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Cartan Model
In mathematics, the Cartan model is a differential graded algebra that computes the equivariant cohomology of a space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless .... References * Stefan Cordes, Gregory Moore, Sanjaye Ramgoolam, ''Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories'', , 1994. Algebraic topology {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Graded Algebra
In mathematics – particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often used to capture information about a topological or geometric space. Explicitly, a differential graded algebra is a graded associative algebra with a chain complex structure that is compatible with the algebra structure. In geometry, the de Rham algebra of differential forms on a manifold has the structure of a differential graded algebra, and it encodes the de Rham cohomology of the manifold. In algebraic topology, the singular cochains of a topological space form a DGA encoding the singular cohomology. Moreover, American mathematician Dennis Sullivan developed a DGA to encode the rational homotopy type of topological spaces. __TOC__ Definitions Let A_\bullet = \bigoplus\nolimits_ A_i be a \mathbb-graded algebra, with product \cdot, equipped with a map d\colon A_\bullet \to A_ ... [...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) o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
