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Alexandrov Space
In geometry, Alexandrov spaces with curvature ≥ ''k'' form a generalization of Riemannian manifolds with sectional curvature ≥ ''k'', where ''k'' is some real number. By definition, these spaces are locally compact complete length spaces where the lower curvature bound is defined via comparison of geodesic triangles in the space to geodesic triangles in standard constant-curvature Riemannian surfaces. One can show that the Hausdorff dimension of an Alexandrov space with curvature ≥ ''k'' is either a non-negative integer or infinite. One can define a notion of "angle" (see Comparison triangle#Alexandrov angles) and "tangent cone" in these spaces. Alexandrov spaces with curvature ≥ ''k'' are important as they form the limits (in the Gromov–Hausdorff metric) of sequences of Riemannian manifolds with sectional curvature ≥ ''k'', as described by Gromov's compactness theorem. Alexandrov spaces with curvature ≥ ''k'' were introduced by the Russian mathematician Aleks ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov's Compactness Theorem (geometry)
In the mathematical field of metric geometry, Mikhael Gromov proved a fundamental compactness theorem for sequences of metric spaces. In the special case of Riemannian manifolds, the key assumption of his compactness theorem is automatically satisfied under an assumption on Ricci curvature. These theorems have been widely used in the fields of geometric group theory and Riemannian geometry. Metric compactness theorem The Gromov–Hausdorff distance defines a notion of distance between any two metric spaces, thereby setting up the concept of a sequence of metric spaces which converges to another metric space. This is known as Gromov–Hausdorff convergence. Gromov found a condition on a sequence of compact metric spaces which ensures that a subsequence converges to some metric space relative to the Gromov–Hausdorff distance: Let be a sequence of compact metric spaces with uniformly bounded diameter. Suppose that for every positive number there is a natural number and, for eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Geometry
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a Conceptual metaphor , metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Conjecture
In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured by Henri Poincaré in 1904, the theorem concerns spaces that locally look like ordinary three-dimensional space but which are finite in extent. Poincaré hypothesized that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. Attempts to resolve the conjecture drove much progress in the field of geometric topology during the 20th century. The eventual proof built upon Richard S. Hamilton's program of using the Ricci flow to solve the problem. By developing a number of new techniques and results in the theory of Ricci flow, Grigori Perelman was able to modify and complete Hamilton's program. In papers posted to the arXiv reposi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grigori Perelman
Grigori Yakovlevich Perelman (, ; born 13June 1966) is a Russian mathematician and geometer who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. In 2005, Perelman resigned from his research post in Steklov Institute of Mathematics and in 2006 stated that he had quit professional mathematics, owing to feeling disappointed over the ethical standards in the field. He lives in seclusion in Saint Petersburg and has declined requests for interviews since 2006. In the 1990s, partly in collaboration with Yuri Burago, Mikhael Gromov, and Anton Petrunin, he made contributions to the study of Alexandrov spaces. In 1994, he proved the soul conjecture in Riemannian geometry, which had been an open problem for the previous 20 years. In 2002 and 2003, he developed new techniques in the analysis of Ricci flow, and proved the Poincaré conjecture and Thurston's geometrization conjecture, the former of which had been a famous op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Leonidovich Gromov
Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; ; 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 Institut des Hautes Études Scientifiques 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". Early years, education and career Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union. His father Leonid Gromov was Russian-Slavic and his mother Lea was of Jewish heritage. Both 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 ol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuri Burago
Yuri Dmitrievich Burago (; born 21 June 1936) is a Russian mathematician. He works in differential and convex geometry. Education and career Burago studied at Leningrad University, where he obtained his Ph.D. and Habilitation degrees. His advisors were Victor Zalgaller and Aleksandr Aleksandrov. Yuri is a creator (with his students Perelman and Petrunin, and M. Gromov) of what is known now as Alexandrov Geometry. Also brought geometric inequalities to the state of art. Burago is the head of the Laboratory of Geometry and Topology that is part of the St. Petersburg Department of Steklov Institute of Mathematics. He took part in a report for the United States Civilian Research and Development Foundation for the Independent States of the former Soviet Union. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandrov Topology
In general topology, an Alexandrov topology is a topology in which the intersection of an ''arbitrary'' family of open sets is open (while the definition of a topology only requires this for a ''finite'' family). Equivalently, an Alexandrov topology is one whose open sets are the upper sets for some preorder on the space. Spaces with an Alexandrov topology are also known as Alexandrov-discrete spaces or finitely generated spaces. The latter name stems from the fact that their topology is uniquely determined by the family of all finite subspaces. This makes them a generalization of finite topological spaces. Alexandrov-discrete spaces are named after the Russian topologist Pavel Alexandrov. They should not be confused with Alexandrov spaces from Riemannian geometry introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov. Characterizations of Alexandrov topologies Alexandrov topologies have numerous characterizations. In a topological space X, the followi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandr Danilovich Aleksandrov
Aleksandr Danilovich Aleksandrov (; 4 August 1912 – 27 July 1999) was a Soviet and Russian mathematician, physicist, philosopher and mountaineer. Personal life Aleksandr Aleksandrov was born in 1912 in Volyn, Ryazan Oblast. His father was a headmaster of a secondary school in St Petersburg and his mother a teacher at said school, thus the young Alekandrov spent a majority of his childhood in the city. His family was old Russian nobility—students noted ancestral portraits which hung in his office. His sisters were Soviet botanist Vera Danilovna Aleksandrov (RU) and Maria Danilovna Aleksandrova, author of the first monograph on gerontopsychology in the USSR. In 1937, he married a student of the Faculty of Physics, Marianna Leonidovna Georg. Together they had two children: Daria (b. 1948) and Daniil (RU) (b. 1957). In 1980, he married Svetlana Mikhailovna Vladimirova (nee Bogacheva). In 1951 he became a member of the Communist Party. Alekandrov had a personal love for po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifold, manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport. Any smooth surface in three-dimensional Eucl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparison Triangle
In metric geometry, comparison triangles are constructions used to define higher bounds on curvature in the framework of locally geodesic metric spaces, thereby playing a similar role to that of higher bounds on sectional curvature in Riemannian geometry. Definitions Comparison triangles Let M_^ = \mathbb^2 be the euclidean plane, M_^ = \mathbb^2 be the unit 2-sphere, and M_^ = \mathbb^2 be the hyperbolic plane. For k > 0, let M_^ and M_^ denote the spaces obtained, respectively, from M_^ and M_^ by multiplying the distance by \frac. For any k\in \R, M_^ is the unique complete, simply-connected, 2-dimensional Riemannian manifold of constant sectional curvature k. Let X be a metric space. Let T be a geodesic triangle in X, i.e. three points p, q and r and three geodesic segments , q/math>, , r/math> and , p/math>. A comparison triangle T* in M_^ for T is a geodesic triangle in M_^ with vertices p', q' and r' such that d(p,q) = d(p',q'), d(p,r) = d(p',r') and d(r,q) = d(r',q') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
