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Kirwan Map
In differential geometry, the Kirwan map, introduced by British mathematician Frances Kirwan, is the homomorphism :H^*_G(M) \to H^*(M /\!/_p G) where *M is a Hamiltonian G-space; i.e., a symplectic manifold acted by a Lie group ''G'' with a moment map \mu: M \to ^*. *H^*_G(M) is the equivariant cohomology ring of M; i.e.. the cohomology ring of the homotopy quotient EG \times_G M of M by G. *M /\!/_p G = \mu^(p)/G is the symplectic quotient of M by G at a regular central value p \in Z(^*) of \mu. It is defined as the map of equivariant cohomology induced by the inclusion \mu^(p) \hookrightarrow M followed by the canonical isomorphism H_G^*(\mu^(p)) = H^*(M /\!/_p G). A theorem of Kirwan says that if M is compact, then the map is surjective in rational coefficients. The analogous result holds between the K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frances Kirwan
Dame Frances Clare Kirwan, (born 21 August 1959) is a British mathematician, currently Savilian Professor of Geometry at the University of Oxford. Her fields of specialisation are algebraic and symplectic geometry. Education Kirwan was educated at Oxford High School, and studied maths as an undergraduate at Clare College in the University of Cambridge. She took a D.Phil at Oxford in 1984, with the dissertation title ''The Cohomology of Quotients in Symplectic and Algebraic Geometry'', which was supervised by Michael Atiyah. Research Kirwan's research interests include moduli spaces in algebraic geometry, geometric invariant theory (GIT), and in the link between GIT and moment maps in symplectic geometry. Her work endeavours to understand the structure of geometric objects by investigation of their algebraic and topological properties. She introduced the Kirwan map. From 1983 to 1985 she held a junior fellowship at Harvard. From 1983 to 1986 she held a Fellowship at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" and () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German meaning "similar" to meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra. The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory. A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. (see below). Each of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Group Action
In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums. Formal definition Let ''M'' be a manifold with symplectic form ω. Suppose that a Lie group ''G'' acts on ''M'' via symplectomorphisms (that is, the action of each ''g'' in ''G'' preserves ω). Let \mathfrak be the Lie algebra of ''G'', \mathfrak^* its dual, and :\langle \, \cdot, \cdot\rangle : \mathfrak^* \times \mathfrak \to \mathbb the pairing between the two. Any ξ in \mathfrak induces a vector field ρ(ξ) on ''M'' describing the infinitesimal action of ξ. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment Map
In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums. Formal definition Let ''M'' be a manifold with symplectic form ω. Suppose that a Lie group ''G'' acts on ''M'' via symplectomorphisms (that is, the action of each ''g'' in ''G'' preserves ω). Let \mathfrak be the Lie algebra of ''G'', \mathfrak^* its dual, and :\langle, \rangle : \mathfrak^* \times \mathfrak \to \mathbf the pairing between the two. Any ξ in \mathfrak induces a vector field ρ(ξ) on ''M'' describing the infinitesimal action of ξ. To be precise, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Cohomology Ring
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) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Quotient
In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group , is a specific bundle over a classifying space , such that every bundle with the given structure group over is a pullback by means of a continuous map . Existence of a universal bundle In the CW complex category When the definition of the classifying space takes place within the homotopy category of CW complexes, existence theorems for universal bundles arise from Brown's representability theorem. For compact Lie groups We will first prove: :Proposition. Let be a compact Lie group. There exists a contractible space on which acts freely. The projection is a -principal fibre bundle. Proof. There exists an injection of into a unitary group for big enough. If we find then we can take to be . The construction of is given in classifying space for . The following Theorem is a corollary of the above Proposition. :Theorem. If is a paracompact manifold and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Quotient
In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums. Formal definition Let ''M'' be a manifold with symplectic form ω. Suppose that a Lie group ''G'' acts on ''M'' via symplectomorphisms (that is, the action of each ''g'' in ''G'' preserves ω). Let \mathfrak be the Lie algebra of ''G'', \mathfrak^* its dual, and :\langle, \rangle : \mathfrak^* \times \mathfrak \to \mathbf the pairing between the two. Any ξ in \mathfrak induces a vector field ρ(ξ) on ''M'' describing the infinitesimal action of ξ. To be precise, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices. K-theory involves the construction of families of ''K''-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the Grothendieck–Riemann–Roch theorem, Bott periodicity, the Atiyah ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
