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Pedro Ontaneda
Pedro Ontaneda Portal is a Peruvian-American mathematician specializing in topology and differential geometry. He is a distinguished professor at Binghamton University, a unit of the State University of New York. Education and career Ontaneda received his Ph.D. in 1994 from Stony Brook University (another unit of SUNY), advised by Lowell Jones. Subsequently he taught at the Federal University of Pernambuco in Brazil. He moved to Binghamton University in 2005. Mathematical contributions Ontaneda's work deals with the geometry and topology of aspherical spaces, with particular attention to the relationship between exotic structures and negative or non-positive curvature on manifolds. Classical examples of Riemannian manifolds of negative curvature are given by real hyperbolic manifolds, or more generally by locally symmetric spaces of rank 1. One of Ontaneda's most celebrated contributions is the construction of manifolds that admit negatively curved Riemannian metrics but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Torsion (mechanics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a Set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and Homotopy, homotopies. A property that is invariant under such deformations is a to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by . The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology (in particular the results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface, and more complex phenomena in three-dimensional topology), and combinatorial group theory. In a very influential (over 1000 citations ) chapter from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others. Definition Let G be a fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American People Of Peruvian Descent
American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States of America ** American ancestry, people who self-identify their ancestry as "American" ** American English, the set of varieties of the English language native to the United States ** Native Americans in the United States, indigenous peoples of the United States * American, something of, from, or related to the Americas, also known as "America" ** Indigenous peoples of the Americas * American (word), for analysis and history of the meanings in various contexts Organizations * American Airlines, U.S.-based airline headquartered in Fort Worth, Texas * American Athletic Conference, an American college athletic conference * American Recordings (record label), a record label that was previously known as Def American * American University, in Washington, D.C. Sports teams S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Place Of Birth Missing (living People)
Place may refer to: Geography * Place (United States Census Bureau), defined as any concentration of population ** Census-designated place A census-designated place (CDP) is a Place (United States Census Bureau), concentration of population defined by the United States Census Bureau for statistical purposes only. CDPs have been used in each decennial census since 1980 as the counte ..., a populated area lacking its own municipal government * "Place", a type of street or road name ** Often implies a dead end (street) or cul-de-sac * Place, based on the Cornish word "plas" meaning mansion * Place, a populated place, an area of human settlement ** Incorporated place (see municipal corporation), a populated area with its own municipal government * Location (geography), an area with definite or indefinite boundaries or a portion of space which has a name in an area Placenames * Placé, a commune in Pays de la Loire, Paris, France * Plače, a small settlement in Slov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Living People
Purpose: Because living persons may suffer personal harm from inappropriate information, we should watch their articles carefully. By adding an article to this category, it marks them with a notice about sources whenever someone tries to edit them, to remind them of WP:BLP (biographies of living persons) policy that these articles must maintain a neutral point of view, maintain factual accuracy, and be properly sourced. Recent changes to these articles are listed on Special:RecentChangesLinked/Living people. Organization: This category should not be sub-categorized. Entries are generally sorted by family name In many societies, a surname, family name, or last name is the mostly hereditary portion of one's personal name that indicates one's family. It is typically combined with a given name to form the full name of a person, although several give .... Maintenance: Individuals of advanced age (over 90), for whom there has been no new documentation in the last ten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Year Of Birth Missing (living People)
A year is a unit of time based on how long it takes the Earth to orbit the Sun. In scientific use, the tropical year (approximately 365 solar days, 5 hours, 48 minutes, 45 seconds) and the sidereal year (about 20 minutes longer) are more exact. The modern calendar year, as reckoned according to the Gregorian calendar, approximates the tropical year by using a system of leap years. The term 'year' is also used to indicate other periods of roughly similar duration, such as the lunar year (a roughly 354-day cycle of twelve of the Moon's phasessee lunar calendar), as well as periods loosely associated with the calendar or astronomical year, such as the seasonal year, the fiscal year, the academic year, etc. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by changes in weather, the hours of daylight, and, consequently, vegetation and soil fertility. In temperate and subpolar regions around the planet, four seasons a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anosov Diffeomorphism
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contraction". Anosov systems are a special case of Axiom A systems. Anosov diffeomorphisms were introduced by Dmitri Anosov, Dmitri Victorovich Anosov, who proved that their behaviour was in an appropriate sense ''generic'' (when they exist at all). Overview Three closely related definitions must be distinguished: * If a differentiable map (mathematics), map ''f'' on ''M'' has a Hyperbolic set, hyperbolic structure on the tangent bundle, then it is called an Anosov map. Examples include the Bernoulli map, and Arnold's cat map. * If the map is a diffeomorphism, then it is called an Anosov diffeomorphism. * If a flow (mathematics), flow on a manifold splits the tangent bundle into three invariant subbundles, with one subbundle that is exponen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a Set (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolization Theorem
In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture. Statement One form of the hyperbolization theorem states: If ''M'' is a compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of ''M'' has a complete hyperbolic structure of finite volume. The Mostow rigidity theorem implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique. The conditions that the manifold ''M'' should be irreducible and atoroidal are necessary, as hyperbolic manifolds have these properties. However the condition that the manifold be Haken is unnecessarily strong. Thurston's hyperbolization conjecture states that a closed irreducible atoroidal 3-manifold with infinite fundamental group is hyperbolic, and this follows from Perelman's proof of the Thurst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniformization Theorem
In mathematics, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces. Since every Riemann surface has a universal cover which is a simply connected Riemann surface, the uniformization theorem leads to a classification of Riemann surfaces into three types: those that have the Riemann sphere as universal cover ("elliptic"), those with the plane as universal cover ("parabolic") and those with the unit disk as universal cover ("hyperbolic"). It further follows that every Riemann surface admits a Riemannian metric of constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case. The uniformization theorem also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane where every point is a saddle point. Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane. The hyperboloid model of hyperbolic geometry provides a representation of events one temporal unit into the future in Minkowski space, the basis of special relativity. Eac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pontryagin Classes
In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Definition Given a real vector bundle E over M, its k-th Pontryagin class p_k(E) is defined as :p_k(E) = p_k(E, \Z) = (-1)^k c_(E\otimes \Complex) \in H^(M, \Z), where: *c_(E\otimes \Complex) denotes the 2k-th Chern class of the complexification E\otimes \Complex = E\oplus iE of E, *H^(M, \Z) is the 4k-cohomology group of M with integer coefficients. The rational Pontryagin class p_k(E, \Q) is defined to be the image of p_k(E) in H^(M, \Q), the 4k-cohomology group of M with rational coefficients. Properties The total Pontryagin class :p(E)=1+p_1(E)+p_2(E)+\cdots\in H^*(M,\Z), is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e., :2p(E\oplus F)=2p(E)\smile p(F) for two vector bundles E and F over M. In terms of the individual Pontry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
