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Mikhail Gromov (mathematician)
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] |
Boksitogorsk
Boksitogorsk () is a town and the administrative center of Boksitogorsky District in Leningrad Oblast, Russia, located on the banks of the Pyardomlya River in the basin of the Syas River, east of St. Petersburg. Population: History The settlement of Boksity () was established in 1929 to house the workers of the local bauxite mine. It was a part of Tikhvinsky District of Leningrad Oblast. In December 1934, the construction of a bauxite plant started. In 1935, the settlement was granted urban-type settlement status and given its present name. In 1940, the population neared 10,000 and a school, kindergarten, nursery, ambulatory and drugstore, several canteens, and shops were built. In 1950, Boksitogorsk was granted town status and on July 25, 1952 it became the administrative center of Boksitogorsky District. Administrative and municipal status Within the subdivisions of Russia#Administrative divisions, framework of administrative divisions, Boksitogorsk serves as th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov–Hausdorff Convergence
In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff distance. Gromov–Hausdorff distance The Gromov–Hausdorff distance was introduced by David Edwards in 1975, and it was later rediscovered and generalized by Mikhail Gromov in 1981. This distance measures how far two compact metric spaces are from being isometric. If ''X'' and ''Y'' are two compact metric spaces, then ''dGH'' (''X'', ''Y'') is defined to be the infimum of all numbers ''d''''H''(''f''(''X''), ''g''(''Y'')) for all (compact) metric spaces ''M'' and all isometric embeddings ''f'' : ''X'' → ''M'' and ''g'' : ''Y'' → ''M''. Here ''d''''H'' denotes Hausdorff distance between subsets in ''M'' and the ''isometric embedding'' is understood in the global sense, i.e. it must preserve all distances, not only infinitesimally small ones; for example no c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filling Area Conjecture
In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points. Definitions and statement of the conjecture Every smooth surface or curve in Euclidean space is a metric space, in which the (intrinsic) distance between two points of is defined as the infimum of the lengths of the curves that go from to ''along'' . For example, on a closed curve C of length , for each point of the curve there is a unique other point of the curve (called the antipodal of ) at distance from . A compact surface fills a closed curve if its border (also called boundary, denoted ) is the curve . The filling is said to be isometric if for any two points of the boundary curve , the distance between them along is the same (not less) than the distance along the boundary. In other words, to fill a curve isometrically is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Essential Manifold
In geometry, an essential manifold is a special type of closed manifold. The notion was first introduced explicitly by Mikhail Gromov. Definition A closed manifold ''M'' is called essential if its fundamental class 'M''defines a nonzero element in the homology of its fundamental group , or more precisely in the homology of the corresponding Eilenberg–MacLane space ''K''(, 1), via the natural homomorphism :H_n(M)\to H_n(K(\pi,1)), where ''n'' is the dimension of ''M''. Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise. Examples *All closed surfaces (i.e. 2-dimensional manifolds) are essential with the exception of the 2-sphere ''S2''. *Real projective space ''RPn'' is essential since the inclusion *:\mathbb^n \to \mathbb^\infty :is injective in homology, where ::\mathbb^\infty = K(\mathbb_2, 1) :is the Eilenberg–MacLane space of the finite cyclic group of order 2. *All comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Dimension
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος (''asumptōtos'') which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve. There are three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique''. For curves given by the graph of a function , horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to Vertical asymptotes are vertical lines near which th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bishop–Gromov Inequality
In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness theorem. Statement Let M be a complete ''n''-dimensional Riemannian manifold whose Ricci curvature satisfies the lower bound : \mathrm \geq (n-1) K for a constant K\in \R. Let M_K^n be the complete ''n''-dimensional simply connected space of constant sectional curvature K (and hence of constant Ricci curvature (n-1)K); thus M_K^n is the ''n''-sphere of radius 1/\sqrt if K>0, or ''n''-dimensional Euclidean space if K=0, or an appropriately rescaled version of ''n''-dimensional hyperbolic space if K<0. Denote by the ball of radius ''r'' around a point ''p'', defined with respect to the [...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 cor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outer Space (mathematics)
In the mathematical subject of geometric group theory, the Culler–Vogtmann Outer space or just Outer space of a free group ''F''''n'' is a topological space consisting of the so-called "marked metric graph structures" of volume 1 on ''F''''n''. The Outer space, denoted ''X''''n'' or ''CV''''n'', comes equipped with a natural action of the group of outer automorphisms Out(''F''''n'') of ''F''''n''. The Outer space was introduced in a 1986 paper of Marc Culler and Karen Vogtmann, and it serves as a free group analog of the Teichmüller space of a hyperbolic surface. Outer space is used to study homology and cohomology groups of Out(''F''''n'') and to obtain information about algebraic, geometric and dynamical properties of Out(''F''''n''), of its subgroups and individual outer automorphisms of ''F''''n''. The space ''X''''n'' can also be thought of as the set of isometry types of minimal free discrete isometric actions of ''F''''n'' on R-trees ''T'' such that the quotient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov Product
In mathematics, the Gromov product is a concept in the theory of metric spaces named after the mathematician Mikhail Gromov. The Gromov product can also be used to define ''δ''-hyperbolic metric spaces in the sense of Gromov. Definition Let (''X'', ''d'') be a metric space and let ''x'', ''y'', ''z'' ∈ ''X''. Then the Gromov product of ''y'' and ''z'' at ''x'', denoted (''y'', ''z'')''x'', is defined by :(y, z)_ = \frac1 \big( d(x, y) + d(x, z) - d(y, z) \big). Motivation Given three points ''x'', ''y'', ''z'' in the metric space ''X'', by the triangle inequality there exist non-negative numbers ''a'', ''b'', ''c'' such that d(x,y) = a + b, \ d(x,z) = a + c, \ d(y,z) = b + c. Then the Gromov products are (y,z)_x = a, \ (x,z)_y = b, \ (x,y)_z = c. In the case that the points ''x'', ''y'', ''z'' are the outer nodes of a tripod then these Gromov products are the lengths of the edges. In the hyperbolic, spherical or euclidean plane, the Gromov produc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Volume
In the mathematical field of geometric topology, the simplicial volume (also called Gromov norm) is a measure of the topological complexity of a manifold. More generally, the simplicial norm measures the complexity of homology classes. Given a closed and oriented manifold, one defines the simplicial norm by minimizing the sum of the absolute values of the coefficients over all singular chains homologous to a given cycle. The simplicial volume is the simplicial norm of the fundamental class.. It is named after Mikhail Gromov, who introduced it in 1982. With William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurst ..., he proved that the simplicial volume of a finite volume hyperbolic manifold is proportional to the hyperbolic volume. The simplicial volume is equal to twice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
