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P-compact Group
In mathematics, in particular algebraic topology, a ''p''-compact group is a homotopical version of a compact Lie group, but with all the local structure concentrated at a single prime ''p''. This concept was introduced in , making precise earlier notions of a mod p finite loop space. A p-compact group has many Lie-like properties like maximal tori and Weyl groups, which are defined purely homotopically in terms of the classifying space, but with the important difference that the Weyl group, rather than being a finite reflection group over the integers, is now a finite ''p''-adic reflection group. They admit a classification in terms of root data, which mirrors the classification of compact Lie groups, but with the integers replaced by the ''p''-adic integers. Definition A ''p''-compact group is a pointed space ''BG'', with is local with respect to mod ''p'' homology, and such the pointed loop space ''G = ΩBG'' has finite mod ''p'' homology. One sometimes also refer to the ''p ... [...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 points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre (geometry), centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. spherical Earth, The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in any direction, so mos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Neighbourhood (mathematics), neighborhood that is homeomorphic to an open (topology), open subset of n-dimensional Euclidean space. One-dimensional manifolds include Line (geometry), lines and circles, but not Lemniscate, lemniscates. Two-dimensional manifolds are also called Surface (topology), surfaces. Examples include the Plane (geometry), plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as Graph of a function, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Groups
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. Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and category theory (specifically the study of higher categories). Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated, or Hausdorff, or a CW complex. In the same vein as above, a "map" is a continuous function, possibly with some extra constraints. Often, one works with a pointed space -- that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology Of Lie Groups
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
J A Todd
John Arthur Todd (23 August 1908 – 22 December 1994) was an English mathematician who specialised in geometry. Biography He was born in Liverpool, and went up to Trinity College, Cambridge in 1925. He did research under H.F. Baker, and in 1931 took a position at the University of Manchester. He became a lecturer at Cambridge in 1937. He remained at Cambridge for the rest of his working life. Work The Todd class in the theory of the higher-dimensional Riemann–Roch theorem is an example of a characteristic class (or, more accurately, a reciprocal of one) that was discovered by Todd in work published in 1937. It used the methods of the Italian school of algebraic geometry. The Todd–Coxeter process for coset enumeration is a major method of computational algebra, and dates from a collaboration with H.S.M. Coxeter in 1936. In 1953 he and Coxeter discovered the Coxeter–Todd lattice. In 1954 he and G. C. Shephard classified the finite complex reflection groups. Honours I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geoffrey Colin Shephard
Geoffrey Colin Shephard is a mathematician who works on convex geometry and reflection groups. He asked Shephard's problem on the volumes of projected convex bodies, posed another problem on polyhedral nets, proved the Shephard–Todd theorem in invariant theory of finite groups, began the study of complex polytopes, and classified the complex reflection groups. Shephard earned his Ph.D. in 1954 from Queens' College, Cambridge, under the supervision of J. A. Todd. He was a professor of mathematics at the University of East Anglia The University of East Anglia (UEA) is a public research university in Norwich, England. Established in 1963 on a campus west of the city centre, the university has four faculties and 26 schools of study. The annual income of the institution f ... until his retirement. University of East Anglia School of M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Gerard Dwyer
William Gerard Dwyer (born 1947) is an American mathematician specializing in algebraic topology and group theory. For many years he was a professor at the University of Notre Dame, where he is the William J. Hank Family Professor Emeritus. Life He was born in 1947 in Jersey City, New Jersey. Career Dwyer completed his B.A. at Boston College in 1969. He completed his Ph.D. at the Massachusetts Institute of Technology in 1973. His doctoral thesis was on ''Strong Convergence of the Eilenberg-Moore Spectral Sequence'' and his doctoral advisor was Daniel Kan. Afterwards he taught at Yale University and visited the Institute for Advanced Study in Princeton, New Jersey before joining the faculty at the University of Notre Dame. In 1998 Dwyer was an invited speaker at the International Congress of Mathematicians in Berlin. In 2007 he was awarded a Doctor Honoris Causa degree by the University of Warsaw. He was elected a Fellow of the American Mathematical Society in 2012. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Data
In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, published in 1970. Definition A root datum consists of a quadruple :(X^\ast, \Phi, X_\ast, \Phi^\vee), where * X^\ast and X_\ast are free abelian groups of finite rank together with a perfect pairing between them with values in \mathbb which we denote by ( , ) (in other words, each is identified with the dual of the other). * \Phi is a finite subset of X^\ast and \Phi^\vee is a finite subset of X_\ast and there is a bijection from \Phi onto \Phi^\vee, denoted by \alpha\mapsto\alpha^\vee. * For each \alpha, (\alpha, \alpha^\vee)=2. * For each \alpha, the map x\mapsto x-(x,\alpha^\vee)\alpha induces an automorphism of the root datum (in other words it maps \Phi to \Phi and the induced action on X_\ast maps \Phi^\vee to \Phi^\vee) The elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
