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Mumford Conjecture (other)
There are several conjectures in mathematics by David Mumford. * Mumford's conjecture about reductive groups, now called Haboush's theorem. * The Mumford conjecture on the cohomology of the stable mapping class group, proved by Ib Madsen and Michael Weiss. * The Manin-Mumford conjecture In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has ... about Jacobians of curves, proved by Michel Raynaud. {{mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Mumford
David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University. Early life Mumford was born in Worth, West Sussex in England, of an English father and American mother. His father William started an experimental school in Tanzania and worked for the then newly created United Nations. He attended Phillips Exeter Academy, where he received a Westinghouse Science Talent Search prize for his relay-based computer project. Mumford then went to Harvard University, where he became a student of Oscar Zariski. At Harvard, he became a Putnam Fellow in 1955 and 1956. He completed his PhD in 1961, with a thesis entitled ''Existence of the moduli scheme for curves of any genus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haboush's Theorem
In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group ''G'' over a field ''K'', and for any linear representation ρ of ''G'' on a ''K''- vector space ''V'', given ''v'' ≠ 0 in ''V'' that is fixed by the action of ''G'', there is a ''G''-invariant polynomial ''F'' on ''V'', without constant term, such that :''F''(''v'') ≠ 0. The polynomial can be taken to be homogeneous, in other words an element of a symmetric power of the dual of ''V'', and if the characteristic is ''p''>0 the degree of the polynomial can be taken to be a power of ''p''. When ''K'' has characteristic 0 this was well known; in fact Weyl's theorem on the complete reducibility of the representations of ''G'' implies that ''F'' can even be taken to be linear. Mumford's conjecture about the extension to prime characteristic ''p'' was proved by W. J. , about a decade after the problem had been posed by David Mumford, in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mapping Class Group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Motivation Consider a topological space, that is, a space with some notion of closeness between points in the space. We can consider the set of homeomorphisms from the space into itself, that is, continuous maps with continuous inverses: functions which stretch and deform the space continuously without breaking or gluing the space. This set of homeomorphisms can be thought of as a space itself. It forms a group under functional composition. We can also define a topology on this new space of homeomorphisms. The open sets of this new function space will be made up of sets of functions that map compact subsets ''K'' into open subsets ''U'' as ''K'' and ''U'' range throughout our original topological space, completed with their finite intersect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ib Madsen
Ib Henning Madsen (born 12 April 1942, in Copenhagen)Curriculum vitae retrieved 3 February 2013. is a Danish mathematician, a professor of mathematics at the . He is known for (with ) proving the Mumford conjecture on the of the stable , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Weiss (mathematician)
Michael Weiss (born 14 December 1955) is a German mathematician and an expert in algebraic and geometric topology. He is a professor at the University of Münster. Life He completed his PhD in 1982 at the University of Warwick under the supervision of Brian Sanderson. He was then affiliated as a researcher with the Institute of Advanced Scientific Studies near Paris and the universities of Bielefeld, Edinburgh, and Göttingen. In 1999, he joined the faculty of Aberdeen University where he stayed until 2011, when he was awarded a Alexander von Humboldt Professorship at the University of Münster. Academic Work His research is on algebraic topology and differential topology. In work with Ib Madsen, he resolved the Mumford Conjecture about rational characteristic classes of surface bundles in the limit as the genus tends to infinity.Allen Hatcher, A Short Exposition of the Madsen–Weiss Theorem' Building on earlier work of Thomas Goodwillie, he developed Embedding Calculus, a C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manin-Mumford Conjecture
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety ''A'' over a number field ''K''; or more generally (for global fields or more general finitely-generated rings or fields). Integer points on abelian varieties There is some tension here between concepts: ''integer point'' belongs in a sense to affine geometry, while ''abelian variety'' is inherently defined in projective geometry. The basic results, such as Siegel's theorem on integral points, come from the theory of diophantine approximation. Rational points on abelian varieties The basic result, the Mordell–Weil theorem in Diophantine geometry, says that ''A''(''K''), the group of points on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Variety
In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian variety. Introduction The Jacobian variety is named after Carl Gustav Jacobi, who proved the complete version of the Abel–Jacobi theorem, making the injectivity statement of Niels Abel into an isomorphism. It is a principally polarized abelian variety, of dimension ''g'', and hence, over the complex numbers, it is a complex torus. If ''p'' is a point of ''C'', then the curve ''C'' can be mapped to a subvariety of ''J'' with the given point ''p'' mapping to the identity of ''J'', and ''C'' generates ''J'' as a group. Construction for complex curves Over the complex numbers, the Jacobian variety can be realized as the quotient space ''V''/''L'', where ''V'' is the dual of the vector space of all global holomorphic differentials on ''C'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
