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J. H. C. Whitehead
John Henry Constantine Whitehead FRS (11 November 1904 – 8 May 1960), known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai (then known as Madras), in India, and died in Princeton, New Jersey, in 1960. Life J. H. C. (Henry) Whitehead was the son of the Right Rev. Henry Whitehead, Bishop of Madras, who had studied mathematics at Oxford, and was the nephew of Alfred North Whitehead and Isobel Duncan. He was brought up in Oxford, went to Eton and read mathematics at Balliol College, Oxford. After a year working as a stockbroker, at Buckmaster & Moore, he started a PhD in 1929 at Princeton University. His thesis, titled ''The representation of projective spaces'', was written under the direction of Oswald Veblen in 1930. While in Princeton, he also worked with Solomon Lefschetz. He became a fellow of Balliol in 1933. In 1934 he married the concert pianist Barbara Smyth, great-great-granddaughter of Elizabeth Fry and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chennai
Chennai (, ), formerly known as Madras ( the official name until 1996), is the capital city of Tamil Nadu, the southernmost Indian state. The largest city of the state in area and population, Chennai is located on the Coromandel Coast of the Bay of Bengal. According to the 2011 Indian census, Chennai is the sixth-most populous city in the country and forms the fourth-most populous urban agglomeration. The Greater Chennai Corporation is the civic body responsible for the city; it is the oldest city corporation of India, established in 1688—the second oldest in the world after London. The city of Chennai is coterminous with Chennai district, which together with the adjoining suburbs constitutes the Chennai Metropolitan Area, the 36th-largest urban area in the world by population and one of the largest metropolitan economies of India. The traditional and de facto gateway of South India, Chennai is among the most-visited Indian cities by foreign tourists. It was ranked the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Problem
In group theory, a branch of abstract algebra, the Whitehead problem is the following question: Saharon Shelah proved that Whitehead's problem is independent of ZFC, the standard axioms of set theory. Refinement Assume that ''A'' is an abelian group such that every short exact sequence :0\rightarrow\mathbb\rightarrow B\rightarrow A\rightarrow 0 must split if ''B'' is also abelian. The Whitehead problem then asks: must ''A'' be free? This splitting requirement is equivalent to the condition Ext1(''A'', Z) = 0. Abelian groups ''A'' satisfying this condition are sometimes called Whitehead groups, so Whitehead's problem asks: is every Whitehead group free? It should be mentioned that if this condition is strengthened by requiring that the exact sequence :0\rightarrow C\rightarrow B\rightarrow A\rightarrow 0 must split for any abelian group ''C'', then it is well known that this is equivalent to ''A'' being free. ''Caution'': The converse of Whitehead's problem, namely that ever ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biographical Memoirs Of Fellows Of The Royal Society
The ''Biographical Memoirs of Fellows of the Royal Society'' is an academic journal on the history of science published annually by the Royal Society. It publishes obituaries of Fellows of the Royal Society. It was established in 1932 as ''Obituary Notices of Fellows of the Royal Society'' and obtained its current title in 1955, with volume numbering restarting at 1. Prior to 1932, obituaries were published in the ''Proceedings of the Royal Society''. The memoirs are a significant historical record and most include a full bibliography of works by the subjects. The memoirs are often written by a scientist of the next generation, often one of the subject's own former students, or a close colleague. In many cases the author is also a Fellow. Notable biographies published in this journal include Albert Einstein, Alan Turing, Bertrand Russell, Claude Shannon, Clement Attlee, Ernst Mayr, and Erwin Schrödinger. Each year around 40 to 50 memoirs of deceased Fellows of the Royal Soci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fellow Of The Royal Society
Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural science, natural knowledge, including mathematics, engineering science, and medical science". Fellow, Fellowship of the Society, the oldest known scientific academy in continuous existence, is a significant honour. It has been awarded to many eminent scientists throughout history, including Isaac Newton (1672), Michael Faraday (1824), Charles Darwin (1839), Ernest Rutherford (1903), Srinivasa Ramanujan (1918), Albert Einstein (1921), Paul Dirac (1930), Winston Churchill (1941), Subrahmanyan Chandrasekhar (1944), Dorothy Hodgkin (1947), Alan Turing (1951), Lise Meitner (1955) and Francis Crick (1959). More recently, fellowship has been awarded to Stephen Hawking (1974), David Attenborough (1983), Tim Hunt (1991), Elizabeth Blackburn (1992), Tim Berners-Lee (2001), Venki R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Senior Berwick Prize
The Berwick Prize and Senior Berwick Prize are two prizes of the London Mathematical Society awarded in alternating years in memory of William Edward Hodgson Berwick, a previous Vice-President of the LMS. Berwick left some money to be given to the society to establish two prizes. His widow Daisy May Berwick gave the society the money and the society established the prizes, with the first Senior Berwick Prize being presented in 1946 and the first Junior Berwick Prize the following year. The prizes are awarded "in recognition of an outstanding piece of mathematical research ... published by the Society" in the eight years before the year of the award. The Berwick Prize was known as the Junior Berwick Prize up to 1999, and was given its current name for the 2001 award. Senior Berwick Prize winners Source:List of LMS prize winners L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spanier–Whitehead Duality
In mathematics, Spanier–Whitehead duality is a duality theory in homotopy theory, based on a geometrical idea that a topological space ''X'' may be considered as dual to its complement in the ''n''-sphere, where ''n'' is large enough. Its origins lie in Alexander duality theory, in homology theory, concerning complements in manifolds. The theory is also referred to as ''S-duality'', but this can now cause possible confusion with the S-duality of string theory. It is named for Edwin Spanier and J. H. C. Whitehead, who developed it in papers from 1955. The basic point is that sphere complements determine the homology, but not the homotopy type, in general. What is determined, however, is the stable homotopy type, which was conceived as a first approximation to homotopy type. Thus Spanier–Whitehead duality fits into stable homotopy theory. Statement Let ''X'' be a compact neighborhood retract in \R^n. Then X^+ and \Sigma^\Sigma'(\R^n \setminus X) are dual obje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead's Point-free Geometry
In mathematics, point-free geometry is a geometry whose primitive ontological notion is ''region'' rather than point. Two axiomatic systems are set out below, one grounded in mereology, the other in mereotopology and known as ''connection theory''. Point-free geometry was first formulated in Whitehead (1919, 1920), not as a theory of geometry or of spacetime, but of "events" and of an "extension relation" between events. Whitehead's purposes were as much philosophical as scientific and mathematical. Formalizations Whitehead did not set out his theories in a manner that would satisfy present-day canons of formality. The two formal first-order theories described in this entry were devised by others in order to clarify and refine Whitehead's theories. The domain of discourse for both theories consists of "regions." All unquantified variables in this entry should be taken as tacitly universally quantified; hence all axioms should be taken as universal closures. No axiom require ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead's Lemma (Lie Algebras)
In homological algebra, Whitehead's lemmas (named after J. H. C. Whitehead) represent a series of statements regarding representation theory of finite-dimensional, semisimple Lie algebras in characteristic zero. Historically, they are regarded as leading to the discovery of Lie algebra cohomology. One usually makes the distinction between Whitehead's first and second lemma for the corresponding statements about first and second order cohomology, respectively, but there are similar statements pertaining to Lie algebra cohomology in arbitrary orders which are also attributed to Whitehead. The first Whitehead lemma is an important step toward the proof of Weyl's theorem on complete reducibility. Statements Without mentioning cohomology groups, one can state Whitehead's first lemma as follows: Let \mathfrak be a finite-dimensional, semisimple Lie algebra over a field of characteristic zero, ''V'' a finite-dimensional module over it, and f\colon \mathfrak \to V a linear map such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead's Algorithm
Whitehead's algorithm is a mathematical algorithm in group theory for solving the automorphic equivalence problem in the finite rank free group ''Fn''. The algorithm is based on a classic 1936 paper of J. H. C. Whitehead. J. H. C. Whitehead, ''On equivalent sets of elements in a free group'', Ann. of Math. (2) 37:4 (1936), 782–800. It is still unknown (except for the case ''n'' = 2) if Whitehead's algorithm has polynomial time complexity. Statement of the problem Let F_n=F(x_1,\dots, x_n) be a free group of rank n\ge 2 with a free basis X=\. The automorphism problem, or the automorphic equivalence problem for F_n asks, given two freely reduced words w, w'\in F_n whether there exists an automorphism \varphi\in \operatorname(F_n) such that \varphi(w)=w'. Thus the automorphism problem asks, for w, w'\in F_n whether \operatorname(F_n)w=\operatorname(F_n)w'. For w, w'\in F_n one has \operatorname(F_n)w=\operatorname(F_n)w' if and only if \operatorname(F_n) \operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Postnikov System
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree k agrees with the truncated homotopy type of the original space X. Postnikov systems were introduced by, and are named after, Mikhail Postnikov. Definition A Postnikov system of a path-connected space X is an inverse system of spaces :\cdots \to X_n \xrightarrow X_\xrightarrow \cdots \xrightarrow X_2 \xrightarrow X_1 \xrightarrow * with a sequence of maps \phi_n\colon X \to X_n compatible with the inverse system such that # The map \phi_n\colon X \to X_n induces an isomorphism \pi_i(X) \to \pi_i(X_n) for every i\leq n. # \pi_i(X_n) = 0 for i > n. # Each map p_n\colon X_n \to X_ is a fibration, and so the fiber F_n is an Eilenberg–MacLane space, K(\pi_n(X),n). The first two conditions imply that X_1 is also a K(\pi_1(X),1)-space. More generally, if X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Torsion
In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau(f) which is an element in the Whitehead group \operatorname(\pi_1(Y)). These concepts are named after the mathematician J. H. C. Whitehead. The Whitehead torsion is important in applying surgery theory to non- simply connected manifolds of dimension > 4: for simply-connected manifolds, the Whitehead group vanishes, and thus homotopy equivalences and simple homotopy equivalences are the same. The applications are to differentiable manifolds, PL manifolds and topological manifolds. The proofs were first obtained in the early 1960s by Stephen Smale, for differentiable manifolds. The development of handlebody theory allowed much the same proofs in the differentiable and PL categories. The proofs are much harder in the topological category, requiring the theory of Robion Kirby and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Theorem
In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping ''f'' between CW complexes ''X'' and ''Y'' induces isomorphisms on all homotopy groups, then ''f'' is a homotopy equivalence. This result was proved by J. H. C. Whitehead in two landmark papers from 1949, and provides a justification for working with the concept of a CW complex that he introduced there. It is a model result of algebraic topology, in which the behavior of certain algebraic invariants (in this case, homotopy groups) determines a topological property of a mapping. Statement In more detail, let ''X'' and ''Y'' be topological spaces. Given a continuous mapping :f\colon X \to Y and a point ''x'' in ''X'', consider for any ''n'' ≥ 1 the induced homomorphism :f_*\colon \pi_n(X,x) \to \pi_n(Y,f(x)), where π''n''(''X'',''x'') denotes the ''n''-th homotopy group of ''X'' with base point ''x''. (For ''n'' = 0, π0(''X'') just means the set of path components o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
