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73 relations: Academic Press, Alain Connes, Algebraic geometry, Algebraic structure, American Mathematical Society, Associative algebra, Atiyah–Singer index theorem, Banach algebra, Binary operation, C*-algebra, Characteristic class, Chern class, Commutative property, Commutative ring, Compact space, Complex number, Continuous function, Cyclic homology, Duality (mathematics), Dynamical system, Ergodic theory, Foliation, Fred Van Oystaeyen, Function (mathematics), Fuzzy sphere, Gelfand representation, Gennadi Sardanashvily, George Mackey, Grothendieck topology, Group action, Hamiltonian mechanics, Hausdorff space, Heisenberg group, Homogeneous space, JLO cocycle, K-theory, Locally compact space, M-theory, Mathematical Sciences Research Institute, Mathematics, Matilde Marcolli, Measure space, Michael Artin, Moyal product, Noncommutative algebraic geometry, Noncommutative quantum field theory, Noncommutative ring, Noncommutative standard model, Noncommutative topology, Noncommutative torus, ..., Norm (mathematics), Number theory, Particle physics, Phase space, Phase space formulation, Pierre Gabriel, Riemannian manifold, Scheme (mathematics), Serre duality, Sheaf (mathematics), Signature operator, Spectral triple, Spectrum of a C*-algebra, Spectrum of a ring, Springer Science+Business Media, Standard Model, Symplectic manifold, Topological property, Topological space, Topology, Topos, Von Neumann algebra, Wigner–Weyl transform. Expand index (23 more) »
Academic Press
Academic Press is an academic book publisher.
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Alain Connes
Alain Connes (born 1 April 1947) is a French mathematician, currently Professor at the Collège de France, IHÉS, Ohio State University and Vanderbilt University.
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Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
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Algebraic structure
In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.
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American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.
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Associative algebra
In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.
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Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by, states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data).
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Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, i.e. a normed space and complete in the metric induced by the norm.
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Binary operation
In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set.
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C*-algebra
C∗-algebras (pronounced "C-star") are an area of research in functional analysis, a branch of mathematics.
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Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" — and whether it possesses sections.
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Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles.
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Commutative property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.
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Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
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Cyclic homology
In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative algebras which generalize the de Rham (co)homology of manifolds.
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Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself.
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Dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.
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Ergodic theory
Ergodic theory (Greek: έργον ergon "work", όδος hodos "way") is a branch of mathematics that studies dynamical systems with an invariant measure and related problems.
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Foliation
In mathematics, a foliation is a geometric tool for understanding manifolds.
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Fred Van Oystaeyen
Fred Van Oystaeyen (born 1947), also Freddy van Oystaeyen, is a mathematician and emeritus professor of mathematics at the University of Antwerp.
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Function (mathematics)
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
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Fuzzy sphere
In mathematics, the fuzzy sphere is one of the simplest and most canonical examples of non-commutative geometry.
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Gelfand representation
In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) has two related meanings.
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Gennadi Sardanashvily
Gennadi Sardanashvily (Генна́дий Алекса́ндрович Сарданашви́ли; March 13, 1950 - September 1, 2016) was a theoretical physicist, a principal research scientist of Moscow State University.
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George Mackey
George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician.
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Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space.
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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Hamiltonian mechanics
Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics.
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Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.
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Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form \end under the operation of matrix multiplication.
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Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively.
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JLO cocycle
In noncommutative geometry, the JLO cocycle is a cocycle (and thus defines a cohomology class) in entire cyclic cohomology.
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K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme.
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Locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.
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M-theory
M-theory is a theory in physics that unifies all consistent versions of superstring theory.
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Mathematical Sciences Research Institute
The Mathematical Sciences Research Institute (MSRI) is an independent nonprofit mathematical research institution in Berkeley, California.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Matilde Marcolli
Matilde Marcolli is an Italian mathematical physicist.
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Measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes.
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Michael Artin
Michael Artin (born 28 June 1934) is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.
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Moyal product
In mathematics, the Moyal product (also called the star product or Weyl–Groenewold product) is perhaps the best-known example of a phase-space star product.
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Noncommutative algebraic geometry
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients).
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Noncommutative quantum field theory
In mathematical physics, noncommutative quantum field theory (or quantum field theory on noncommutative spacetime) is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functions are noncommutative.
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Noncommutative ring
In mathematics, more specifically abstract algebra and ring theory, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exists a and b in R with a·b ≠ b·a.
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Noncommutative standard model
In theoretical particle physics, the non-commutative Standard Model, mainly due to the French mathematician Alain Connes, uses his noncommutative geometry to devise an extension of the Standard Model to include a modified form of general relativity.
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Noncommutative topology
In mathematics, noncommutative topology is a term used for the relationship between topological and C*-algebraic concepts.
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Noncommutative torus
In mathematics, and more specifically in the theory of C*-algebras, the noncommutative tori Aθ, also known as irrational rotation algebras for irrational values of θ, form a family of noncommutative C*-algebras which generalize the algebra of continuous functions on the 2-torus.
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Norm (mathematics)
In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.
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Number theory
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
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Particle physics
Particle physics (also high energy physics) is the branch of physics that studies the nature of the particles that constitute matter and radiation.
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Phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space.
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Phase space formulation
The phase space formulation of quantum mechanics places the position and momentum variables on equal footing, in phase space.
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Pierre Gabriel
Pierre Gabriel (1 August 1933 – 24 November 2015), also known as Peter Gabriel, was a French mathematician at the universities of Strasbourg (1962-1970), Bonn (1970-1974) and Zürich (1974-1998) who worked on category theory, algebraic groups, and representation theory of algebras.
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Riemannian manifold
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.
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Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.
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Serre duality
In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves).
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Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
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Signature operator
In mathematics, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential forms on an even-dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold if the dimension of the manifold is a multiple of four.
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Spectral triple
In noncommutative geometry and related branches of mathematics and mathematical physics, a spectral triple is a set of data which encodes a geometric phenomenon in an analytic way.
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Spectrum of a C*-algebra
In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. A *-representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and which is invariant under all operators π(x) with x ∈ A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces.
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Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by \operatorname(R), is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Standard Model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (the electromagnetic, weak, and strong interactions, and not including the gravitational force) in the universe, as well as classifying all known elementary particles.
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Symplectic manifold
In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form.
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Topological property
In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms.
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Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
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Topos
In mathematics, a topos (plural topoi or, or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site).
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Von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator.
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Wigner–Weyl transform
In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödinger picture.
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References
[1] https://en.wikipedia.org/wiki/Noncommutative_geometry
