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Raoul Bott
Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous basic 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, 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. 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 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 mathematics from Carnegie Mellon University in Pittsburgh in 1949. His thesis, titled ''Electrical Network Theory'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Budapest
Budapest (, ; ) is the capital and most populous city of Hungary. It is the ninth-largest city in the European Union by population within city limits and the second-largest city on the Danube river; the city has an estimated population of 1,752,286 over a land area of about . Budapest, which is both a city and county, forms the centre of the Budapest metropolitan area, which has an area of and a population of 3,303,786; it is a primate city, constituting 33% of the population of Hungary. The history of Budapest began when an early Celtic settlement transformed into the Roman town of Aquincum, the capital of Lower Pannonia. The Hungarians arrived in the territory in the late 9th century, but the area was pillaged by the Mongols in 1241–42. Re-established Buda became one of the centres of Renaissance humanist culture by the 15th century. The Battle of Mohács, in 1526, was followed by nearly 150 years of Ottoman rule. After the reconquest of Buda in 1686, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nancy Hingston
Nancy Burgess Hingston is a mathematician working in differential geometry. She is a professor emerita of mathematics at The College of New Jersey.. Early life and education Nancy Hingston's father William Hingston was superintendent of the Central Bucks School District in Pennsylvania; her mother was a high school mathematics and computer science teacher. She graduated from the University of Pennsylvania with a double major in mathematics and physics. After a year studying physics as a graduate student, she switched to mathematics, and completed her PhD in 1981 from Harvard University under the supervision of Raoul Bott. Career Before joining TCNJ, she taught at the University of Pennsylvania. She has also been a frequent visitor to the Institute for Advanced Study, and has been involved with the Program for Women and Mathematics at the Institute for Advanced Study since its founding in 1994. Contributions Nancy Hingston made major contributions in Riemannian geometry and Ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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^+( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bott–Taubes Polytope
In geometry, the cyclohedron is a d-dimensional polytope where d 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 is useful in studying 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] |
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 Smooth may refer to: Mathematics * Smooth function, a function that is infinitely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 Wile ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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) 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 spheres. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 In mathematics, an Adams operation, denoted ψ''k'' for natural numbers ''k'', is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduce ... \psi^k on the Thom class \lambda_V of a complex representation 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] |
Eric Weinstein
Eric Ross Weinstein (born October 26, 1965) is an American podcast host and a managing director of Thiel Capital. Education Weinstein received his PhD in mathematical physics from Harvard University in 1992 under the supervision of Raoul Bott. In his dissertation, ''Extension of Self-Dual Yang-Mills Equations Across the Eighth Dimension'', Weinstein showed that the self-dual Yang–Mills equations were not peculiar to dimension four and admitted generalizations to higher dimensions. Career Physics Weinstein left academia after stints at the Massachusetts Institute of Technology and the Hebrew University of Jerusalem. Weinstein was invited to a colloquium by mathematician Marcus du Sautoy at Oxford University's Clarendon Laboratory in May 2013. There he presented his ideas on a theory of everything called ''Geometric Unity''. Physicists expressed skepticism about the theory. Joseph Conlon of Oxford stated that some of the predicted particles would already have been d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Susan Tolman
Susan Tolman is an American mathematician known for her work in symplectic geometry. She is a professor of mathematics at the University of Illinois at Urbana–Champaign, and Lynn M. Martin Professorial Scholar at Illinois. Tolman earned her Ph.D. in 1993 at Harvard University. Her dissertation, ''Group Actions And Cohomology'', was supervised by Raoul Bott. She was awarded a Sloan Research Fellowship The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This program is one of the oldest of its kind in the United States. ... in 1998, and was named Lynn M. Martin Professorial Scholar in 2008–2009. References Year of birth missing (living people) Living people 20th-century American mathematicians 21st-century American mathematicians American women mathematicians University of Illinois Urbana-Champaign faculty Sloan Research Fellows Harvard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty of the University of California, Berkeley (1960–1961 and 1964–1995), where he currently is Professor Emeritus, with research interests in algorithms, numerical analysis and global analysis. Education and career Smale was born in Flint, Michigan and entered the University of Michigan in 1948. Initially, he was a good student, placing into an honors calculus sequence taught by Bob Thrall and earning himself A's. However, his sophomore and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in nuclear physics. However, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale performed poorly in his first years, earning a C average as a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
