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Topic: Gromov-Hausdorff convergence

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Gromov-Hausdorff convergence - Wikipedia, the free encyclopedia Gromov-Hausdorff convergence is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence. The notion of Gromov-Hausdorff convergence was first used by Gromov to prove that any discrete group with polynomial growth is almost nilpotent (i.e. The key ingredient in the proof was almost trivial observation that for the Cayley graph of a group with polynomial growth a sequence of rescalings converges in the pointed Gromov-Hausdorff sense. en.wikipedia.org /wiki/Gromov-Hausdorff_convergence   (326 words)

Mikhail Gromov - Wikipedia, the free encyclopedia Mikhail Leonidovich Gromov Russian: Михаил Леонидович Громов (born December 23, 1943, also known as Mikhael Gromov, Michael Gromov, or Misha Gromov) is a mathematician known for important contributions in many different areas of geometry, especially metric geometry, symplectic geometry, and geometric group theory. Mikhail Gromov studied for a doctorate (1973) in Leningrad, where he was a student of V. For other people named Mikhail Gromov, see Mikhail Gromov (disambiguation). en.wikipedia.org /wiki/Mikhail_Gromov   (157 words)

Convergence Index - www.computer-tutorials-online.com Convergence and Unity is a coalition of the two political parties Democratic Convergence of Catalonia and the Democratic Union of Catalonia in Catalonia Spain. Convergence (evolutionary computing) is a means of modelling the tendency for genetic characteristics of populations to stabilize over time. Catch-up effect is otherwise known as the Theory of convergence in economic theory. www.computer-tutorials-online.com /Convergence   (1205 words)

Hausdorff distance - Wikpedia Hausdorff distance can be defined the same way for closed non-compact subsets of M, but in this case the distance may take infinite value and the topology of F(M) starts to depend on particular metric on M (not only on its topology). Hausdorff distance measures how far two compact subsets of a metric space are from each other. It gives a pre-metric (or pseudometric) on the set of all subsets of M (Hausdorff distance between any two sets and with the same closures is zero). www.bostoncoop.net /~tpryor/wiki/index.php?title=Hausdorff_distance   (230 words)

2003-01-009.tex.html The second is the geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. Geometric convergence for manifolds with boundary} A sequence $(\Mbar _{k},g_{k})$ of compact Riemannian manifolds with boundary $\pa M_{k}$ is said to converge in the $C^{r}$-topology (given $0 The first is the regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. www.math.psu.edu /era-mirror/2003-01-009/2003-01-009.tex.html   (2501 words)

malawi.ca - Hausdorff distance We explain what is Hausdorff distance, an give an algorithm for computing it in linear time for convex polygons. Given two sets of points and, the Hausdorff distance is defined as. h(A,B) is called the directed Hausdorff `distance' from A to B (this func... www.malawi.ca /Hausdorff-distance/reference/fullview/wikipedia/429296   (96 words)

arb-ws9900-2.html Petersen, "Gromov-Hausdorff convergence of metric spaces", in "Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990)," Proc. Gromov, "Metric structures for Riemannian and non-Riemannian Gallot, Hulin and Lafontaine, "Riemannian Geometry," 2. Gromov, "Metric structures for Riemannian and non-Riemannian spaces," appendices by M. Katz, P. Pansu and S. Semmes, Progress in Math. www.math.ethz.ch /analysis+geometry/arb/arb-ws9900/arb-ws9900-2.html   (334 words)

280-01 "On Hausdorff-Gromov convergence and a theorem of Paulin", by M.Bridson and G.Swarup; Enseign. Action on the boundary as a convergence action. Amalgamated free products, HNN extensions and the Combination Theorem. www.math.uiuc.edu /~kapovich/415-01/415-01.html   (343 words)

wikien.info: Main_Page Gromov-Hausdorff convergence is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence. In Riemannian geometry is Gromov's compactness theorem states that the set of Riemannian manifolds with Ricci curvature ≥ c and diameter ≤ D is pre-compact in the Gromov-Hausdorff metric. Gromov's theorem may mean one of a number of results of Mikhail Gromov: Gromov's compactness theoremGromov's Betti number theoremGromov's theorem on almost flat manifoldsGromov's theorem on groups of polynomial growth See also Bishop-Gromov inequalityGromov-Thurston 2pi theorem This is a [dis.. mutluyasam.info /browse.php?title=G/GR/GRO   (11138 words)

CurRes.html Noncommutative metric spaces in the sense of M. Rieffel and noncommutative Gromov-Hausdorff convergence. The relations between the theory of renormalization in algebraic quantum field theory and the non-commutative generalization of Gromov's notion of tangent cone will be studied. The formulation of generally covariant algebraic theories will be studied further and extended to the case of non-globally hyperbolic spacetime using the generally covariant formulation of the renormalization group a la Buchholz and Verch both in the perturbative and the non-perturbative case. www.mat.uniroma2.it /~mp/OA/research/CurRes.html   (787 words)

Citations: On Hausdorff-Gromov convergence and a theorem of Paulin - Bridson, Swarup (ResearchIndex) Citations: On Hausdorff-Gromov convergence and a theorem of Paulin - Bridson, Swarup (ResearchIndex) Bridson and G. Swarup, On Hausdorff-Gromov convergence and a theorem of Paulin, to appear in L'Ens. In effect, we are conjecturing that the word hyperbolic group K = Go OE Z is such that G is hyperbolic iff there.... citeseer.ist.psu.edu /context/470339/0   (207 words)

Cornell Math - MATH 757, Fall 2000 Potential topics include Toponogov's theorem, Bishop-Gromov volume comparison, Gromov-Hausdorff convergence, finiteness theorems (for homotopy or diffeomorphism types), critical point theory for the distance function, Alexandrov spaces, sphere theorems, convexity and soul theorems. While the focus of the course will be Riemannian manifolds, sectional curvature and Ricci curvature, we will also explore how the notion of curvature can be extended to more general metric spaces. Prerequisites: An introduction to Riemannian geometry, 651 and a willingness to accept results from 662 as needed. www.math.cornell.edu /Courses/GradCourses/FA00/757.html   (95 words)

DMS.MPS.a9970509.txt I believe that a quantum Gromov-Hausdorff convergence can often be applied to give a stronger form of convergence. In most such situations of which I am aware, the present notions of convergence are either just heuristic, or quite weak. I expect to apply this theory to the many situations already in view in the physics and mathematics of quantization where one has a sequence of quantum spaces which appear to be converging to another space, either quantum or classical. www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9970509.txt   (413 words)

Papers of Sergei Ivanov Let $n\ge 2$, $M$ and $M_k$ ($k=1,2,...$) be compact Riemannian $n$-manifolds, possibly with boundaries, and let $\{M_k\}$ converge to $M$ with respect to the Gromov-Hausdorff distance. For $n\ge 3$ we give examples of convergence in which $M$ and $M_k$ are diffeomorphic to $S^n$ and $Vol(M_k)\to 0$. A note on converging metrics of curvature bounded above on 2-polyhedra. www.pdmi.ras.ru /~svivanov/papers   (367 words)

IMPRS Working Areas The questions of interest here include the triviality of such spaces (rigidity) and the structure of their boundaries (e.g., Gromov-Hausdorff convergence), among others. Gromov Precompactness Theorem; Quasi-isometric Rigidity; Spaces of Nonpositive Curvature; Spaces of Maps; Isospectral Manifolds Mathematical objects of a given type often come in families depending on continuous parameters. www.imprs-modulispaces.mpg.de /working_areas.html   (1038 words)

CUNYGC DGS Loftin The main result states that if we have a converging sequence of compact manifolds with a uniform upper bound on diameter and lower bound on Ricci curvature, then their limit space has a universal cover. In particular, if a sequence of Riemannian manifolds converge to a metric space, Y, then the universal covers of these Riemannian manifolds need not even converge to a covering of $Y$. She will then present examples demonstrating how the universal covers of converging sequences of metric spaces behave. comet.lehman.cuny.edu /sormani/seminars/s01/sormani01.html   (172 words)

Limits of polyhedra in Gromov-Hausdorff Space (ResearchIndex) 1 Gromov-Hausdorff convergence to non-manifolds (context) - Moore We also show that these spaces are precisely the compact metric spaces which are limits of polyhedra in Gromov's topological moduli spaces M(n; ae) for some choice of ae and n. 0.1: Convergence Of Alexandrov Spaces And Spectrum Of Laplacian - Takashi Shioya (1998) citeseer.ist.psu.edu /ferry95limits.html   (447 words)

convergence :: gada.be Convergence: The International Journal of Research into New Media... Nokia announces convergence devices for mobile TV and the digital... US and France convergence in the Middle East A ver... convergence.gada.be   (458 words)

topfin.txt Typeset by LAMS-* *TEX 1 2 where X and Y are isometrically embedded in some Z. (iii)Let CM be the set of isometry classes of compact metric spaces with the * *Gromov- Hausdorff metric. This set is pr* *ecompact by Gromov's Precompactness Theorem, [G], so Theorem 1 shows that this class con* *tains at most finitely many homeomorphism types. (i)If X and Y are compact subsets of a metric space Z, the Hausdorff distanc* *e between X and Y is dHZ(X; Y) = inf{ffl > 0 hopf.math.purdue.edu /Ferry/topfin.txt   (3542 words)

03syla In particular we will show that volume is continuous under Gromov-Hausdorff convergence for mainfolds with Ricci curvature lower bounds. We will start by introducing Gromov-Hausdorff convergence, proving Gromov's precompactness, then study the properties of the limit spaces. From volume comparison we will prove Cheng's diameter sphere theorem, Milnor's growth of fundamental group, Gromov's bound and optimal bound of the first Betti number for almost nonnegative Ricci curvature, Anderson's theorem on the finiteness of isomorphism class of the fundamental groups. www.math.ucsb.edu /~wei/teach/241/03syla/03syla.html   (251 words)

AIF : Tome 52 fascicle 4 -- 2002 We study the spectral convergence of compact Riemannian manifolds in relation with the Gromov-Hausdorff distance and discuss the geodesic distances and the energy forms of the limit spaces. Mots Clés : Laplace operator, energy form, heat kernel, spectral convergence, Gromov-Hausdorff distance annalif.ujf-grenoble.fr /Vol52/E524_10/E524_10.html   (87 words)

icalendar.cfm?eventid=3267 This should be contrasted with the Hawaii Ring and Menguy's example of a limit space with locally infinite topologic al type. Then present joint work with Guofang Wei proving that if a sequence of manifolds with a uniform lower bound on Ricci curvature converge to a length space $Y$\, then $Y$ has a universal cover. webapps.jhu.edu /eventslist/icalendar.cfm?eventid=3267   (74 words)

GT Vol 7 (2003) Paper 14 (Abstract) Without injectivity radius bounds we must allow for convergence in the Gromov-Hausdorff sense to a space which is not a manifold but only a metric space. Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(M_i,g_i(t), O_i)} such that t in [0,T] are solutions to the Ricci flow and g_i(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an n-dimensional solution to the Ricci flow. www.emis.de /journals/GT/GTVol7/paper14.abs.html   (148 words)

MIT Differential Geometry Seminar, Fall 2004 The methods involve metric geometry involving Gromov-Hausdorff convergence and limits of local covers. Abstract: We will look at how to use the metric geometry work of Gromov and Fukaya to study sequences of solutions of the Ricci flow with bounded curvature, including the case of collapsing sequences. Such results will allow for singularity models for more general types of singularities, including slow forming (type II) infinite time singularities. www-math.mit.edu /~jeffv/DG_Fall_2004.html   (628 words)

Stony Brook Mathematics Areas of current research include comparison geometry; Gromov-Hausdorff convergence; minimal submanifolds and geometric measure theory; Einstein manifolds; Kaehler geometry; manifolds of special holonomy; geometry and topology of low-dimensional manifolds; spin geometry; twistor theory. Areas of current research include Riemann surfaces (Kleinian groups, Teichmuller theory, relations with 3-dimensional topology); complex manifold theory (emphasisizing links with Riemannian geometry, symplectic topology, and algebraic geometry); CR manifolds (cohomology; pseudoconvavity/convexity); real-analytic methods in one complex variable (harmonic measure, Brownian motion); theta functions and their applications to combinatorics and number theory. Areas of current research include Julia and Mandelbrot sets for polynomial maps in one and several complex variables; Tecihmuller theory and Kleinian groups. www.math.sunysb.edu /html/research-areas.shtml   (239 words)

UCSD Geometry Seminar These can be used to prove, in two dimensions, a weak form of a conjectural remark of Donaldson that if the J-flow does not converge then it should blow up over some curves of negative self-intersection. We investigate the relationship between this covering spectrum, the length spectrum, the marked length spectrum and the Laplace spectrum. It will be discussed how these results can be applied to prove properness of the Mabuchi energy for some Kahler classes, and, conjecturally, how they relate to notions of stability due to Tian and Ross-Thomas. math.ucsd.edu /~lni/seminar.html   (858 words)

Tangent Spaces of Metric Spaces The first purpose of the seminar is to understand various aspects of these objects, these include the notions of tangent cones, Gromov-Hausdorff convergence and approximate tangent spaces of rectifiable sets. Rectifiable metric spaces: local structure and regularity of the Hausdorff measure These development have shown that quite general metric spaces often have a kind of differentiable structure and generalized tangent spaces. dmawww.epfl.ch /~troyanov/Seminaire/borel03.htm   (306 words)

gc05.html We will review various equivalent definitions of Gromov-Hausdorff Convergence and then prove Gromov's Precompactness Theorem (7.4.15) mentioned on Sept 22 and prove that Gromov-Hausdorff limits of length spaces are length spaces (7.5.1). Topics: Lipschitz Maps, Hausdorff Measure and Dimension, Length Spaces, Hyperbolic Space, Alexandrov Spaces, Fundamental Groups of these Spaces, Gromov-Hausdorff Convergence, Gromov's Compactness Theorem, Quasi Isometries, Gromov Hyperbolic Spaces, Gromov-Hausdorff Limits, Tangent Cones at Infinity, and Metric Measure Spaces. See "Hausdorff Convergence and Universal Covers" by Sormani and Wei in Transactions of the American Mathematical Society 353 (2001), no. 9, 3585--3602. comet.lehman.cuny.edu /sormani/teaching/gc05.html   (723 words)

OUP: Topics on Analysis in Metric Spaces: Ambrosio The book covers abstract measure theory, Hausdorff measures, Lipschitz functions, covering theorems, lower semicontinuity of the one-dimensional Hausdorff measure, Sobolev spaces of maps between metric spaces, and Gromov-Hausdorff theory, all developed in a general metric setting. The existence of geodesics (and more generally of minimal Steiner connections) is discussed in general metric spaces and, as an application of the Gromov-Hausdorff theory, even in some cases when the ambient space is not locally compact. Supplemented with exercises of varying difficulty it is ideal for a graduate-level short course for applied mathematicians and engineers. www.oup.co.uk /isbn/0-19-852938-4   (400 words)

Global Riemannian Geometry (L24) Additional topics (if time permits) such as Gromov-Hausdorff convergence and critical points of the distance function. Basic knowledge on local Riemannian geometry (curvature, geodesics etc.) and Lie groups. www.maths.cam.ac.uk /CASM/courses/02-03/descriptions/node22.html   (232 words)

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