139 relations: Abstract simplicial complex, Adjoint representation, Algebraic surface, American Mathematical Society, André Haefliger, Augmented triad, Automorphic form, Barycentric subdivision, Betti number, Binomial theorem, Branched covering, Building (mathematics), Calabi–Yau manifold, CAT(k) space, Category theory, Cayley graph, Classifying space, Cocompact group action, Collineation, Compactification (physics), Congruence subgroup, Contractible space, Covering space, Cyclic group, D-brane, David Mumford, Degrees of freedom (physics and chemistry), Diffeomorphism, Dihedral group, Dimensional reduction, Dmitri Tymoczko, Dyad (music), Equal temperament, Equivariant map, Euclidean space, Euler characteristic, Euler characteristic of an orbifold, Fano plane, Finite group, Flag (linear algebra), Free product, Frobenius endomorphism, Frobenius group, Fuchsian group, Fundamental domain, Fundamental group, Galois group, Geometric group theory, Geometric quotient, Geometrization conjecture, ..., Geometry, Gerbe, Girth (graph theory), Glen Bredon, Group (mathematics), Group action, Group extension, Group homomorphism, Groupoid, Guerino Mazzola, Gyration, Hausdorff space, Heawood graph, Henri Poincaré, Herbert Seifert, Hilbert space, Homotopy, Hyperbolic geometry, Hyperbolic space, Ichirō Satake, Image (mathematics), Intrinsic metric, Jean-Pierre Serre, John Stillwell, K3 surface, Kawasaki's Riemann–Roch formula, Kleinian group, Limit (category theory), Linear map, Linear subspace, Link (geometry), Locally compact space, Logarithm, Major chord, Manifold, Möbius strip, Metric space, Mikhail Leonidovich Gromov, Minor chord, Mirror symmetry (string theory), Modular form, Modular group, Moduli space, Music theory, Nerve of a covering, Non-positive curvature, Orbifold notation, Orientifold, Phenomenology (particle physics), Poincaré metric, Projective plane, Projective space, Proper map, Quantum field theory, Quiver diagram, Quotient space (topology), Reflection group, Riemann–Roch theorem, Riemannian manifold, Science (journal), Seifert fiber space, Set (music), Sheaf (mathematics), Sheaf cohomology, Simplicial approximation theorem, Simplicial complex, Simply connected space, Singularity (mathematics), Spacetime, Stratification (mathematics), String (physics), String theory, Superstring theory, Supersymmetry, Sylow theorems, Time (magazine), Topology, Torus, Triad (music), Triangle group, Trivial group, Two-dimensional conformal field theory, Unit disk, Upper half-plane, Vacuum expectation value, Vertex operator algebra, Wallpaper group, William Thurston, 3-manifold. Expand index (89 more) » « Shrink index
In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a simplicial complex, consisting of a family of non-empty finite sets closed under the operation of taking non-empty subsets.
In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space.
In mathematics, an algebraic surface is an algebraic variety of dimension two.
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.
André Haefliger (born 22 May 1929) is a Swiss mathematician who works primarily on topology.
An augmented triad is a chord, made up of two major thirds (an augmented fifth).
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of the topological group.
In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way.
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.
In mathematics, a branched covering is a map that is almost a covering map, except on a small set.
In mathematics, a building (also Tits building, Bruhat–Tits building, named after François Bruhat and Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces.
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics.
In mathematics, a \mathbf space, where k is a real number, is a specific type of metric space.
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group.
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e. a topological space all of whose homotopy groups are trivial) by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG → BG.
In mathematics, an action of a group G on a topological space X is cocompact if the quotient space X/G is a compact space.
In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear.
In physics, compactification means changing a theory with respect to one of its space-time dimensions.
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries.
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map.
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.
In algebra, a cyclic group or monogenous group is a group that is generated by a single element.
In string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named.
David Bryant Mumford (born 11 June 1937) is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory.
In physics, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system.
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.
Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero.
Dmitri Tymoczko is a composer and music theorist.
In music, a dyad (less commonly, doad) is a set of two notes or pitches that, in particular contexts, may imply a chord.
An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio.
In mathematics, equivariance is a form of symmetry for functions from one symmetric space to another.
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
In differential geometry, the Euler characteristic of an orbifold, or orbifold Euler characteristic, is a generalization of the topological Euler characteristic that includes contributions coming from nontrivial automorphisms.
In finite geometry, the Fano plane (after Gino Fano) is the finite projective plane of order 2.
In abstract algebra, a finite group is a mathematical group with a finite number of elements.
In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space V. Here "increasing" means each is a proper subspace of the next (see filtration): If we write the dim Vi.
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties.
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic, an important class which includes finite fields.
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point.
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,'''R''').
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action.
In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.
In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).
In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties \pi: X \to Y such that The notion appears in geometric invariant theory.
In mathematics, Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them.
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
In mathematics, a gerbe is a construct in homological algebra and topology.
In graph theory, the girth of a graph is the length of a shortest cycle contained in the graph.
Glen Eugene Bredon (August 24, 1932 in Fresno, California – May 8, 2000) was an American mathematician who worked in the area of topology.
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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.
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group.
In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways.
Guerino Mazzola (born 1947) is a Swiss mathematician, musicologist, jazz pianist as well as book writer.
In geometry, a gyration is a rotation in a discrete subgroup of symmetries of the Euclidean plane such that the subgroup does not also contain a reflection symmetry whose axis passes through the center of rotational symmetry.
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.
In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood.
Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.
Herbert Karl Johannes Seifert (27 May 1907, Bernstadt – 1 October 1996, Heidelberg) was a German mathematician known for his work in topology.
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.
In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.
(25 December 1927 – 10 October 2014) was a Japanese mathematician working on algebraic groups who introduced the Satake isomorphism and Satake diagrams.
In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.
In the mathematical study of metric spaces, one can consider the arclength of paths in the space.
Jean-Pierre Serre (born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory.
John Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University.
In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.
In differential geometry, Kawasaki's Riemann–Roch formula, introduced by Tetsuro Kawasaki, is the Riemann–Roch formula for orbifolds.
In mathematics, a Kleinian group is a discrete subgroup of PSL(2, '''C''').
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.
In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.
In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space.
In geometry, the link of a vertex of a 2-dimensional simplicial complex is a graph that encodes information about the local structure of the complex at the vertex.
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.
In mathematics, the logarithm is the inverse function to exponentiation.
In music theory, a major chord is a chord that has a root note, a major third above this root, and a perfect fifth above this root note.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
The Möbius strip or Möbius band, also spelled Mobius or Moebius, is a surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary.
In mathematics, a metric space is a set for which distances between all members of the set are defined.
Mikhail Leonidovich Gromov (also Mikhael Gromov, Michael Gromov or Mischa Gromov; Михаи́л Леони́дович Гро́мов; born 23 December 1943), is a French-Russian mathematician known for work in geometry, analysis and group theory.
In music theory, a minor chord is a chord having a root, a minor third, and a perfect fifth.
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds.
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition.
In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant.
In algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects.
Music theory is the study of the practices and possibilities of music.
In topology, the nerve of an open covering is a construction of an abstract simplicial complex from an open covering of a topological space X that captures many of the interesting topological properties in an algorithmic or combinatorial way.
In mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry.
In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature.
In theoretical physics orientifold is a generalization of the notion of orbifold, proposed by Augusto Sagnotti in 1987.
Particle physics phenomenology is the part of theoretical particle physics that deals with the application of theoretical physics to high-energy experiments.
In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature.
In mathematics, a projective plane is a geometric structure that extends the concept of a plane.
In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact.
In theoretical physics, quantum field theory (QFT) is the theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics.
In physics, a quiver diagram is a graph representing the matter content of a gauge theory that describes D-branes on orbifolds.
In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.
In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space.
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles.
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.
Science, also widely referred to as Science Magazine, is the peer-reviewed academic journal of the American Association for the Advancement of Science (AAAS) and one of the world's top academic journals.
A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles.
A set (pitch set, pitch-class set, set class, set form, set genus, pitch collection) in music theory, as in mathematics and general parlance, is a collection of objects.
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space.
In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that are piecewise of the simplest kind.
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration).
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.
In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.
Stratification has several usages in mathematics.
In physics, a string is a physical phenomenon that appears in string theory and related subjects.
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings.
In particle physics, supersymmetry (SUSY) is a theory that proposes a relationship between two basic classes of elementary particles: bosons, which have an integer-valued spin, and fermions, which have a half-integer spin.
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains.
Time is an American weekly news magazine and news website published in New York City.
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.
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
In music, a triad is a set of three notes (or "pitches") that can be stacked vertically in thirds.
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle.
In mathematics, a trivial group is a group consisting of a single element.
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.
In mathematics, the open unit disk (or disc) around P (where P is a given point in the plane), is the set of points whose distance from P is less than 1: The closed unit disk around P is the set of points whose distance from P is less than or equal to one: Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself.
In mathematics, the upper half-plane H is the set of complex numbers with positive imaginary part: The term arises from a common visualization of the complex number x + iy as the point (x,y) in the plane endowed with Cartesian coordinates.
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average, expected value in the vacuum.
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory.
A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern.
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician.
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space.