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∞-Chern–Simons Theory
In mathematics, ∞-Chern–Simons theory (not to be confused with infinite-dimensional Chern–Simons theory) is a generalized formulation of Chern–Simons theory from differential geometry using the formalism of higher category theory, which in particular studies ∞-categories. It is obtained by taking general abstract analogs of all involved concepts defined in any cohesive ∞-topos, for example that of smooth ∞-groupoids. Principal bundles on which Lie groups act are for example replaced by ∞-principal bundles on with group objects in ∞-topoi act.Definition in Schreiber 2013, 1.2.6.5.2 The theory is named after Shiing-Shen Chern and James Simons, who first described Chern–Simons forms in 1974, although the generalization was not developed by them. See also * ∞-Chern–Weil theory Literature * * * * {{cite arXiv , arxiv=1301.2580 , author1=Domenico Fiorenza , author2=Hisham Sati , author3=Urs Schreiber Urs Schreiber (born 1974) is a mathematician sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Group Object
In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is a topological space such that the group operations are continuous. Definition Formally, we start with a category ''C'' with finite products (i.e. ''C'' has a terminal object 1 and any two objects of ''C'' have a product). A group object in ''C'' is an object ''G'' of ''C'' together with morphisms *''m'' : ''G'' × ''G'' → ''G'' (thought of as the "group multiplication") *''e'' : 1 → ''G'' (thought of as the "inclusion of the identity element") *''inv'' : ''G'' → ''G'' (thought of as the "inversion operation") such that the following properties (modeled on the group axioms – more precisely, on the definition of a group used in universal algebra) are satisfied * ''m'' is associative, i.e. ''m'' (''m'' × id''G'') = ''m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jim Stasheff
James Dillon Stasheff (born January 15, 1936, New York City) is an American mathematician, a professor emeritus of mathematics at the University of North Carolina at Chapel Hill. He works in algebraic topology and algebra as well as their applications to physics. Biography Stasheff did his undergraduate studies in mathematics at the University of Michigan, graduating in 1956. Stasheff then began his graduate studies at Princeton University; his notes for a 1957 course by John Milnor on characteristic classes first appeared in mimeographed form and later in 1974 in revised form book with Stasheff as a co-author. After his second year at Princeton, he moved to Oxford University on a Marshall Scholarship. Two years later in 1961, with a pregnant wife, needing an Oxford degree to get reimbursed for his return trip to the US, and yet still feeling attached to Princeton, he split his thesis into two parts (one topological, the other algebraic) and earned two doctorates, a D.Ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Urs Schreiber
Urs Schreiber (born 1974) is a mathematician specializing in the connection between mathematics and theoretical physics (especially string theory) and currently working as a researcher at New York University Abu Dhabi. He was previously a researcher at the Czech Academy of Sciences, Institute of Mathematics, Department for Algebra, Geometry and Mathematical Physics. Education Schreiber obtained his doctorate from the University of Duisburg-Essen in 2005 with a thesis supervised by Robert Graham and titled ''From Loop Space Mechanics to Nonabelian Strings''. Work Schreiber's research fields include the mathematical foundation of quantum field theory. Schreiber is a co-creator of the ''n''Lab, a wiki for research mathematicians and physicists working in higher category theory In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
∞-Chern–Weil Theory
In mathematics, ∞-Chern–Weil theory is a generalized formulation of Chern–Weil theory from differential geometry using the formalism of higher category theory. The theory is named after Shiing-Shen Chern and André Weil, who first constructed the Chern–Weil homomorphism in the 1940s, although the generalization was not developed by them. Generalization There are three equivalent ways to describe the k-th Chern class of complex vector bundles of rank n, which is as a: * (1-categorical) natural transformation ,\operatorname(n)Rightarrow ,K(\mathbb,2k)/math> * homotopy class of a continuous map \operatorname(n)\rightarrow K(\mathbb,2k) * singular cohomology class in H^(\operatorname(n),\mathbb) \operatorname(n) is the classifying space for the unitary group \operatorname(n) and K(\mathbb,2k) is an Eilenberg–MacLane space, which represent the set of complex vector bundles of rank n with ,\operatorname(n)cong\operatorname_\mathbb^n(-) and singular cohomology with ,K(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern–Simons Form
In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose. Definition Given a manifold and a Lie algebra valued 1-form \mathbf over it, we can define a family of ''p''-forms: In one dimension, the Chern–Simons 1-form is given by :\operatorname \mathbf In three dimensions, the Chern–Simons 3-form is given by :\operatorname \left \mathbf \wedge \mathbf-\frac \mathbf \wedge \mathbf \wedge \mathbf \right= \operatorname \left d\mathbf \wedge \mathbf + \frac \mathbf \wedge \mathbf \wedge \mathbf\right In five dimensions, the Chern–Simons 5-form is given by : \begin & \operatorname \left \mathbf\wedge\mathbf \wedge \mathbf-\frac \mathbf \wedge\mathbf\wedge\mathbf\wedge\mathbf +\frac \mathbf \wedge \mathbf \wedge \mathbf \wedge \mathbf \wedge\mathbf \right\\ pt= & \ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jim Simons
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Jim or James Simons may refer to: *Jim Simons (mathematician) (born 1938), mathematician and hedge fund manager *Jim Simons (golfer) (1950–2005), American golfer *Jimmy Simons (born 1970), Dutch footballer *Jimmy Simons, co-winner of 2001 Primetime Emmy Award for Outstanding Comedy Series *James Simons, preceded by John Calhoun Sheppard as speaker of the South Carolina House of Representatives See also *James Simmons (other) *James Simon (other) James or Jim Simon may refer to: * James Simon (composer) (1880–1944), German composer, pianist and musicologist * James Simon (journalist), journalism professor at Fairfield University, Fairfield, Connecticut * James D. Simon (1897–1982), Lou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite-dimensional Chern–Simons Theory
In mathematics, infinite-dimensional Chern–Simons theory (not to be confused with ∞-Chern–Simons theory) is a generalization of Chern–Simons theory to manifolds with infinite dimensions. These are not modeled with finite-dimensional Euclidean spaces, but infinite-dimensional topological vector spaces, for example Hilbert, Banach and Fréchet spaces, which lead to Hilbert, Banach and Fréchet manifolds respectively. Principal bundles, which in finite-dimensional Chern–Simons theory are considered with (compact) Lie groups as gauge groups, are then fittingly considered with Hilbert Lie, Banach Lie and Fréchet Lie groups as gauge groups respectively, which also makes their total spaces into a Hilbert, Banach and Fréchet manifold respectively. These are called Hilbert, Banach and Fréchet principal bundles respectively. The theory is named after Shiing-Shen Chern and James Simons James Harris Simons (; born 25 April 1938) is an American mathematician, billionaire hed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with # An action of G on P, analogous to (x, g)h = (x, gh) for a product space. # A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) \mapsto x. Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (x,e). Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, X \times G \to G that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space. A common example of a princi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
