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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] [Amazon] |
Baum And Connes
Baum is a German surname meaning "tree" (not to be confused with the French language, French surname Baume (surname), Baume). Notable people with this surname include: * Andreas Baum (born 1978), German politician * Bernie Baum (1929–1993), American songwriter * Bruce Baum (born 1952), American comedian * Carl Edward Baum (1940–2010), American electrical engineer * Carol Baum, American film producer * Christina Baum (born 1956), German politician * Dale Baum, (born 1943), American historian and professor * Walter Emerson Baum#Family life, Edgar Schofield Baum (1916–2006), American artist, physician, and WW2 combat medical officer * Fran Baum, Australian social scientist * Frank Baum (footballer) (born 1956), German footballer * Frank Joslyn Baum (1883–1958), American lawyer, soldier, writer and film producer * Friedrich Baum (1727–1777), colonel in British service during the American revolutionary war * Gerhart Baum (1932–2025), German lawyer and politician, mini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Atiyah–Singer Index Theorem
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics. History The index problem for elliptic differential operators was posed by Israel Gel'fand. He noticed the homotopy invariance of the index, and asked for a formula for it by means of topological invariants. Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem. Friedrich Hirzebruch and Armand Borel had proved the integrality of the  genus of a spin manifold, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gennadi Kasparov
Gennadi () is a Greece, Greek village, seat of the municipal unit of South Rhodes, on the island of Rhodes, South Aegean region. In 2021 its population was 1,224. The village is 64 km from the town of Rhodes (city), Rhodes and 27 km from ancient Lindos and 65 km from the Rhodes International Airport, Airport of Rhodes. It is an agriculture place with a bit of tourism located on the south east side of Rhodes coast. References External linksSouth Rhodes website Populated places in Rhodes {{SouthAegean-geo-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Poincaré Duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology (mathematics), homology and cohomology group (mathematics), groups of manifolds. It states that if ''M'' is an ''n''-dimensional Orientability, oriented closed manifold (Compact space, compact and without boundary), then the ''k''th cohomology group of ''M'' is Group isomorphism, isomorphic to the th homology group of ''M'', for all integers ''k'' : H^k(M) \cong H_(M). Poincaré duality holds for any coefficient ring (mathematics), 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 th Betti numbers of a closed (i.e., compact and witho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Georges Skandalis
Georges Skandalis (; 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 counterexamples to a generalization of the Baum–C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 per ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 is derived . 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 often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Paul Baum (mathematician)
Paul Frank Baum 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 2012-03-14. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Although many examples of morphisms are structure-preserving maps, morphisms need not to be maps, but they can be composed in a way that is similar to function composition. Morphisms and objects are constituents of a category. Morphisms, also called ''maps'' or ''arrows'', relate two objects called the ''source'' and the ''target'' of the morphism. There is a partial operation, called ''composition'', on the morphisms of a category that is defined if the target of the first morphism equals the source of the second morphism. The composition of morphisms behaves like function composition ( associativity of composition when it is defined, and existence of an identity morphism for every object). Morphisms and categories recur in much of co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
