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Differential Structure
In mathematics, an ''n''- dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold. If ''M'' is already a topological manifold, it is required that the new topology be identical to the existing one. Definition For a natural number ''n'' and some ''k'' which may be a non-negative integer or infinity, an ''n''-dimensional ''C''''k'' differential structure is defined using a ''C''''k''-atlas, which is a set of bijections called charts between a collection of subsets of ''M'' (whose union is the whole of ''M''), and a set of open subsets of \mathbb^: :\varphi_:M\supset W_\rightarrow U_\subset\mathbb^ which are ''C''''k''-compatible (in the sense defined below): Each such map provides a way in which certain subsets of the manifold may be viewed as being like open subsets of \mathbb^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word ''homeomorphism'' comes from the Greek words '' ὅμοιος'' (''homoios'') = similar or same and '' μορφή'' (''morphē'') = shape or form, introduced to mathematics by Henri Poincaré in 1895. Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Michel Kervaire
Michel André Kervaire (26 April 1927 – 19 November 2007) was a French mathematician who made significant contributions to topology and algebra. He introduced the Kervaire semi-characteristic. He was the first to show the existence of topological ''n''-manifolds with no differentiable structure (using the Kervaire invariant), and (with John Milnor) computed the number of exotic spheres in dimensions greater than four. He is also well known for fundamental contributions to high-dimensional knot theory. The solution of the Kervaire invariant problem was announced by Michael Hopkins in Edinburgh on 21 April 2009. Education He was the son of André Kervaire (a French industrialist) and Nelly Derancourt. After completing high school in France, Kervaire pursued his studies at ETH Zurich (1947–1952), receiving a Ph.D. in 1955. His thesis, entitled ''Courbure intégrale généralisée et homotopie'', was written under the direction of Heinz Hopf and Beno Eckmann. Career Kervaire ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook University and one of the five mathematicians to have won the Fields Medal, the Wolf Prize, and the Abel Prize (the others being Serre, Thompson, Deligne, and Margulis.) Early life and career Milnor was born on February 20, 1931, in Orange, New Jersey. His father was J. Willard Milnor and his mother was Emily Cox Milnor. As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and also proved the Fáry–Milnor theorem when he was only 19 years old. Milnor graduated with an A.B. in mathematics in 1951 after completing a senior thesis, titled "Link groups", under the supervision of Robert H. Fox. He remained at Princeton to pursue graduate studies and received his Ph.D. in mathematics in 1954 after com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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PL Structure
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation. An isomorphism of PL manifolds is called a PL homeomorphism. Relation to other categories of manifolds PL, or more precisely PDIFF, sits between DIFF (the category of smooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, the Generalized Poincaré conjecture is true in PL (with the possible exception of dimension 4, where it is equivalent to DIFF), but is false generally in DIFF — but is "worse behaved" than TOP, as elaborated in surgery theory. Smooth manifolds Smooth manifolds have canonical PL structures — they are uniquely ''triangulizable,'' by Whitehead's the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Laurent C
Laurent may refer to: *Laurent (name), a French masculine given name and a surname **Saint Laurence (aka: Saint ''Laurent''), the martyr Laurent **Pierre Alphonse Laurent, mathematician **Joseph Jean Pierre Laurent, amateur astronomer, discoverer of minor planet (51) Nemausa *Laurent, South Dakota, a proposed town for the Deaf to be named for Laurent Clerc See also *Laurent series, in mathematics, representation of a complex function ''f(z)'' as a power series which includes terms of negative degree, named for Pierre Alphonse Laurent *Saint-Laurent (other) *Laurence Laurence is an English and French given name (usually female in French and usually male in English). The English masculine name is a variant of Lawrence and it originates from a French form of the Latin ''Laurentius'', a name meaning "man from ... (name), feminine form of "Laurent" * Lawrence (other) {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Robion Kirby
Robion Cromwell Kirby (born February 25, 1938) is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology. Together with Laurent C. Siebenmann he invented the Kirby–Siebenmann invariant for classifying the piecewise linear structures on a topological manifold. He also proved the fundamental result on the Kirby calculus, a method for describing 3-manifolds and smooth 4-manifolds by surgery on framed links. Along with his significant mathematical contributions, he has over 50 doctoral students and his problem list. He received his Ph.D. from the University of Chicago in 1965. He soon became an assistant professor at UCLA. While there he developed his " torus trick" which enabled him to solve, in dimensions greater than four (with additional joint work with Siebenmann), four of John Milnor's seven most important problems in geometric topology. In 1971, he was awarded the Oswald Veblen Prize in Geometry by the Americ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Obstruction Theory
In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants. In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the existence of certain fields of linear independent vectors. Obstruction theory turns out to be an application of cohomology theory to the problem of constructing a cross-section of a bundle. In homotopy theory The older meaning for obstruction theory in homotopy theory relates to the procedure, inductive with respect to dimension, for extending a continuous mapping defined on a simplicial complex, or CW complex. It is traditionally called ''Eilenberg obstruction theory'', after Samuel Eilenberg. It involves cohomology groups with coefficients in homotopy groups to define obstructions to extensions. For example, with a mapping from a simplicial complex ''X'' to another, ''Y'', defined initially on the 0-skeleton of ''X'' (the vertice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Edwin E
The name Edwin means "rich friend". It comes from the Old English elements "ead" (rich, blessed) and "ƿine" (friend). The original Anglo-Saxon form is Eadƿine, which is also found for Anglo-Saxon figures. People * Edwin of Northumbria (died 632 or 633), King of Northumbria and Christian saint * Edwin (son of Edward the Elder) (died 933) * Eadwine of Sussex (died 982), King of Sussex * Eadwine of Abingdon (died 990), Abbot of Abingdon * Edwin, Earl of Mercia (died 1071), brother-in-law of Harold Godwinson (Harold II) * Edwin (director) (born 1978), Indonesian filmmaker * Edwin (musician) (born 1968), Canadian musician * Edwin Abeygunasekera, Sri Lankan Sinhala politician, member of the 1st and 2nd State Council of Ceylon * Edwin Ariyadasa (1922-2021), Sri Lankan Sinhala journalist * Edwin Austin Abbey (1852–1911) British artist * Edwin Eugene Aldrin (born 1930), although he changed it to Buzz Aldrin, American astronaut * Edwin Howard Armstrong (1890–1954), American ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Tibor Radó
Tibor Radó (June 2, 1895 – December 29, 1965) was a Hungarian mathematician who moved to the United States after World War I. Biography Radó was born in Budapest and between 1913 and 1915 attended the Polytechnic Institute, studying civil engineering. In World War I, he became a First Lieutenant in the Hungarian Army and was captured on the Russian Front. He escaped from a Siberian prisoner camp and, traveling thousands of miles across Arctic wasteland, managed to return to Hungary. He received a doctorate from the Franz Joseph University in 1923. He taught briefly at the university and then became a research fellow in Germany for the Rockefeller Foundation. In 1929, he moved to the United States and lectured at Harvard University and the Rice Institute before obtaining a faculty position in the Department of Mathematics at Ohio State University in 1930. In 1935 he was granted American citizenship. In World War II he was a science consultant to the United Sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Generalized Poincaré Conjecture
In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere a sphere. More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or differentiable (Diff). Then the statement is :Every homotopy sphere (a closed ''n''-manifold which is homotopy equivalent to the ''n''-sphere) in the chosen category (i.e. topological manifolds, PL manifolds, or smooth manifolds) is isomorphic in the chosen category (i.e. homeomorphic, PL-isomorphic, or diffeomorphic) to the standard ''n''-sphere. The name derives from the Poincaré conjecture, which was made for (topological or PL) manifolds of dimension 3, where being a homotopy sphere is equivalent to being simply connected and closed. The generalized Poincaré conjecture is known to be true or false in a number of instances, due to the work of many distinguished topologists, including the Fields medal awardees John Milnor, Steve Smale, M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |