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Michael Atiyah
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the Fields Medal in 1966 and the Abel Prize in 2004. Life Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at the University of Oxford and the University of Cambridge and in the United States at the Institute for Advanced Study. He was the President of the Royal Society (1990–1995), founding director of the Isaac Newton Institute (1990–1996), master of Trinity College, Cambridge (1990–1997), chancellor of the University of Leicester (1995–2005), and the President of the Royal Society of Edinburgh (2005–2008). From 1997 until his death, he was an honorary professor in the University of Edinburgh. Atiyah's mathematical collaborators included Raoul Bott, Friedrich Hirzebruch and Isadore Sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hampstead
Hampstead () is an area in London, which lies northwest of Charing Cross, and extends from Watling Street, the A5 road (Roman Watling Street) to Hampstead Heath, a large, hilly expanse of parkland. The area forms the northwest part of the London Borough of Camden, a borough in Inner London which for the purposes of the London Plan is designated as part of Central London. Hampstead is known for its intellectual, liberal, artistic, musical, and literary associations. It has some of the most expensive housing in the London area. Hampstead has more millionaires within its boundaries than any other area of the United Kingdom.Wade, David"Whatever happened to Hampstead Man?" ''The Daily Telegraph'', 8 May 2004 (retrieved 3 March 2016). History Toponymy The name comes from the Old English, Anglo-Saxon words ''ham'' and ''stede'', which means, and is a cognate of, the Modern English "homestead". To 1900 Early records of Hampstead can be found in a grant by King Ethelred the Unread ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Leicester
, mottoeng = So that they may have life , established = , type = public research university , endowment = £20.0 million , budget = £326 million , chancellor = David Willetts , vice_chancellor = Nishan Canagarajah , head_label = Visitor , head = The King , academic_staff = 1,705 (2018/19) , administrative_staff = 2,205 (2018/19) , students = () , undergrad = () , postgrad = () , city = Leicester , country = England, UK , coordinates = , campus = Urban parkland , colours = , website = , logo = UniOfLeicesterLogo.svg , logo_size = 250px , affiliations = ACUAMBA EMUA EUA Sutton 30 M5 UniversitiesUniversities UK The University of Leicester ( ) is a public research university based in Leicester, England. The main campus is south of the city centre, adjacent to Victoria Park. The university's predecessor, University College, Leicester, gained university status in 1957. The university had an income of £323.1 million in 2019/20, of which £5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atiyah–Hirzebruch Spectral Sequence
In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet, it relates the generalized cohomology groups : E^i(X) with 'ordinary' cohomology groups H^j with coefficients in the generalized cohomology of a point. More precisely, the E_2 term of the spectral sequence is H^p(X;E^q(pt)), and the spectral sequence converges conditionally to E^(X). Atiyah and Hirzebruch pointed out a generalization of their spectral sequence that also generalizes the Serre spectral sequence, and reduces to it in the case where E=H_. It can be derived from an exact couple that gives the E_1 page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with E. In detail, assume X to be the total space of a Serre fibration with fibre F and base space B. The filtration of B by its n-skeletons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floer Homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds. Floer homology is typically defined by associating to the object of interest an infinite-dimensional manifold and a real valued function on it. In the sympl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atiyah–Bott Fixed-point Theorem
In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds ''M'', which uses an elliptic complex on ''M''. This is a system of elliptic differential operators on vector bundles, generalizing the de Rham complex constructed from smooth differential forms which appears in the original Lefschetz fixed-point theorem. Formulation The idea is to find the correct replacement for the Lefschetz number, which in the classical result is an integer counting the correct contribution of a fixed point of a smooth mapping f\colon M \to M. Intuitively, the fixed points are the points of intersection of the graph of ''f'' with the diagonal (graph of the identity mapping) in M\times M, and the Lefschetz number thereby becomes an intersection number. The Atiyah–Bott theorem is an equation in which the LHS must be the outcome of a global topological (homological) calcul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 In topology, a branch of mathematics, Borel's theorem, due to , says the cohomology ring of a classifying space or a classifying stack is a polynomial ring. See also *Atiyah–Bott formula In algebraic geometry, the Atiyah–Bott formula s ..., 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] |
Flip (mathematics)
In algebraic geometry, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up along a relative canonical ring. In dimension 3 flips are used to construct minimal models, and any two birationally equivalent minimal models are connected by a sequence of flops. It is conjectured that the same is true in higher dimensions. The minimal model program The minimal model program can be summarised very briefly as follows: given a variety X, we construct a sequence of contractions X = X_1\rightarrow X_2 \rightarrow \cdots \rightarrow X_n , each of which contracts some curves on which the canonical divisor K_ is negative. Eventually, K_ should become nef (at least in the case of nonnegative Kodaira dimension), which is the desired result. The major technical problem is that, at some stage, the variety X_i may become 'too singular', in the sense that the canonical divisor K_ is no longer a Cartier divisor, so the intersection numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atiyah Conjecture On Configurations
In mathematics, the Atiyah conjecture on configurations is a conjecture introduced by stating that a certain ''n'' by ''n'' matrix depending on ''n'' points in R3 is always non-singular. See also *Berry–Robbins problem In mathematics, the Berry–Robbins problem asks whether there is a continuous map from configurations of ''n'' points in R3 to the flag manifold ''U''(''n'')/''T'n'' that is compatible with the action of the symmetric group on ''n'' points. It ... References * * {{Portal bar, Mathematics Conjectures Unsolved problems in geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atiyah Conjecture
In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of l^2-Betti numbers. History In 1976, Michael Atiyah introduced l^2-cohomology of manifolds with a free co-compact action of a discrete countable group (e.g. the universal cover of a compact manifold together with the action of the fundamental group by deck transformations.) Atiyah defined also numbers as von Neumann dimensions of the resulting groups, and computed several examples, which all turned out to be rational numbers. He therefore asked if it is possible for l^2-Betti numbers to be irrational. Since then, various researchers asked more refined questions about possible values of l^2-Betti numbers, all of which are customarily referred to as "Atiyah conjecture". Results Many positive results were proven by Peter Linnell. For example, if the group acting is a free group, then the l^2-Betti numbers are integers. The most general question open as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atiyah Algebroid
In mathematics, the Atiyah algebroid, or Atiyah sequence, of a principal '' G''-bundle '' P'' over a manifold '' M'', where '' G'' is a Lie group, is the Lie algebroid of the gauge groupoid of '' P''. Explicitly, it is given by the following short exact sequence of vector bundles over '' M'': : 0 \to P\times_G \mathfrak g\to TP/G \to TM\to 0. It is named after Michael Atiyah, who introduced the construction to study the existence theory of complex analytic connections. It plays a crucial example in the integrability of (transitive) Lie algebroids, and it has applications in gauge theory and geometric mechanics. Definitions As a sequence For any fiber bundle P over a manifold M, the differential d\pi of the projection \pi: P \to M defines a short exact sequence : 0 \to VP \to TP \xrightarrow \pi^* TM\to 0 of vector bundles over P, where the vertical bundle VP is the kernel of d\pi. If '' P'' is a principal '' G''-bundle, then the group '' G'' acts on the vector bundles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Founded in 1546 by Henry VIII, King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any college at either Cambridge or University of Oxford, Oxford. Trinity has some of the most distinctive architecture in Cambridge with its Trinity Great Court, Great Court said to be the largest enclosed courtyard in Europe. Academically, Trinity performs exceptionally as measured by the Tompkins Table (the annual unofficial league table of Cambridge colleges), coming top from 2011 to 2017. Trinity was the top-performing college for the 2020-21 undergraduate exams, obtaining the highest percentage of good honours. Members of Trinity have been awarded 34 Nobel Prizes out of the 121 received by members of Cambridge University (the highest of any college at either Oxford or Cambridge). Members of the college have received four Fields Medals, one Turing Award and one Abel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manchester Grammar School
The Manchester Grammar School (MGS) in Manchester, England, is the largest independent school (UK), independent day school for boys in the United Kingdom. Founded in 1515 as a Grammar school#free tuition, free grammar school next to Manchester Cathedral, Manchester Parish Church, it moved in 1931 to its present site at Rusholme. In accordance with its founder's wishes, MGS remains a predominantly academic school and belongs to the Headmasters' and Headmistresses' Conference. In the post-war period, MGS was a direct-grant grammar school. It chose to become an independent school in 1976 after the Labour Party (UK), Labour government abolished the Direct grant grammar school, Direct Grant System. Fees for 2016–2017 were £11,970 per annum. Motto, coat of arms and school badges The school's motto is ''wikt:sapere aude, Sapere Aude'' ("Dare to be Wise"), which was also the motto of the County Borough Council, council of the former County Borough of Oldham (now, with the same coat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
