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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

Problemy Gil′berta, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0250804

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

V. I. Arnol′d, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR 156 (1964), 9–12 (Russian). MR 0163026

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 193645

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

Anri Puankare and Anri Puankare, Izbrannye trudy v trekh tomakh, Klassiki Nauki, Izdat. “Nauka”, Moscow, 1972 (Russian). Tom II: 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. MR 0384460

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

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.

V. Arnol′d, On some problems in singularity theory, Geometry and analysis, Indian Acad. Sci., Bangalore, 1980, pp. 1–9. MR 592248

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

V. Arnol′d, Problems on singularities and dynamical systems, Developments in mathematics: the Moscow school, Chapman & Hall, London, 1993, pp. 251–274. MR 1264427

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, DOI 10.1007/978-1-4612-0859-4_{1}

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, DOI 10.12775/TMNA.1994.027

Vladimir Igorevich Arnol′d, 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.

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, DOI 10.1070/rm1998v053n01ABEH000005

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, DOI 10.1002/jnm.1660010404

18.

V. I. Arnold, The Russian edition of David Hilbert's works (Russian), Priroda 1999, no. 4, 114-121.

V. Arnold, M. Atiyah, P. Lax, and B. Mazur (eds.), Mathematics: frontiers and perspectives, American Mathematical Society, Providence, RI, 2000. MR 1754762

V. I. Arnol′d, 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.

Klaus Barner, Paul Wolfskehl und der Wolfskehl-Preis, Mitt. Dtsch. Math.-Ver. 3 (1997), 4–11 (German, with German summary). MR 1477102, DOI 10.1515/dmvm-1997-0304

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, DOI 10.1090/fic/024

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, DOI 10.1007/bf02791532

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

Keith Devlin, The millennium problems, Basic Books, New York, 2002. The seven greatest unsolved mathematical puzzles of our time. MR 1930195

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.

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

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.

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, DOI 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.

V. F. R. Jones, Ten problems, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 79–91. MR 1754769

Jean-Michel Kantor, Hilbert’s problems and their sequels, Math. Intelligencer 18 (1996), no. 1, 21–30. MR 1381576, DOI 10.1007/BF03024812

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

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.

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, DOI 10.1007/s10240-003-0011-5

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, DOI 10.1017/S0143385703000786

Gregory Margulis, Problems and conjectures in rigidity theory, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 161–174. MR 1754775

B. Mazur, The theme of $p$-adic variation, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 433–459. MR 1754790

Shigefumi Mori, Rational curves on algebraic varieties, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 189–195. MR 1754777

Laurent Niederman, Exponential stability for small perturbations of steep integrable Hamiltonian systems, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 593–608. MR 2054052, DOI 10.1017/S014338570300049X

Constance Reid, Hilbert, Springer-Verlag, New York-Berlin, 1970 (German). With an appreciation of Hilbert’s mathematical work by Hermann Weyl. MR 0270884

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

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

Steve Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998), no. 2, 7–15. MR 1631413, DOI 10.1007/BF03025291

Steve Smale, Mathematical problems for the next century, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 271–294. MR 1754783

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

D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems, Nonlinearity 17 (2004), no. 5, 1803–1841. MR 2086152, DOI 10.1088/0951-7715/17/5/014

Benjamin H. Yandell, The honors class, A K Peters, Ltd., Natick, MA, 2002. Hilbert’s problems and their solvers. MR 1880187

S.-T. Yau, Review of geometry and analysis, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 353–401. MR 1754787, DOI 10.1016/s0378-3758(98)00182-7

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.