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GKM Variety
In algebraic geometry, a GKM variety is a complex algebraic variety equipped with a torus action that meets certain conditions. The concept was introduced by Mark Goresky, Robert Kottwitz, and Robert MacPherson in 1998. The torus action of a GKM variety must be ''skeletal'': both the set of fixed points of the action, and the number of one-dimensional orbits of the action, must be finite. In addition, the action must be ''equivariantly formal'', a condition that can be phrased in terms of the torus' rational cohomology. See also *equivariant cohomology In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordi ... References {{algebraic-geometry-stub Algebraic geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus Action
In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus ''T'' is called a ''T''-variety. In differential geometry, one considers an action of a real or complex torus on a manifold (or an orbifold). A normal algebraic variety with a torus acting on it in such a way that there is a dense orbit is called a toric variety (for example, orbit closures that are normal are toric varieties). Linear action of a torus A linear action of a torus can be simultaneously diagonalized, after extending the base field if necessary: if a torus ''T'' is acting on a finite-dimensional vector space ''V'', then there is a direct sum decomposition: :V = \bigoplus_ V_ where *\chi: T \to \mathbb_m is a group homomorphism, a character of ''T''. *V_ = \, ''T''-invariant subspace called the weight subspace of weight \chi. The decomposition exists because the linear action determines (and is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Western Ontario
The University of Western Ontario (UWO), also known as Western University or Western, is a Public university, public research university in London, Ontario, London, Ontario, Canada. The main campus is located on of land, surrounded by residential neighbourhoods and the Thames River (Ontario), Thames River bisecting the campus's eastern portion. The university operates twelve academic faculties and schools. It is a member of the U15 Group of Canadian Research Universities, U15, a group of research-intensive universities in Canada. The university was founded on 7 March 1878 by Bishop Isaac Hellmuth of the Diocese of Huron, Anglican Diocese of Huron as the Western University of London, Ontario. It incorporated Huron University College, Huron College, which had been founded in 1863. The first four faculties were Arts, Divinity, Law and Medicine. The university became non-denominational in 1908. Beginning in 1919, the university had affiliated with several denominational colleges. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mark Goresky
Robert Mark Goresky is a Canadian mathematician who invented intersection homology with his advisor and life partner Robert MacPherson. Career Goresky received his Ph.D. from Brown University in 1976. His thesis, titled ''Geometric Cohomology and Homology of Stratified Objects'', was written under the direction of MacPherson. Many of the results in his thesis were published in 1981 by the American Mathematical Society. He has taught at the University of British Columbia in Vancouver, and Northeastern University. Awards Goresky received a Sloan Research Fellowship in 1981. He received the Coxeter–James Prize in 1984. In 2002, Goresky and MacPherson were jointly awarded the Leroy P. Steele Prize for Seminal Contribution to Research by the American Mathematical Society. In 2012 Goresky became a fellow of the American Mathematical Society. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Kottwitz
Robert Edward Kottwitz (born 1950 in Lynn, Massachusetts) is an American mathematician. Kottwitz studied at the University of Washington (B.A.) and then went to Harvard University, where he received his Ph.D. in 1977 under the supervision of Phillip Griffiths and John T. Tate (''Orbital Integrals on _3''). In 1976 he was assistant professor and later professor at the University of Washington and went in 1989 as a professor to the University of Chicago. He was several times at the Institute for Advanced Study in Princeton, New Jersey (for example, in 1976 and 1977). Kottwitz works in the Langlands program, including harmonic analysis on ''p''-adic Lie groups and automorphic forms and the general linear groups and Shimura varieties. He is a fellow of the American Academy of Arts and Sciences and the American Mathematical Society (AMS). He was an invited speaker at the International Congress of Mathematicians in Berlin in 1998 (Harmonic analysis on semisimple Lie ''p''-adic algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert MacPherson (mathematician)
Robert Duncan MacPherson (born May 25, 1944) is an American mathematician at the Institute for Advanced Study and Princeton University. He is best known for the invention of intersection homology with Mark Goresky, whose thesis he directed at Brown University, and who became his life partner. MacPherson previously taught at Brown University, the University of Paris, and the Massachusetts Institute of Technology. In 1983 he gave a plenary address at the International Congress of Mathematicians in Warsaw. Education and career Educated at Swarthmore College and Harvard University, MacPherson received his PhD from Harvard in 1970. His thesis, written under the direction of Raoul Bott, was entitled ''Singularities of Maps and Characteristic Classes''. Among his many PhD students are Kari Vilonen and Mark Goresky. Honors and awards In 1992, MacPherson was awarded the NAS Award in Mathematics from the National Academy of Sciences. In 2002 he and Goresky were awarded the Leroy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbit (group Theory)
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any set with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Cohomology
In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space X with action of a topological group G is defined as the ordinary cohomology ring with coefficient ring \Lambda of the homotopy quotient EG \times_G X: :H_G^*(X; \Lambda) = H^*(EG \times_G X; \Lambda). If G is the trivial group, this is the ordinary cohomology ring of X, whereas if X is contractible, it reduces to the cohomology ring of the classifying space BG (that is, the group cohomology of G when ''G'' is finite.) If ''G'' acts freely on ''X'', then the canonical map EG \times_G X \to X/G is a homotopy equivalence and so one gets: H_G^*(X; \Lambda) = H^*(X/G; \Lambda). Definitions It is also possible to define the equivariant cohomology H_G^*(X;A) of X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
