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Length Structures: Path Metric Spaces 1 A. 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. . . . . . 30 C. Homotopy counting Lipschitz maps . . . . . . . . 35 D. Dilatation of sphere-valued mappings . . .. . . . 41 E+ Degrees of short maps between compact and noncompact manifolds 55 3 Metric Structures on Families of Metric Spaces 71 A. Lipschitz and Hausdorff distance .. .. .. ... 71 B. The noncompactcase .· . . . 85 C. 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 113 A. A review of measures and mm spaces . 1. . . . ..13 B. OAconvergence of mm spaces 116 C. Geometry of measures in metric spaces ... . . 124 D. Basic geometry of the space X . . . . . . . 129 E. Concentration phenomenon . . . . . . . . . . . . . . . . . 140 F. Geometric invariants of measures related to concentration . ... 181 G. Concentration, spectrum, and the spectral diameter 190 H. Observable distance HA on the space X and concentration XA n X-i X. ...: . ........- II.. 200 . The Lipschitz order on X, pyramids, and asymptotic con- centration . ... 212 J. Concentration versus dissipation 221 4 Loewner Rediscovered 239 A. First, some history (in dimension 2) . 239 B. Next, some questions in dimensions > 3 .. . . . 244 C. Norms on homology and Jacobi varieties . . .... 245 D. An application of geometric integration theory . . . . 261 "E" Unstable systolic inequalities and filling 26. . F+ Finer inequalities and systoles of universal spaces 269 5 Manifolds with Bounded Ricci Curvature 273 "A. Precompactness . . 273 B. Growth of fundamental groups . . . . . . . . . . . . . . . 279 C. The first Bettinumber . . , . . . . . . . . . . . 284 D. Small loops .. .. 288 E+ Applications of the packing inequalities . 294 F4 On the nilpotencyofr .. ..... . . . . 295 G4 Simplicial volume and entropy . . . ... . .. 302 H+ Generalized simplicial norms and the metrization of homotopy theory ....... 307 I+ Ricci curvature beyond coverings . . 316 6 Isoperinietric Inequalities and Amenability 321 A. Quasiregular mappings... . ...: . . 321 B. Isoperimetric dimension of a manifold . . 322 C. Computations of isoperimetric dimension , . .327 D. Generalized quasiconformality 3 . . . .. . . . . 336 E+ The Varopoulos isoperimetric inequality . . . . . . 346 "7 Morse Theory and Minimal Models 351 A. Application of NMorse theory to loop spaces .. 351 B. Dilatation of mappings between simply connected manifolds . . . . . . . . . 357 8+ Pinching and Collapse 365 A. Invariant classes of metrics and the stability problem . . .J. . . . . . . 365 B Sign and the meaning of curvature 369 C. Elenmntary geometry of collapse . . . 375 Di Convergence without collapse 384 E Basic features of collapse . 390 A "Quasiconvex" Domains inR 393 B Mjetric Spaces and Mappings Seen at Many Scales 401 1. Basic concepts and examples . . . . . 402 1. Euclidean spaces, hyperbolic spaces, and ideas from analysis 402 2 Quasinmetrics, the doubling condition, and examples of metric spaces . ... 404 1 Doubling measures and regular metric spaces, deformations of geometryi Riesz products and Riemann surfaces .., 411 4. Quasisymmetric mappings and deformations of geometry from do ubling measures 417 5. Rest and recaptuaton . . . . . 422 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 .... 435 9. 116older continuity and nmean oscillation .. 437 10, Vanishing mean oscillation .... . 439 11. Bounded mean oscillation . 43 IIL Rigidity and structure ... . . . 445 12. Differentiability almost everywhere . 445 13. Pause for reflection 448 14. Almost flat curves 448 15. Mappings that almost preserve distances . . . . 452 16. Almost flat hypersurfaces 45 17.ý The A, condition for doubling measures . ... . 458 18. Quasisymmnetric mappings and doubling measures 6. 42 19M NMtric doubling measures . . .. . 464 20. Bi-Lipschitz embeddings .. . 468 21. A, weights . 470 22, Interlude: bi-Lpschitz mappings between Cantor sets . 471 23. Another moment of reflection .471 24. Rectifiability 4 71 25 Uniform rectifiability 475 26. Stories from the past .. :::. . , . . . . 477 27 Regular mappings. . . .. , , . 479 28 Big pieces of bi-Lipschitz mappings ... . . . 480 29. Quantitative smoothness for Lipschitz functions 482 30. Smoothness of uniformly rectifiable sets ,. 488 31. Comments about geometric complexity . . . ., . . . 490 IV. An introduction to realvariable method 491 32. The aximal function . . . 491 33 Covering emmas . . . 493 34. Lebesgue points . ...... 495 35 Differentiability almost everywhere . *. .. 497 36. Finding Lipschitz pieces inside functions .. 502 37, Maximal functions and snapshots . , 505 38. Dyvadic cubes . . ... . . . . . 505 39. The Calder6n Zygmund approximation . 507 40. The John-Nirenberg theorem ..... . . . 508 41 Reverse H6lder inequalities . ... . . . . 511l 42 Two useful lemmas 513 43. Better methods for small oscillations . . . 515 44. Real-variable methods and geometry . . . 517 C Paul Levy's Isoperimetric Inequality 519 D Systolical y Free Manifolds 531