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Branch of mathematics
Noncommutative geometry (NCG) is a branch of mathematics that studies geometric ideas through noncommutative algebras. In ordinary geometry, a space can
Noncommutative_geometry
Branch of mathematics
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
Quantum mechanical model based on mathematical matrices
hand, and noncommutative geometry on the other hand. It quickly led to the discovery of other important links between noncommutative geometry and various
Matrix_theory_(physics)
French mathematician (born 1947)
known for his contributions to the study of operator algebras and noncommutative geometry. He was a professor at the Collège de France, Institut des Hautes
Alain_Connes
Framework of superstring theory
hand, and noncommutative geometry on the other hand. It quickly led to the discovery of other important links between noncommutative geometry and various
M-theory
Conjecture on zeros of the zeta function
has described a relationship between the Riemann hypothesis and noncommutative geometry, and showed that a suitable analog of the Selberg trace formula
Riemann_hypothesis
Overview of and topical guide to geometry
geometry Noncommutative algebraic geometry Noncommutative geometry Ordered geometry Parabolic geometry Plane geometry Projective geometry Quantum geometry Riemannian
Outline_of_geometry
Model (best known as Spectral Standard Model), is a model based on noncommutative geometry that unifies a modified form of general relativity with the Standard
Noncommutative_standard_model
Lebanese physicist (born 1953)
bridge between noncommutative geometry and quantum gravity. At its heart, noncommutative geometry (NCG) generalizes ordinary geometry by allowing the
Ali_Chamseddine
Branch of algebra
identities. Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory, which provide many natural
Ring_theory
Mathematics study in geometry
In mathematics, derived noncommutative algebraic geometry, the derived version of noncommutative algebraic geometry, is the geometric study of derived
Derived noncommutative algebraic geometry
Derived_noncommutative_algebraic_geometry
Quantum field theory using noncommutative mathematics
theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functions are noncommutative. One commonly studied version
Noncommutative quantum field theory
Noncommutative_quantum_field_theory
Property of a mathematical space
back to René Descartes, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley, William
Dimension
mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry. The quantum
Noncommutative projective geometry
Noncommutative_projective_geometry
University of Antwerp. He has pioneered work on noncommutative geometry, in particular noncommutative algebraic geometry. In 1972, Fred Van Oystaeyen obtained his
Fred_Van_Oystaeyen
Concept in theoretical mathematical physics
described mathematically using the noncommutative geometry of Connes, quantum geometry, or quantum groups. Any noncommutative algebra with at least four generators
Quantum_spacetime
Description of gravity using discrete values
less important theories include causal dynamical triangulation, noncommutative geometry, and twistor theory. One of the difficulties of formulating a quantum
Quantum_gravity
In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra A {\displaystyle
Quantum_differential_calculus
Algebraic structure
right Ore domain. Derived algebraic geometry Noncommutative geometry Noncommutative algebraic geometry Noncommutative harmonic analysis Representation theory
Noncommutative_ring
Set of mathematical concepts in quantum gravity
to reconstruct the geometry of space-time from "first principles" is Discrete Lorentzian quantum gravity. Noncommutative geometry Quantum spacetime Tomasiello
Quantum_geometry
Noncommutative geometric structure
spectral behaviour of operators and have found applications in the noncommutative geometry of French mathematician Alain Connes. In heuristic terms, a singular
Singular_trace
C*-algebras. Noncommutative topology is related to analytic noncommutative geometry. The premise behind noncommutative topology is that a noncommutative C*-algebra
Noncommutative_topology
Theoretical object in mathematics
symmetries in projective geometry and the combinatorics of simplicial complexes. F1 has been connected to noncommutative geometry and to a possible proof
Field_with_one_element
Hypothetical physical concept
developed a geometric framework known as noncommutative geometry in which spacetime is extended via noncommutative operator algebras. When combined with
Theory_of_everything
Italian mathematician and physicist
Arithmetic Noncommutative Geometry. American Mathematical Society. ISBN 978-0-8218-3833-4. Connes, Alain; Marcolli, Matilde (2008). Noncommutative Geometry, Quantum
Matilde_Marcolli
Theory of subatomic structure
hand, and noncommutative geometry on the other hand. It quickly led to the discovery of other important links between noncommutative geometry and various
String_theory
In noncommutative geometry and related branches of mathematics and mathematical physics, a spectral triple is a set of data which encodes a geometric
Spectral_triple
Branch of mathematics
differential geometry topics Noncommutative geometry Projective differential geometry Synthetic differential geometry Systolic geometry Gauge theory (mathematics)
Differential_geometry
Study of geometry using a coordinate system
In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts
Analytic_geometry
American mathematician
notion in noncommutative geometry and as a tool for classifying C*-algebras. For example, in 1981 he showed that if Aθ denotes the noncommutative torus of
Marc_Rieffel
Branch of differential geometry and differential topology
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds
Symplectic_geometry
Infinitely detailed mathematical structure
in the Menger sponge, the shape is called affine self-similar. Fractal geometry relates to the mathematical branch of measure theory by their Hausdorff
Fractal
Fundamental object of geometry
considered fundamental in mainstream geometry and topology, there are some systems that forgot it, e.g. noncommutative geometry and pointless topology. A "pointless"
Point_(geometry)
Type of non-Euclidean geometry
mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate
Hyperbolic_geometry
American mathematician
his contributions to quantum logic, representation theory, and noncommutative geometry. Mackey earned his B.A. at Rice University in 1938 and obtained
George_Mackey
Straight figure with zero width and depth
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature. It is a special case of a curve
Line_(geometry)
British mathematician
research interests center around index theorems, coarse geometry, operator algebras, noncommutative geometry, and the Novikov conjecture in differential topology
John_Roe_(mathematician)
Cocycle in an entire cyclic cohomology group
In noncommutative geometry, the Jaffe–Lesniewski–Osterwalder (JLO) cocycle (named after Arthur Jaffe, Andrzej Lesniewski, and Konrad Osterwalder) is a
JLO_cocycle
C*-algebras, the noncommutative tori Aθ, also known as irrational rotation algebras for irrational values of θ, form a family of noncommutative C*-algebras
Noncommutative_torus
Theory in number theory
Anabelian geometry is a theory in arithmetic geometry which describes the way in which the algebraic fundamental group of a certain arithmetic variety
Anabelian_geometry
Geometry of the surface of a sphere
Spherical geometry or spherics (from Ancient Greek σφαιρικά) is the geometry of the two-dimensional surface of a sphere or the n-dimensional surface of
Spherical_geometry
Theory of quantum gravity merging quantum mechanics and general relativity
(2012). "Intersecting Quantum Gravity with Noncommutative Geometry – a Review". Symmetry, Integrability and Geometry: Methods and Applications. 8: 18. arXiv:1203
Loop_quantum_gravity
Geometry of quantum systems (e.g., noncommutative geometry and supergeometry) is mainly phrased in algebraic terms of modules and algebras. Connections
Connection (algebraic framework)
Connection_(algebraic_framework)
local arithmetic dynamics Noncommutative algebra Noncommutative algebraic geometry a direction in noncommutative geometry studying the geometric properties
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative
Cyclic_homology
Straight line segment that passes through the centre of a circle
In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It
Diameter
Branch of geometry
geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry
Convex_geometry
Branch of mathematics
classical algebraic geometry Important publications in algebraic geometry List of algebraic surfaces Noncommutative algebraic geometry A witness of this
Algebraic_geometry
Russian American mathematician (born 1957)
American mathematician who works in representation theory and in noncommutative geometry. He is known for his contributions to geometric representation
Victor_Ginzburg
Euclidean geometry without distance and angles
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance
Affine_geometry
Branch of functional analysis
the philosophy of noncommutative geometry, which tries to study various non-classical and/or pathological objects by noncommutative operator algebras
Operator_algebra
Geometric space with four dimensions
ordinary space is called Euclidean space because it corresponds to Euclid's geometry, which was originally abstracted from the spatial experiences of everyday
Four-dimensional_space
Two geometries based on axioms closely related to those specifying Euclidean geometry
non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the
Non-Euclidean_geometry
Mathematics of varieties with integer coordinates
geometry. The extensive development of algebraic geometry in the 20th century produced powerful tools to study these equations. Diophantine geometry is
Diophantine_geometry
Banach bundle, called the trivial bundle Banach bundles in differential geometry Fell, M.G., Doran, R.S.: "Representations of *-Algebras, Locally Compact
Banach bundle (non-commutative geometry)
Banach_bundle_(non-commutative_geometry)
American theoretical physicist
In their paper on string theory and noncommutative geometry, Seiberg and Witten studied certain noncommutative quantum field theories that arise as limits
Edward_Witten
Classical theory of gravitation
Riemann–Cartan geometry, which possesses a locally gauged Lorentz symmetry, while general relativity is formulated within the framework of Riemannian geometry, which
Einstein–Cartan_theory
2000, arXiv:hep-th/0206192 John Madore, An introduction to Noncommutative Differential Geometry and its Physical Applications, London Mathematical Society
Fuzzy_sphere
Attraction of masses and energy
the idea of altering geometry only joined the story of gravity once mechanics required the Lorentz transformations. Geometry was an ancient science
Gravity
Geometry without the parallel postulate
Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally
Absolute_geometry
Chinese American mathematician
Simons Fellow in Mathematics. Yu's research interests include noncommutative geometry, higher index theory of elliptic operators, K-theory, and geometric
Guoliang_Yu
Branch of mathematics
discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes. Homological algebra began to be studied in its
Homological_algebra
Extended physical object in string theory
pure mathematics for insight into homological mirror symmetry and noncommutative geometry. The word "brane" originated in 1987 as a contraction of "membrane"
Brane
American mathematician
includes special functions, especially applications from and to noncommutative algebraic geometry, as well as earlier work in random matrix theory, coding theory
Eric_M._Rains
Concept in mathematical category theory
motivation for the notion was its use in noncommutative algebraic geometry; in this formalism, noncommutative spaces are defined as sheaves on Q-categories
Q-category
Branch of mathematics
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is
Geometry
Geometry without using coordinates
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic
Synthetic_geometry
Relationship between two lines that meet at a right angle
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of
Perpendicular
Category of mathematics papers in ArXiv
algebraic deformations, Hopf algebras, category theory, topology, noncommutative geometry, and quantum groups within quantum mechanics and quantum field
Quantum_algebra
Branch of geometry that studies combinatorial properties and constructive methods
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric
Discrete_geometry
Theory proposed by Roger Penrose
recent proposal in this direction by Penrose in 2015 was based on noncommutative geometry on twistor space and referred to as palatial twistor theory. The
Twistor_theory
Branch of algebraic geometry
arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around
Arithmetic_geometry
Mathematical model of the physical space
Euclidean geometry is a mathematical system attributed to Euclid, an ancient Greek mathematician, which he described in his textbook on geometry, Elements
Euclidean_geometry
Non-Euclidean geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel
Elliptic_geometry
Type of geometry
planes over noncommutative rings had likely desensitized Dirac. In more advanced work, Dirac used extensive drawings in projective geometry to understand
Projective_geometry
Study of complex manifolds and several complex variables
geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry
Complex_geometry
Polish mathematician (born 1956)
works primarily focus on analysis, algebraic K-theory, noncommutative geometry, and algebraic geometry. Wodzicki was born in Bytom, Poland in 1956. He received
Mariusz_Wodzicki
Mathematical concept
In mathematics, noncommutative residue, defined independently by M. Wodzicki (1984) and Guillemin (1985), is a certain trace on the algebra of pseudodifferential
Noncommutative_residue
Theory for associative algebras over rings
L. Allegretti, Differential Forms on Noncommutative Spaces. An elementary introduction to noncommutative geometry which uses Hochschild homology to generalize
Hochschild_homology
Theory of gravitation as curved spacetime
seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity
General_relativity
Indian mathematician
theory of Schrödinger operators, quantum stochastic calculus, noncommutative geometry, and, more broadly, in mathematical physics. Kalyan Bidhan Sinha
Kalyan_Bidhan_Sinha
specializes in geometry. Some notable geometers and their main fields of work, chronologically listed, are: Baudhayana (fl. c. 800 BC) – Euclidean geometry Manava
List_of_geometers
Group where ab = ba does not always hold
these play an important role in gauge theory. Associative algebra Noncommutative geometry Niels Henrik Abel David Madore: Orders of non abelian simple groups
Non-abelian_group
Deals with digitized models or images of objects of the 2D or 3D Euclidean space
Digital geometry deals with discrete sets (usually discrete point sets) considered to be digitized models or images of objects of the 2D or 3D Euclidean
Digital_geometry
Geometric model of the physical space
In geometry, a three-dimensional space (3D space) is a mathematical space in which three values (termed coordinates) are required to determine the position
Three-dimensional_space
Italian mathematician (born 1963)
known for her work with Matilde Marcolli on Arakelov theory and noncommutative geometry.[NG] Consani was born January 9, 1963, in Chiavari. She earned
Caterina_Consani
Approach to quantum gravity using discrete spacetime
main proponent of the program. He has coined the slogan "Order + Number = Geometry" to characterize the above argument. The program provides a theory in which
Causal_sets
Contraction of an astronomical object due to the influence of its gravity
black hole, so the spacetime region outside will still have a well-behaved geometry, with strong but finite curvature, that is expected to evolve towards a
Gravitational_collapse
Branch of computer science
Computational geometry is a branch of computer science devoted to the study of algorithms that can be stated in terms of geometry. Some purely geometrical
Computational_geometry
Field theory in physics that aims to unify the fundamental forces and particles
of gravity, general relativity, using a field to describe the curving geometry of four-dimensional (4D) spacetime. In the years following the creation
Unified_field_theory
Unified field theory
(pseudo-)Riemannian manifold, or even a supersymmetric manifold or orbifold or even a noncommutative space. The construction can be outlined, roughly, as follows. One starts
Kaluza–Klein_theory
German mathematician (1826–1866)
made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous
Bernhard_Riemann
Historical development of geometry
Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") arose as the field of knowledge dealing with spatial relationships. Geometry
History_of_geometry
Professor of mathematics
Boris Tsygan is a mathematician prominent in noncommutative geometry who is currently a professor of mathematics at Northwestern University. He performed
Boris_Tsygan
French mathematician
Mathématiques de Jussieu in Paris. His fields of interest are noncommutative geometry, ergodic theory, Dirichlet problem, non-commutative residue. Katz
Éric_Leichtnam
Geometric model of the planar projection of the physical universe
Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem
Euclidean_plane
Hypothetical elementary particle that mediates gravity
theory Supergravity Unified-field-theoric and quantum-mechanical Noncommutative geometry Semiclassical gravity Superfluid vacuum theory Logarithmic BEC
Graviton
American theoretical physicist
formulations of string theory), for his work on Dirichlet branes and on noncommutative geometry in string theory, and for the development of the statistical approach
Michael_R._Douglas
British mathematician and physicist
researchers today. He also pioneered a quantum groups approach to noncommutative geometry and the use of such methods as a route to quantum gravity, leading
Shahn_Majid
Relation between sides of a right triangle
theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of
Pythagorean_theorem
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