Spaces and Questions

  • Misha Gromov
Part of the Modern Birkhäuser Classics book series (MBC)


Our Euclidean intuition, probably inherited from ancient primates, might have grown out of the first seeds of geometry in the motor control systems of early animals who were brought up to sea and then to land by the Cambrian explosion half a billion years ago. The primates' brain had been idling for 30–40 million years. Suddenly, in a flash of one million years, it exploded into growth under the relentless pressure of sexual-social competition and sprouted a massive neocortex (70% neurons in humans) with an inexplicable capability for language, sequential reasoning and generation of mathematical ideas. Then Man came and laid down space on papyrus in a string of axioms, lemmas and theorems around 300 B.C. in Alexandria.


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  1. [ABCKT]
    J. Amorós, M. Burger, K. Corlette, D. Kotschick, D. Toledo, Fundamental Groups of Compact Kähler Manifolds, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1996.Google Scholar
  2. [B]
    V. Bangert, Existence of a complex line in tame almost complex tori, Duke Math. J. 94 (1998), 29–40.CrossRefMATHMathSciNetGoogle Scholar
  3. [BeEE]
    J.K. Beem, P.E. Ehrlich, K.L. Easley, Global Lorentzian Geometry, 2. ed., Monograph + Textbooks in Pure and Appl. Mathematics 202, Marcel Dekker Inc., NY, 1996.Google Scholar
  4. [Ber]
    M. Berger, Riemannian geometry during the second half of the twentieth century, Jahrbericht der Deutschen Math. Vereinigung 100 (1998), 45–208.MATHGoogle Scholar
  5. [Br]
    R. Bryant, Recent advances in the theory of holonomy, Sém. N. Bourbaki, vol. 1998–99, juin 1999.Google Scholar
  6. [CC]
    J. Cheeger, T. Colding, On the structure of spaces with Ricci curvature bounded below, III, preprint.Google Scholar
  7. [DG]
    G. D’Ambra, M. Gromov, Lectures on transformation groups: Geometry and dynamics, Surveys in Differential Geometry 1 (1991), 19–111.MathSciNetGoogle Scholar
  8. [Gl]
    M. Gromov, with Appendices by M. Katz, P. Pansu, and S. Semmes, Metric Structures for Riemannian and Non-Riemannian Spaces, based on “Structures Métriques des Variétés Riemanniennes” (J. LaFontaine, P. Pansu, eds.), English Translation by Sean M. Bates, Birkhäuser, Boston—Basel—Berlin (1999).Google Scholar
  9. [G2]
    M. Gromov, Asymptotic invariants of infinite groups. Geometric group theory, Vol. 2, Proc. Symp. Sussex Univ., Brighton, July 14–l9, 1991. London Math. Soc. Lecture Notes 182 (Niblo, Roller, ed.), Cambridge Univ. Press, Cambridge (1993) 1–295.Google Scholar
  10. [G3]
    M. Gromov, Carnot-Caratheodory spaces seen from within sub-Riemannian geometry, Proc. Journées nonholonomes; Géométrie sousriemannienne, théorie du contrôle, robotique, Paris, June 30—July 1, 1992 (A. Bellaiche, ed.), Birkhäuser, Basel, Prog. Math. 144 (1996), 79–323.Google Scholar
  11. [G4]
    M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. 1 (1999), 109–197.CrossRefMATHMathSciNetGoogle Scholar
  12. [G5]
    M. Gromov, Hyperbolic groups, in “Essays in Group Theory, Mathematical Sciences Research Institute Publications 8 (1978), 75–263, Springer-Verlag.MathSciNetGoogle Scholar
  13. [G6]
    M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, in “Functional Analysis on the Eve of the 21st Century” (Gindikin, Simon, et al., eds.) Volume II. in honor of the eightieth birthday of I.M. Gelfand, Proc. Conf. Rutgers Univ., New Brunswick, NJ, USA, Oct. 24–27, 1993, Birkhäuser, Basel, Prog. Math. 132 (1996), 1–213Google Scholar
  14. [G7]
    M. Gromov, Partial Differential Relations, Springer-Verlag (1986), Ergeb. der Math. 3. Folge, Bd. 9.Google Scholar
  15. [G8]
    M. Gromov, Random walk in random groups, in preparation.Google Scholar
  16. [G9]
    M. Gromov, Sign and geometric meaning of curvature, Rend. Sem. Math. Fis. Milano 61 (1991), 9–123.CrossRefMATHMathSciNetGoogle Scholar
  17. [G10]
    M. Gromov, Soft and hard symplectic geometry, Proc. ICM-1986 (Berkeley), AMS 1,2 (1987), 81–98.MathSciNetGoogle Scholar
  18. [G11]
    M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps, to appear in Journ. of Geometry and Path. Physics.Google Scholar
  19. [GuS]
    V. Guba, M. Sapir, Diagram Groups, Memoirs of the AMS, November, 1997.Google Scholar
  20. [K]
    P. Kanerva, Sparse Distributed Memory, Cambridge, Mass. MIT Press, 1988.MATHGoogle Scholar
  21. [Kl]
    F. Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, Teil 1, Berlin, Verlag von Julius Springer, 1926.Google Scholar
  22. [LS]
    U. Lang, V. Schroeder, Kirszbraun’s theorem and metric spaces of Bounded Curvature, GAFA 7 (1997), 535–560.CrossRefMATHMathSciNetGoogle Scholar
  23. [Lu]
    A. Lubotzky, Discrete groups, expanding graphs and invariant measures, Birkhäuser Progress in Mathematics 125, 1994).Google Scholar
  24. [MS]
    D. McDuff, D. Salamon, J-holomorphic Curves and Quantum Cohomology, Univ. Lect. Series n° 6, A.M.S. Providence, 1994.MATHGoogle Scholar
  25. [Mc]
    M. McQuillan, Holomorphic curves on hyperplane sections of 3-folds, GAFA 9:2 (1999), 370–392.CrossRefMATHMathSciNetGoogle Scholar
  26. [Mi]
    V.D. Milman, The heritage of P. Lévy in geometrical functional analysis, in “Colloque P. Lévy sur les Processus Stochastiques, Astérisque 157–158 (1988), 273–301.MathSciNetGoogle Scholar
  27. [N]
    J.R. Newman, The World of Mathematics, Vol. 1, Simon and Schuster, NY, 1956.MATHGoogle Scholar
  28. [P]
    G. Perelman, Spaces with curvature bounded below, Proc. ICM (S. Chatterji, ed.), Birkhäuser Verlag, Basel (1995), 517–525.Google Scholar
  29. [Pe]
    P. Petersen, Aspects of global Riemannian geometry, BAMS 36:3 (1999), 297–344.CrossRefMATHGoogle Scholar
  30. [S]
    D. Spring, Convex Integration Theory, Birkhäuser Verlag, Basel—Boston— Berlin, 1998.CrossRefMATHGoogle Scholar
  31. [V]
    A.V. Vasiliev, Nikolai Ivanovich Lobachevski, Moscow, Nauka, 1992 (in Russian).Google Scholar

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© Birkhäuser, Springer Basel AG 2010

Authors and Affiliations

  • Misha Gromov
    • 1
    • 2
  1. 1.Bures sur YvetteFrance
  2. 2.The Courant Institute Mathematical SciencesNYUNYUSA

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