Mikhail Gromov (mathematician)
   HOME
*





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]  


Boksitogorsk
Boksitogorsk (russian: Бокситого́рск) 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 framework of administrative divisions, Boksitogorsk serves as the administrat ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Almost Flat Manifold
In mathematics, a smooth compact manifold ''M'' is called almost flat if for any \varepsilon>0 there is a Riemannian metric g_\varepsilon on ''M'' such that \mbox(M,g_\varepsilon)\le 1 and g_\varepsilon is \varepsilon-flat, i.e. for the sectional curvature of K_ we have , K_, 0 such that if an ''n''-dimensional manifold admits an \varepsilon_n-flat metric with diameter \le 1 then it is almost flat. On the other hand, one can fix the bound of sectional curvature and get the diameter going to zero, so the almost-flat manifold is a special case of a collapsing manifold, which is collapsing along all directions. According to the Gromov–Ruh theorem, ''M'' is almost flat if and only if it is infranil. In particular, it is a finite factor of a nilmanifold In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Filling Radius
In Riemannian geometry, the filling radius of a Riemannian manifold ''X'' is a metric invariant of ''X''. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form. The filling radius of a simple loop ''C'' in the plane is defined as the largest radius, ''R'' > 0, of a circle that fits inside ''C'': :\mathrm(C\subset \mathbb^2) = R. Dual definition via neighborhoods There is a kind of a dual point of view that allows one to generalize this notion in an extremely fruitful way, as shown by Gromov. Namely, we consider the \varepsilon-neighborhoods of the loop ''C'', denoted :U_\varepsilon C \subset \mathbb^2. As \varepsilon>0 increases, the \varepsilon-neighborhood U_\varepsilon C swallows up more and more of the interior of the loop. The ''last' ...
[...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 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 to fill ...
[...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 metric geometry, asymptotic dimension of a metric space is a large-scale analog of Lebesgue covering dimension. The notion of asymptotic dimension was introduced by Mikhail Gromov in his 1993 monograph ''Asymptotic invariants of infinite groups'' in the context of geometric group theory, as a quasi-isometry invariant of finitely generated groups. As shown by Guoliang Yu, finitely generated groups of finite homotopy type with finite asymptotic dimension satisfy the Novikov conjecture. Asymptotic dimension has important applications in geometric analysis and index theory. Formal definition Let X be a metric space and n\ge 0 be an integer. We say that \operatorname(X)\le n if for every R\ge 1 there exists a uniformly bounded cover \mathcal U of X such that every closed R-ball in X intersects at most n+1 subsets from \mathcal U. Here 'uniformly bounded' means that \sup_ \operatorname(U) <\infty . We then define the ''asymptotic dimension'' \operatorname(X)
[...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 B(p,r) 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 corres ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Gromov's Inequality For Complex Projective Space
In Riemannian geometry, Gromov's optimal stable 2- systolic inequality is the inequality : \mathrm_2^n \leq n! \;\mathrm_(\mathbb^n), valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained by the symmetric Fubini–Study metric, providing a natural geometrisation of quantum mechanics. Here \operatorname is the stable 2-systole, which in this case can be defined as the infimum of the areas of rational 2-cycles representing the class of the complex projective line \mathbb^1 \subset \mathbb^n in 2-dimensional homology. The inequality first appeared in as Theorem 4.36. The proof of Gromov's inequality relies on the Wirtinger inequality for exterior 2-forms. Projective planes over division algebras \mathbb In the special case n=2, Gromov's inequality becomes \mathrm_2^2 \leq 2 \mathrm_4(\mathbb^2). This inequality can be thought of as an analog of Pu's inequality for the real projective plane \mathbb^2. In both cases, th ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

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 ''F''''n'' on R-trees ''T'' such that the q ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

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 certain 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 representing a 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, 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 the Thurston norm Thurston also used the simplicial volume to prove that hyperbolic volume decreases under hyperbolic Dehn surgery In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a giv ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]