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Raoul Bott
Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous foundational 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. Early life Bott was born in Budapest, Kingdom of Hungary (1920–1946), Hungary, the son of Margit Kovács and Rudolph Bott. His father was of Austrian descent, and his mother was of Hungarian Jewish descent; Bott was raised a Catholic by his mother and stepfather in Bratislava, Czechoslovakia, now the capital of Slovakia. Bott grew up in Czechoslovakia and spent his working life in the United States. His family emigrated to Canada in 1938, and subsequently he served in the Canadian Forces, Canadian Army in Europe during World War II. Career Bott later went to college at McGill University in Montreal, where he studied electrical engineering. He then earned a PhD in math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Budapest
Budapest is the Capital city, capital and List of cities and towns of Hungary, most populous city of Hungary. It is the List of cities in the European Union by population within city limits, tenth-largest city in the European Union by population within city limits and the List of cities and towns on the river Danube, second-largest city on the river Danube. The estimated population of the city in 2025 is 1,782,240. This includes the city's population and surrounding suburban areas, over a land area of about . Budapest, which is both a List of cities and towns of Hungary, city and Counties of Hungary, municipality, forms the centre of the Budapest metropolitan area, which has an area of and a population of 3,019,479. It is a primate city, constituting 33% of the population of Hungary. The history of Budapest began when an early Celts, Celtic settlement transformed into the Ancient Rome, Roman town of Aquincum, the capital of Pannonia Inferior, Lower Pannonia. The Hungarian p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Daniel Quillen
Daniel Gray Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978. From 1984 to 2006, he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. Education and career Quillen was born in Orange, New Jersey, and attended Newark Academy. He entered Harvard University, where he earned both his AB, in 1961, and his PhD in 1964; the latter completed under the supervision of Raoul Bott, with a thesis in partial differential equations. He was a Putnam Fellow in 1959. Quillen obtained a position at the Massachusetts Institute of Technology after completing his doctorate. He also spent a number of years at several other universities. He visited France twice: first as a Sloan Fellow in Paris, during the academic year 1968–69, where he was greatly influenced by Grothendieck, and then, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Atiyah–Bott Formula
In algebraic geometry, the Atiyah–Bott formula says the cohomology ring :\operatorname^*(\operatorname_G(X), \mathbb_l) of the moduli stack of principal bundles is a free algebra, free supercommutative algebra, graded-commutative algebra on certain homogeneous generators. The original work of Michael Atiyah and Raoul Bott concerned the integral cohomology ring of \operatorname_G(X). See also *Borel's theorem, which says that the cohomology ring of a classifying stack is a polynomial ring. Notes References * * Theorems in algebraic geometry {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Virasoro Group
In abstract algebra, the Virasoro group or Bott–Virasoro group (often denoted by ''Vir'') is an infinite-dimensional Lie group defined as the universal central extension of the group of diffeomorphisms of the circle. The corresponding Lie algebra is the Virasoro algebra, which has a key role in conformal field theory (CFT) and string theory. The group is named after Miguel Ángel Virasoro and Raoul Bott. Background An orientation-preserving diffeomorphism of the circle S^1, whose points are labelled by a real coordinate x subject to the identification x\sim x+2\pi, is a smooth map f:\mathbb\to\mathbb:x\mapsto f(x) such that f(x+2\pi)=f(x)+2\pi and f'(x)>0. The set of all such maps spans a group, with multiplication given by the composition of diffeomorphisms. This group is the universal cover of the group of orientation-preserving diffeomorphisms of the circle, denoted as \widetilde^+(S^1). Definition The Virasoro group is the universal central extension of \widetilde^+(S^1). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bott–Taubes Polytope
In geometry, the cyclohedron is a -dimensional polytope where can be any non-negative integer. It was first introduced as a combinatorial object by Raoul Bott and Clifford Taubes and, for this reason, it is also sometimes called the Bott–Taubes polytope. It was later constructed as a polytope by Martin Markl and by Rodica Simion. Rodica Simion describes this polytope as an associahedron of type B. The cyclohedron appears in the study of knot invariants. Construction Cyclohedra belong to several larger families of polytopes, each providing a general construction. For instance, the cyclohedron belongs to the generalized associahedra that arise from cluster algebra, and to the graph-associahedra, a family of polytopes each corresponding to a graph. In the latter family, the graph corresponding to the d-dimensional cyclohedron is a cycle on d+1 vertices. In topological terms, the configuration space of d+1 distinct points on the circle S^1 is a (d+1)-dimensional manifold, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bott–Samelson Resolution
In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities. It was introduced by in the context of compact Lie groups. The algebraic formulation is independently due to and . Definition Let ''G'' be a connected reductive complex algebraic group, ''B'' a Borel subgroup and ''T'' a maximal torus contained in ''B''. Let w \in W = N_G(T)/T. Any such ''w'' can be written as a product of reflections by simple roots. Fix minimal such an expression: :\underline = (s_, s_, \ldots, s_) so that w = s_ s_ \cdots s_. (''ℓ'' is the length of ''w''.) Let P_ \subset G be the subgroup generated by ''B'' and a representative of s_. Let Z_ be the quotient: :Z_ = P_ \times \cdots \times P_/B^\ell with respect to the action of B^\ell by :(b_1, \ldots, b_\ell) \cdot (p_1, \ldots, p_\ell) = (p_1 b_1^, b_1 p_2 b_2^, \ldots, b_ p_\ell b_\ell^). It is a smooth projective variety. Writing X_w = \overline / B = (P_ \cdots P_)/B for the Schubert v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Network Synthesis
Network synthesis is a design technique for linear electrical circuits. Synthesis starts from a prescribed impedance function of frequency or frequency response and then determines the possible networks that will produce the required response. The technique is to be compared to network analysis in which the response (or other behaviour) of a given circuit is calculated. Prior to network synthesis, only network analysis was available, but this requires that one already knows what form of circuit is to be analysed. There is no guarantee that the chosen circuit will be the closest possible match to the desired response, nor that the circuit is the simplest possible. Network synthesis directly addresses both these issues. Network synthesis has historically been concerned with synthesising passive networks, but is not limited to such circuits. The field was founded by Wilhelm Cauer after reading Ronald M. Foster's 1924 paper '' A reactance theorem''. Foster's theorem provided a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bott–Chern Cohomology
In complex geometry in mathematics, Bott–Chern cohomology is a cohomology theory for complex manifolds. It serves as a bridge between de Rham cohomology, which is defined for real manifolds which in particular underlie complex manifolds, and Dobeault cohomology, which is its analogue for complex manifolds. A direct comparison between these cohomology theories through canonical maps is not possible, but Bott–Chern cohomology canonically maps into both. A similiar cohomology theory, into which both map and which hence also serves as a bridge is Aeppli cohomology. Bott–Chern cohomology is named after Raoul Bott and Shiing-Chen Chern, who introduced it in 1965. Definition For a complex manifold X, its ''Bott–Chern cohomology'' is given by:Bott & Chern 1965, p. 74Angella & Tomassini 2014, p. 1 & 1.1. Bott-Chern cohomologyAngella 2015, p. 5 : H_\mathrm^(X) :=\ker(\mathrm_)/\operatorname(\partial_\overline\partial_) =\left(\ker(\partial_)\cap\ker(\overline\partial_)\right)/ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bott Residue Formula
In mathematics, the Bott residue formula, introduced by , describes a sum over the fixed points of a holomorphic vector field of a compact complex manifold. Statement If ''v'' is a holomorphic vector field on a compact complex manifold ''M'', then : \sum_\frac = \int_M P(i\Theta/2\pi) where *The sum is over the fixed points ''p'' of the vector field ''v'' *The linear transformation ''A''''p'' is the action induced by ''v'' on the holomorphic tangent space at ''p'' *''P'' is an invariant polynomial function of matrices of degree dim(''M'') *Θ is a curvature matrix of the holomorphic tangent bundle See also *Atiyah–Bott fixed-point theorem * Holomorphic Lefschetz fixed-point formula References * *{{Citation , last1=Griffiths , first1=Phillip , author1-link=Phillip Griffiths , last2=Harris , first2=Joseph , author2-link=Joe Harris (mathematician) , title=Principles of algebraic geometry , publisher=John Wiley & Sons John Wiley & Sons, Inc., commonly known as Wiley ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bott Periodicity Theorem
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory. There are corresponding period-8 phenomena for the matching theories, (real number, real) KO-theory and (quaternionic) KSp-theory, associated to the real orthogonal group and the quaternionic symplectic group, respectively. The J-homomorphism is a homomorphism from the homotopy groups of orthogonal groups to stable homotopy groups of spheres, which causes the period 8 Bott periodicity to be visible in the stable homotopy groups of sph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bott Cannibalistic Class
In mathematics, the Bott cannibalistic class, introduced by , is an element \theta_k(V) of the representation ring of a compact Lie group that describes the action of the Adams operation \psi^k on the Thom class \lambda_V of a complex representation Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ... V. The term "cannibalistic" for these classes was introduced by . References * * Representation theory K-theory {{mathematics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Loring W
Loring may refer to: People Given name * Loring Woart Bailey (1839–1925), American-Canadian geologist, botanist and university professor * Loring M. Black Jr. (1886–1956), American lawyer and politician * Loring Buzzell (1927–1959), American music publisher * Loring Christie (1885–1941), Canadian diplomat * Loring Coes (1812–1906), American inventor, industrialist and politician * Loring Danforth (born 1949), American professor of anthropology * Loring D. Dewey (1791–1867), early 19th-century Presbyterian minister * Loring Mandel (1928–2020), American playwright and screenwriter * Loring McMillen (1928–1991), American historian * Loring Miner (1860–1935), American physician * Loring Schuler (1886–1968), American journalist and editor * Loring Smith (1890–1981), American actor * Loring W. Tu, Taiwanese-American mathematician Surname * Loring (surname), includes a list of people with this surname Places United States *Loring, Alaska, a census-designate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
