Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.


Full text of review: PDF   This review is available free of charge.
Book Information:

Editor: Vladimir I. Arnold
Title: Arnold's problems
Additional book information: edited by Vladimir I. Arnold, Springer-Verlag, Berlin; PHASIS, Moscow, 2004, xvi+639 pp., ISBN 3-540-20614-0, EUR 99.95

References [Enhancements On Off] (What's this?)

  • 1. \cyr Problemy Gil′berta, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0250804
  • 2. V. I. Arnol′d, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk 18 (1963), no. 6 (114), 91–192 (Russian). MR 0170705
  • 3. V. I. Arnol′d, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR 156 (1964), 9–12 (Russian). MR 0163026
  • 4. Vladimir Arnol′d, Sur une propriété topologique des applications globalement canoniques de la mécanique classique, C. R. Acad. Sci. Paris 261 (1965), 3719–3722 (French). MR 0193645
  • 5. V. I. Arnol′d, A stability problem and ergodic properties of classical dynamical systems, Proc. Internat. Congr. Math. (Moscow, 1966) Izdat. “Mir”, Moscow, 1968, pp. 387–392 (Russian). MR 0239217
  • 6. Anri Puankare and Anri Puankare, \cyr Izbrannye trudy v trekh tomakh, Izdat. “Nauka”, Moscow, 1972 (Russian). \cyr Tom II: \cyr Novye metody nebesnoĭ mekhaniki. Topologiya. Teoriya chisel. [Volume II: New methods in celestial mechanics. Topology. Number theory]; Edited by N. N. Bogoljubov, V. I. Arnol′d and I. B. Pogrebysskiĭ; Translated from the French by V. K. Abalakin, A. A. Brjandinskaja, A. N. Bogoljubov, A. V. Černavskiĭ and Ju. N. Sudarev; With commentaries by G. A. Merman, I. B. Pogrebysskiĭ, A. V. Černavskiĭ, V. A. Zorič, V. I. Arnol′d and Ju. I. Manin; With a biographical sketch by P. S. Aleksandrov; \cyr Seriya “\cyr Klassiki Nauki”. MR 0384460
  • 7. V. I. Arnol′d, Some open problems in the theory of singularities, Singularities, Part 1 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, R.I., 1983, pp. 57–69. Translated from the Russian. MR 713046
  • 8. V. I. Arnold, Some problems in the theory of differential equations (Russian), Unsolved Problems of Mechanics and Applied Mathematics, Moscow Univ. Press, Moscow, 1977, pp. 3-9.
  • 9. V. Arnol′d, On some problems in singularity theory, Geometry and analysis, Indian Acad. Sci., Bangalore, 1980, pp. 1–9. MR 592248
    V. Arnol′d, On some problems in singularity theory, Proc. Indian Acad. Sci. Math. Sci. 90 (1981), no. 1, 1–9. MR 653941, https://doi.org/10.1007/BF02867012
  • 10. V. I. Arnol′d, Ten problems, Theory of singularities and its applications, Adv. Soviet Math., vol. 1, Amer. Math. Soc., Providence, RI, 1990, pp. 1–8. MR 1089668
  • 11. V. Arnol′d, Problems on singularities and dynamical systems, Developments in mathematics: the Moscow school, Chapman & Hall, London, 1993, pp. 251–274. MR 1264427
  • 12. V. I. Arnol′d, Mathematical problems in classical physics, Trends and perspectives in applied mathematics, Appl. Math. Sci., vol. 100, Springer, New York, 1994, pp. 1–20. MR 1277190, https://doi.org/10.1007/978-1-4612-0859-4_1
  • 13. Vladimir I. Arnol′d, Sur quelques problèmes de la théorie des systèmes dynamiques, Topol. Methods Nonlinear Anal. 4 (1994), no. 2, 209–225 (French). MR 1350971
  • 14. Vladimir Igorevich Arnol′d, \cyr Izbrannoe-60, Izdatel′stvo FAZIS, Moscow, 1997 (Russian). MR 1647728
  • 15. V. I. Arnold, From superpositions to the KAM theory (Russian), Selecta-60, PHASIS, Moscow, 1997, pp. 727-740.
  • 16. V. I. Arnol′d, On the teaching of mathematics, Uspekhi Mat. Nauk 53 (1998), no. 1(319), 229–234 (Russian); English transl., Russian Math. Surveys 53 (1998), no. 1, 229–236. MR 1618209, https://doi.org/10.1070/rm1998v053n01ABEH000005
  • 17. V. I. Arnold, From Hilbert’s superposition problem to dynamical systems, The Arnoldfest (Toronto, ON, 1997) Fields Inst. Commun., vol. 24, Amer. Math. Soc., Providence, RI, 1999, pp. 1–18. MR 1733564
    V. I. Arnol′d, From Hilbert’s superposition problem to dynamical systems [MR1733564], Amer. Math. Monthly 111 (2004), no. 7, 608–624. MR 2080045, https://doi.org/10.2307/4145164
  • 18. V. I. Arnold, The Russian edition of David Hilbert's works (Russian), Priroda 1999, no. 4, 114-121.
  • 19. V. Arnold, M. Atiyah, P. Lax, and B. Mazur (eds.), Mathematics: frontiers and perspectives, American Mathematical Society, Providence, RI, 2000. MR 1754762
  • 20. V. I. Arnol′d, \cyr Zadachi Arnol′da, Izdatel′stvo FAZIS, Moscow, 2000 (Russian, with Russian summary). With a preface by M. B. Sevryuk and V. B. Filippov. MR 1832295
  • 21. V. I. Arnold, From Hilbert's problem on superpositions to dynamical systems (Russian), Mathematical Events of the 20-th Century, PHASIS, Moscow, 2003, pp. 19-51.
  • 22. Klaus Barner, Paul Wolfskehl und der Wolfskehl-Preis, Mitt. Dtsch. Math.-Ver. 3 (1997), 4–11 (German, with German summary). MR 1477102
    Klaus Barner, Paul Wolfskehl and the Wolfskehl Prize, Notices Amer. Math. Soc. 44 (1997), no. 10, 1294–1303. MR 1474453
  • 23. Edward Bierstone, Boris Khesin, Askold Khovanskii, and Jerrold E. Marsden (eds.), The Arnoldfest, Fields Institute Communications, vol. 24, American Mathematical Society, Providence, RI, 1999. MR 1733563
  • 24. J. Bourgain, Harmonic analysis and combinatorics: how much may they contribute to each other?, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 13–32. MR 1754764
  • 25. Felix E. Browder (ed.), Mathematical developments arising from Hilbert problems, Proceedings of Symposia in Pure Mathematics, Vol. XXVIII, American Mathematical Society, Providence, R. I., 1976. MR 0419125
  • 26. Keith Devlin, The millennium problems, Basic Books, New York, 2002. The seven greatest unsolved mathematical puzzles of our time. MR 1930195
  • 27. S. K. Donaldson, Polynomials, vanishing cycles and Floer homology, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 55–64. MR 1754767
  • 28. J. Ewing, From the AMS secretary, Notices Amer. Math. Soc. 51 (2004), no. 7, 818-823.
  • 29. Ivor Grattan-Guinness, A sideways look at Hilbert’s twenty-three problems of 1900, Notices Amer. Math. Soc. 47 (2000), no. 7, 752–757. MR 1769581
  • 30. Jeremy J. Gray, The Hilbert challenge, Oxford University Press, Oxford, 2000. MR 1828558
  • 31. D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 8 (1902), no. 10, 437-479.
  • 32. David Hilbert, Mathematical problems, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 4, 407–436. Reprinted from Bull. Amer. Math. Soc. 8 (1902), 437–479. MR 1779412, https://doi.org/10.1090/S0273-0979-00-00881-8
  • 33. A. Jackson, Million-dollar mathematics prizes announced, Notices Amer. Math. Soc. 47 (2000), no. 8, 877-879.
  • 34. V. F. R. Jones, Ten problems, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 79–91. MR 1754769
  • 35. Jean-Michel Kantor, Hilbert’s problems and their sequels, Math. Intelligencer 18 (1996), no. 1, 21–30. MR 1381576, https://doi.org/10.1007/BF03024812
  • 36. P.-L. Lions, On some challenging problems in nonlinear partial differential equations, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 121–135. MR 1754772
  • 37. S. H. Lui, An interview with Vladimir Arnol′d, Notices Amer. Math. Soc. 44 (1997), no. 4, 432–438. MR 1435453
  • 38. O. Makarova, An interview with Vladimir Arnold (Russian), Shkol$'$noe Obozrenie 2000, no. 1, 24-27.
  • 39. Jean-Pierre Marco and David Sauzin, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems, Publ. Math. Inst. Hautes Études Sci. 96 (2002), 199–275 (2003). MR 1986314, https://doi.org/10.1007/s10240-003-0011-5
  • 40. Jean-Pierre Marco and David Sauzin, Wandering domains and random walks in Gevrey near-integrable systems, Ergodic Theory Dynam. Systems 24 (2004), no. 5, 1619–1666. MR 2104598, https://doi.org/10.1017/S0143385703000786
  • 41. Gregory Margulis, Problems and conjectures in rigidity theory, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 161–174. MR 1754775
  • 42. B. Mazur, The theme of 𝑝-adic variation, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 433–459. MR 1754790
  • 43. Shigefumi Mori, Rational curves on algebraic varieties, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 189–195. MR 1754777
  • 44. Laurent Niederman, Exponential stability for small perturbations of steep integrable Hamiltonian systems, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 593–608. MR 2054052, https://doi.org/10.1017/S014338570300049X
  • 45. Constance Reid, Hilbert, With an appreciation of Hilbert’s mathematical work by Hermann Weyl, Springer-Verlag, New York-Berlin, 1970 (German). MR 0270884
    Constance Reid, Hilbert, Copernicus, New York, 1996. Reprint of the 1970 original. MR 1391242
  • 46. Peter Sarnak, Some problems in number theory, analysis and mathematical physics, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 261–269. MR 1754782
  • 47. David Sauzin, Nekhoroshev estimates and instability for Gevrey class Hamiltonians, Dynamical systems. Part I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003, pp. 199–217. MR 2071234
  • 48. Steve Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998), no. 2, 7–15. MR 1631413, https://doi.org/10.1007/BF03025291
  • 49. Steve Smale, Mathematical problems for the next century, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 271–294. MR 1754783
  • 50. Richard P. Stanley, Positivity problems and conjectures in algebraic combinatorics, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 295–319. MR 1754784
  • 51. D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems, Nonlinearity 17 (2004), no. 5, 1803–1841. MR 2086152, https://doi.org/10.1088/0951-7715/17/5/014
  • 52. Benjamin H. Yandell, The honors class, A K Peters, Ltd., Natick, MA, 2002. Hilbert’s problems and their solvers. MR 1880187
  • 53. S.-T. Yau, Review of geometry and analysis, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 353–401. MR 1754787
    S.-T. Yau, Review of geometry and analysis, Asian J. Math. 4 (2000), no. 1, 235–278. Kodaira’s issue. MR 1803723, https://doi.org/10.4310/AJM.2000.v4.n1.a16

Review Information:

Reviewer: Mikhail B. Sevryuk
Affiliation: Institute of Energy Problems of Chemical Physics
Email: sevryuk@mccme.ru
Journal: Bull. Amer. Math. Soc. 43 (2006), 101-109
Published electronically: June 1, 2005
Review copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.