''*. A polarization of ''g'' at ''f'' is a subspace ''h'' of maximal dimension subjec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dixmier Conjecture
In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any endomorphism of a Weyl algebra is an automorphism. Tsuchimoto in 2005, and independently Belov-Kanel and Kontsevich in 2007, showed that the Dixmier conjecture is stably equivalent to the Jacobian conjecture In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an ''n''-dimensional space to itself has Jacobian determinant which is a non-zero con .... References Abstract algebra Conjectures Unsolved problems in mathematics {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicolas Bourbaki
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics. The series is known collectively as the ''Éléments de mathématique'' (''Elements of Mathematics''), the group's central work. Topics treated in the series include set theory, abstract algebra, topology, analysis, Lie groups and Lie algebras. Bourbaki was founded in response to the effects of the First World War which caused the death of a generation of French mathematicians; as a result, young university instructors were forced to use dated texts. While teaching at the University of Strasbourg, Henri Cartan complained to his colleague André Weil of the inadequac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows: *The ring L^\infty(\mathbb R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space L^2(\mathbb R) of square-integrable functions. *The algebra \mathcal B(\mathcal H) of all bounded operators on a Hilbert space \mathcal H is a von Neumann ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous linear operators on a complex Hilbert space with two additional properties: * ''A'' is a topologically closed set in the norm topology of operators. * ''A'' is closed under the operation of taking adjoints of operators. Another important class of non-Hilbert C*-algebras includes the algebra C_0(X) of complex-valued continuous functions on ''X'' that vanish at infinity, where ''X'' is a locally compact Hausdorff space. C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaston Julia
Gaston Maurice Julia (3 February 1893 – 19 March 1978) was a French Algerian mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoit Mandelbrot; the Julia and Mandelbrot fractals are closely related. He founded, independently with Pierre Fatou, the modern theory of holomorphic dynamics. Military service Julia was born in the Algerian town of Sidi Bel Abbes, at the time governed by the French. During his youth, he had an interest in mathematics and music. His studies were interrupted at the age of 21, when France became involved in World War I and Julia was conscripted to serve with the army. During an attack he suffered a severe injury, losing his nose. His many operations to remedy the situation were all unsuccessful, and for the rest of his life he resigned himself to wearing a leather strap around the area where his nose had been. Career in mathematics Julia gained attention for his mathematical work at the age o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michèle Vergne
Michèle Vergne (born August 29, 1943, in L’Isle-Adam, Val d´Oise) is a French mathematician, specializing in analysis and representation theory. Life and work Michèle Vergne studied from 1962 to 1966 at the École Normal Supérieure de jeunes filles, which today is part of the ENS. She wrote her diploma thesis in 1966 with Claude Chevalley, entitled "Variété des algèbres de Lie nilpotentes" and her doctoral thesis in 1971 under the supervision of Jacques Dixmier ("Recherches sur les groupes et les algèbres de Lie") at the University of Paris. She is currently Directeur de Recherche at CNRS. Vergne worked in the construction of unitary representations of Lie groups using coadjoint orbits of the Lie algebras. She proved a generalized Poisson summation formula (called the Poisson-Plancherel formula), which is the integral of a function on adjoint orbits with their Fourier transformation integrals on coadjoint "quantized" orbits. Further, she studied the index the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invited Speaker At The International Congress Of Mathematicians
This is a list of International Congresses of Mathematicians Plenary and Invited Speakers. Being invited to talk at an International Congress of Mathematicians has been called "the equivalent, in this community, of an induction to a hall of fame." The current list of Plenary and Invited Speakers presented here is based on the ICM's post-WW II terminology, in which the one-hour speakers in the morning sessions are called "Plenary Speakers" and the other speakers (in the afternoon sessions) whose talks are included in the ICM published proceedings are called "Invited Speakers". In the pre-WW II congresses the Plenary Speakers were called "Invited Speakers". By congress year 1897, Zürich *Jules Andrade *Léon Autonne * Émile Borel * N. V. Bougaïev * Francesco Brioschi * Hermann Brunn *Cesare Burali-Forti * Charles Jean de la Vallée Poussin * Gustaf Eneström * Federigo Enriques * Gino Fano *Zoel García de Galdeano *Francesco Gerbaldi * Paul Gordan *Jacques Hadamard * Adolf Hurw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vanderbilt University. He was awarded the Fields Medal in 1982. Career Source: Academic career timeline: (1966–1970) – Bachelor's degree from the École Normale Supérieure (now part of Paris Sciences et Lettres University). (1973) – doctorate from Pierre and Marie Curie University, Paris, France (1970–1974) – appointment at the French National Centre for Scientific Research, Paris (1975) – Queen's University at Kingston, Ontario, Canada (1976–1980) – the University of Paris VI (1979 – present) – the Institute of Advanced Scientific Studies, Bures-sur-Yvette, France (1981–1984) – the French National Centre for Scientific Research, Paris (1984–2017) – the , Paris (2003–2011) – Vanderbilt University, Na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michel Duflo
Michel Duflo (born 15 August 1943) is a French mathematician who works in the representation theory of Lie groups. Life From 1962, Duflo studied at the École normale supérieure and received a doctorate under the supervision of Jacques Dixmier. Currently, he is an emeritus professor at the University of Paris VII (Denis Diderot) at the Institut de Mathématiques de Jussieu, and at the École normale supérieure. Duflo has worked on the orbit method of Alexander Kirillov. He introduced the Duflo isomorphism, an isomorphism between the center of the enveloping algebra of a finite-dimensional Lie algebra and the invariants of its symmetric algebra. In 1974 he was an invited speaker at the International Congress of Mathematicians in Vancouver (''Inversion formula and invariant differential operators on solvable Lie groups''). Duflo received the Prix Le Conte of the French Academy of Sciences; in 1986 he became a corresponding member of the Academy. His students include Laure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicole Berline
Nicole Berline (born 1944) is a French mathematician. Life and work Berline studied from 1963 to 1966 at the and she was as an exchange student at the Moscow State University in Moscow in 1966/67. In 1967, she taught at the ENS de Jeunes filles and in 1971, she worked for the CNRS (Attachée de recherches). In 1974 she received her doctorate at the University of Paris under the supervision of Jacques Dixmier (). In 1976/77 she was a visiting professor at the University of California, Berkeley. In 1977 she became a professor at the University of Rennes 1 and she has taught at the Ecole Polytechnique since 1984. She worked in the index theory of elliptic differential operators along the lines of the Atiyah-Singer index theorem and symplectic geometry. Publications *With Ezra Getzler, Michèle Vergne Michèle Vergne (born August 29, 1943, in L’Isle-Adam, Val d´Oise) is a French mathematician, specializing in analysis and representation theory. Life and work Michèle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
