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Systolic Freedom
In differential geometry, systolic freedom refers to the fact that closed Riemannian manifolds may have arbitrarily small volume regardless of their systolic invariants. That is, systolic invariants or products of systolic invariants do not in general provide universal (i.e. curvature-free) lower bounds for the total volume of a closed Riemannian manifold. Systolic freedom was first detected by Mikhail Gromov in an I.H.É.S. preprint in 1992 (which eventually appeared as ), and was further developed by Mikhail Katz, Michael Freedman and others. Gromov's observation was elaborated on by . One of the first publications to study systolic freedom in detail is by . Systolic freedom has applications in quantum error correction. survey the main results on systolic freedom. Example The complex projective plane admits Riemannian metrics of arbitrarily small volume, such that every essential surface is of area at least 1. Here a surface is called "essential" if it cannot be contracte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. The family ''g''''p'' of inner products is called a metric tensor, Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take ''g'' to be Smoothness, smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :g\left(\frac,\frac\right):U\to\mathbb are smooth functions. These functions are commonly designated as g_. With further restrictions on the g_, one could also consider Lipschitz continuity, Lipschitz Riemannian metrics or Measurable function, measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The definition of length (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. In ancient times, volume is measured using similar-shaped natural containers and later on, standardized containers. Some simple three-dimensional shapes can have its volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. Zero-, one- and two-dimensional objects have no volume; in fourth and higher dimensions, an analogous concept to the normal vo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Systolic Geometry
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slower-paced Introduction to systolic geometry. The notion of systole The ''systole'' of a compact metric space ''X'' is a metric invariant of ''X'', defined to be the least length of a noncontractible loop in ''X'' (i.e. a loop that cannot be contracted to a point in the ambient space ''X''). In more technical language, we minimize length over free loops representing nontrivial conjugacy classes in the fundamental group of ''X''. When ''X'' is a graph, the invariant is usually referred to as the girth, ever since the 1947 article on girth by W. T. Tutte. Possibly inspired by Tutte's article, Loewner started thinking about systolic questions on surfaces in the la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Gromov (mathematician)
Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; russian: link=no, Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of IHÉS in France and a professor of mathematics at New York University. Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry". Biography Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union. His Russian father Leonid Gromov and his Jewish mother Lea Rabinovitz were pathologists. His mother was the cousin of World Chess Champion Mikhail Botvinnik, as well as of the mathematician Isaak Moiseevich Rabinovich. Gromov was born during World War II, and his mother, who worked as a medical doctor in the Soviet Army, had to leave the front line in order to give birth to him. When Gromov was nine years old, his mother ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Institut Des Hautes Études Scientifiques
The Institut des hautes études scientifiques (IHÉS; English: Institute of Advanced Scientific Studies) is a French research institute supporting advanced research in mathematics and theoretical physics. It is located in Bures-sur-Yvette, just south of Paris. It is an independent research institute in a partnership with the University of Paris-Saclay. History The IHÉS was founded in 1958 by businessman and mathematical physicist Léon Motchane with the help of Robert Oppenheimer and Jean Dieudonné as a research centre in France, modeled on the renowned Institute for Advanced Study in Princeton, United States. The strong personality of Alexander Grothendieck and the broad sweep of his revolutionizing theories were a dominating feature of the first ten years at the IHÉS. René Thom received an invitation from IHÉS in 1963 and after his appointment remained there until his death in 2002. Dennis Sullivan is remembered as one who had a special talent for encouraging fruitf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Katz
Mikhail "Mischa" Gershevich Katz (born 1958, in Chișinău)Curriculum vitae retrieved 2011-05-23. is an Israeli , a professor of mathematics at . His main interests are , and |
Michael Freedman
Michael Hartley Freedman (born April 21, 1951) is an American mathematician, at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the 4-dimensional generalized Poincaré conjecture. Freedman and Robion Kirby showed that an exotic ℝ4 manifold exists. Life and career Freedman was born in Los Angeles, California, in the United States. His father, Benedict Freedman, was an American Jewish aeronautical engineer, musician, writer, and mathematician. His mother, Nancy Mars Freedman, performed as an actress and also trained as an artist. His parents cowrote a series of novels together. . He entered the University of California, Berkeley, but dropped out after two semesters. In the same year he wrote a letter to Ralph Fox, a Princeton professor at the time, and was admitted to graduate school so in 1968 he continued his studies at Princeton University where he received Ph.D. degree in 1973 fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Error Correction
Quantum error correction (QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault tolerant quantum computing that can reduce the effects of noise on stored quantum information, faulty quantum gates, faulty quantum preparation, and faulty measurements. Classical error correction employs redundancy. The simplest albeit inefficient approach is the repetition code. The idea is to store the information multiple times, and—if these copies are later found to disagree—take a majority vote; e.g. suppose we copy a bit in the one state three times. Suppose further that a noisy error corrupts the three-bit state so that one of the copied bits is equal to zero but the other two are equal to one. Assuming that noisy errors are independent and occur with some sufficiently low probability ''p'', it is most likely that the error is a single-bit error and the tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Projective Plane
In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1,Z_2,Z_3)\neq (0,0,0) where, however, the triples differing by an overall rescaling are identified: :(Z_1,Z_2,Z_3) \equiv (\lambda Z_1,\lambda Z_2, \lambda Z_3);\quad \lambda\in \mathbf,\qquad \lambda \neq 0. That is, these are homogeneous coordinates in the traditional sense of projective geometry. Topology The Betti numbers of the complex projective plane are :1, 0, 1, 0, 1, 0, 0, ..... The middle dimension 2 is accounted for by the homology class of the complex projective line, or Riemann sphere, lying in the plane. The nontrivial homotopy groups of the complex projective plane are \pi_2=\pi_5=\mathbb. The fundamental group is trivial and all other higher homotopy groups are those of the 5-sphere, i.e. torsion. Algebraic geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov's Systolic Inequality For Essential Manifolds
In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1983;see it can be viewed as a generalisation, albeit non-optimal, of Loewner's torus inequality and Pu's inequality for the real projective plane. Technically, let ''M'' be an essential Riemannian manifold of dimension ''n''; denote by sys''π''1(''M'') the homotopy 1-systole of ''M'', that is, the least length of a non-contractible loop on ''M''. Then Gromov's inequality takes the form : \left(\operatorname_1(M)\right)^n \leq C_n \operatorname(M), where ''C''''n'' is a universal constant only depending on the dimension of ''M''. Essential manifolds A closed manifold is called ''essential'' if its fundamental class defines a nonzero element in the homology of its fundamental group, or more precisely in the homology of the corre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
