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Baum–Connes Conjecture
In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the operator K-theory, K-theory of the reduced C*-algebra of a group theory, group and the K-homology of the classifying space of proper actions of that group. The conjecture sets up a correspondence between different areas of mathematics, with the K-homology of the classifying space being related to geometry, differential operator theory, and homotopy theory, while the K-theory of the group's reduced C*-algebra is a purely analytical object. The conjecture, if true, would have some older famous conjectures as consequences. For instance, the surjectivity part implies the Kaplansky's conjectures#Group rings, Kadison–Kaplansky conjecture for discrete torsion-free groups, and the injectivity is closely related to the Novikov conjecture. The conjecture is also closely related to index theory, as the assembly map \mu is a sort of index, and it plays a major role in Alain Connes' ... [...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] |
Second-countable Space
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mathcal = \_^ of open subsets of T such that any open subset of T can be written as a union of elements of some subfamily of \mathcal. A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have. Many "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (R''n'') with its usual topology is second-countable. Although the usual base of open balls is uncountable, one can restrict to the collection of all open balls with rational radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis. Properties Second-countability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indefinite Orthogonal Group
In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimension of a vector space, dimensional real vector space that leave invariant a nondegenerate form, nondegenerate, symmetric bilinear form of signature of a quadratic form, signature , where . It is also called the pseudo-orthogonal group or generalized orthogonal group. The dimension of the group is . The indefinite special orthogonal group, is the subgroup of consisting of all elements with determinant 1. Unlike in the definite case, is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected and , which has 2 components – see ' for definition and discussion. The signature of the form determines the group up to isomorphism; interchanging ''p'' with ''q'' amounts to replacing the metric by its negative, and so gives the same group. If either ''p'' or ''q'' equals zero, then the group is isomorphic to the ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gennadi Kasparov
Gennadi ( gr, Γεννάδι) is a Greek village, seat of the municipal unit of South Rhodes, on the island of Rhodes, South Aegean region. In 2011 its population was 671. Overview The village is 64 km from the town of Rhodes and 27 km from ancient Lindos Lindos (; grc-gre, Λίνδος) is an archaeological site, a fishing village and a former municipality on the island of Rhodes, in the Dodecanese, Greece. Since the 2011 local government reform it is part of the municipality Rhodes, of which it ... and 65 km from the Airport of Rhodes. It is an agriculture place with a bit of tourism located on the south east side of Rhodes coast. References External links South Rhodes website Populated places in Rhodes {{SouthAegean-geo-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact and without boundary), then the ''k''th cohomology group of ''M'' is isomorphic to the (n-k)th homology group of ''M'', for all integers ''k'' :H^k(M) \cong H_(M). Poincaré duality holds for any coefficient ring, so long as one has taken an orientation with respect to that coefficient ring; in particular, since every manifold has a unique orientation mod 2, Poincaré duality holds mod 2 without any assumption of orientation. History A form of Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of Betti numbers: The ''k''th and (n-k)th Betti numbers of a closed (i.e., compact and without boundary) orientable ''n''-manifold are equal. The ''cohomology'' concept was at that time about 40 y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georges Skandalis
Georges Skandalis ( el, Γεώργιος Σκανδάλης; born 5 November 1955, in Athens) is a Greek and French mathematician, known for his work on noncommutative geometry and operator algebras. After following secondary education and ''classes préparatoires scientifiques'' in the parisian ''Lycée Louis-le-Grand'', Skandalis studied from 1975 at 1979 at ''l’École Normale Supérieure de la rue d’Ulm'' with ''agrégation'' in 1977. From 1979 he was an at the University of Paris VI, where under Alain Connes in 1986 he earned his doctorate (''doctorat d´État''). From 1980 to 1988 he was ''attaché de recherches'' and then ''chargé de recherches'' at CNRS and as of 1988 Professor at the University of Paris VII (in the ''Institut de Mathématiques de Jussieu''). He works on operator algebras, K-theory of operator algebras, groupoids, locally compact quantum groups and singular foliations. In 2002 with Nigel Higson and Vincent Lafforgue, Skandalis published counterexam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vincent Lafforgue
Vincent Lafforgue (born 20 January 1974) is a French mathematician who is active in algebraic geometry, especially in the Langlands program, and a CNRS " Directeur de Recherches" at the Institute Fourier in Grenoble. He is the younger brother of Fields Medalist Laurent Lafforgue. Awards Lafforgue was awarded the 2000 EMS Prize for his contribution to the K-theory of operator algebras: the proof of the Baum–Connes conjecture for discrete co-compact subgroups of \mathrm(3,\mathbb), \mathrm(3,\mathbb), \mathrm(3,\mathbb_p) and some other locally compact groups, and of more general objects. He participated in the International Mathematical Olympiad and wrote two perfect papers in 1990 and 1991, making him one of only three French mathematicians to win two gold medals (besides Joseph Najnudel, 1997–98, and Aurélien Fourré, 2020-21). Lafforgue was an Invited Speaker of the ICM in 2002 in Beijing, China and a Plenary Speaker of the ICM in 2018 in Rio de Janeiro, Brazil. He ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nigel Higson
Nigel David Higson (born 1963) is a Canadian math professor at Pennsylvania State University who received the 1996 Coxeter–James Prize. His doctorate came from Dalhousie University in 1985, under the supervision of Peter Fillmore. He works in the fields of operator algebra and K-theory. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. In 2012 he was chosen as one of the inaugural Fellows of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, .... retrieved 2015-06-12. References < ...
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KK-theory
In mathematics, ''KK''-theory is a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician Gennadi Kasparov in 1980. It was influenced by Atiyah's concept of Fredholm modules for the Atiyah–Singer index theorem, and the classification of extensions of C*-algebras by Lawrence G. Brown, Ronald G. Douglas, and Peter Arthur Fillmore in 1977. In turn, it has had great success in operator algebraic formalism toward the index theory and the classification of nuclear C*-algebras, as it was the key to the solutions of many problems in operator K-theory, such as, for instance, the mere calculation of ''K''-groups. Furthermore, it was essential in the development of the Baum–Connes conjecture and plays a crucial role in noncommutative topology. ''KK''-theory was followed by a series of similar bifunctor constructions such as the ''E''-theory and the bivariant periodic c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Baum (mathematician)
Paul Frank Baum (born 1936) is an American mathematician, the Evan Pugh Professor of Mathematics at Pennsylvania State University. He is known for formulating the Baum–Connes conjecture with Alain Connes in the early 1980s. Baum studied at Harvard University, earning a bachelor's degree ''summa cum laude'' in 1958. He went on to Princeton University for his graduate studies, completing his Ph.D. in 1963 under the supervision of John Coleman Moore and Norman Steenrod. He was several times a visiting scholar at the Institute for Advanced Study (1964–65, 1976–77, 2004) After several visiting positions and an assistant professorship at Princeton, he moved to Brown University in 1967, and remained there until 1987 when he moved to Penn State. He became a distinguished professor in 1991 and was given his named chair in 1996.Curriculum vitae retrieved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
