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About Misha Gromov.

It is hard work to go through the book, but it is worth the effort. Liam marked it as to-read Jun 12, To see what your friends thought of this book, please sign up. It is a major accomplishment of Misha Gromov to have written this exposition. Consider the story of Exercise number 1 from Gromov-Ballmann-Schroeder, which became an infamous open problem and stayed so until very recently! Sort order.

Misha Gromov. Books by Misha Gromov. Trivia About Metric Structures No trivia or quizzes yet. The new wave began with seminal papers by Svarc and Milnor on the growth of groups and the spectacular proof of the rigidity of lattices by Mostow. This progress was followed by the creation of the asymptotic metric theory of infinite groups by Gromov.

Metric theory has undergone a dramatic phase transition in the last decades when its focus moved from the foundations of real analysis to Riemannian geometry and algebraic topology, to the theory of infinite groups and probability theory. The new wave began with seminal papers by. Metric Structures for Riemannian and Non-Riemannian Spaces (Modern Birkhäuser Classics) 1st ed. Corr. 2nd printing 3rd printing Edition.

This distance organizes Riemannian manifolds of all possible topological types into a single connected moduli space, where convergence allows the collapse of dimension with unexpectedly rich geometry, as revealed in the work of Cheeger, Fukaya, Gromov and Perelman. Also, Gromov found metric structure within homotopy theory and thus introduced new invariants controlling combinatorial complexity of maps and spaces, such as the simplicial volume, which is responsible for degrees of maps between manifolds.

The first stages of the new developments were presented in Gromov's course in Paris, which turned into the famous "Green Book" by Lafontaine and Pansu The present English translation of that work has been enriched and expanded with new material to reflect recent progress. Length structures B, Path metric spaces 6 C,. Examnples of path metric spaces , 10 D. Are-wise isometries 22 2 Degree and Dilatation 27 A. Topological review 27 B. Elementary properties of dilatations for spheres.

Homotopy counting Lipschitz maps 35 D. Lipschitz and Hausdorff distance 71 B. The Hausdorff-Lipschitz metric, quasi-isometries, and word metrics 89 De First-order metric invariants and ultralimits 94 ES Convergence with control 98 31 onvergence and Concentration of Metrics and Measures A. A review of measures and mm spaces 1. OAconvergence of mm spaces C. Geometry of measures in metric spaces D. Basic geometry of the space X E. Concentration phenomenon F. Geometric invariants of measures related to concentration G.

Concentration, spectrum, and the spectral diameter H. The Lipschitz order on X, pyramids, and asymptotic con- centration J. Concentration versus dissipation 4 Loewner Rediscovered A. First, some history in dimension 2 B.

Norms on homology and Jacobi varieties D. An application of geometric integration theory "E" Unstable systolic inequalities and filling Precompactness B. Growth of fundamental groups C.

The first Bettinumber , D. Quasiregular mappings : B. Isoperimetric dimension of a manifold C.

Metric Structures for Riemannian and Non-Riemannian Spaces - Wikipedia audio article

Computations of isoperimetric dimension , D. Application of NMorse theory to loop spaces B. Invariant classes of metrics and the stability problem J.

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Basic concepts and examples 1. Euclidean spaces, hyperbolic spaces, and ideas from analysis 2 Quasinmetrics, the doubling condition, and examples of metric spaces 1 Doubling measures and regular metric spaces, deformations of geometryi Riesz products and Riemann surfaces..

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Quasisymmetric mappings and deformations of geometry from do ubling measures 5. Rest and recaptuaton S Analysis on general spaces 23 6.

HMder continuous functions on metric spaces 23 7. Metric spaces which are doubling , 30 8. Spaces of homogeneous type 9. Bounded mean oscillation 43 IIL Rigidity and structure Differentiability almost everywhere Pause for reflection Almost flat curves Mappings that almost preserve distances Almost flat hypersurfaces 45 Quasisymmnetric mappings and doubling measures 6.

Bi-Lipschitz embeddings A, weights 22, Interlude: bi-Lpschitz mappings between Cantor sets Another moment of reflection. Rectifiability 4 71 25 Uniform rectifiability