The Mathematical Intelligencer

, Volume 20, Issue 2, pp 7–15 | Cite as

Mathematical problems for the next century

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References

  1. Abraham, R. and Marsden, J. (1978).Foundations of Mechanics. Addison-Wesley Publishing Co., Reading, Mass.MATHGoogle Scholar
  2. Babin, A.V. and Vishik, M.I. (1983). Attractors of partial differential evolution equations and their dimension.Russian Math. Surveys 38, 151- 213.CrossRefMATHMathSciNetGoogle Scholar
  3. Barvinok, A. and Vershik, A. (1993). Polynomial-time, computable approximation of families of semi-algebraic sets and combinatorial complexity.Amer. Math. Soc. Trans. 155, 1–17.MATHGoogle Scholar
  4. Bass, H., Connell, E., and Wright, D. (1982). The Jacobian conjecture: reduction on degree and formal expansion of the inverse.Bull. Amer. Math. Soc. (2) 7, 287–330.CrossRefMATHMathSciNetGoogle Scholar
  5. BCSS: Blum, L, Cucker, F., Shub, M., and Smale, S. (1997).Complexity and Real Computation, Springer-Verlag.Google Scholar
  6. Blum, L, Shub, M., and Smale, S. (1989). On a theory of computation and complexity over the real numbers: NP-completness, recursive functions and universal machines.Bulletin of the Amer. Math. Soc. (2)21, 1–46.CrossRefMathSciNetGoogle Scholar
  7. Blum, L. and Smale, S. (1993). The Gödel incompleteness theorem and decidability over a ring. Pages 321-339 in M. Hirsch, J. Marsden, and M. Shub (editors),From Topology to Computation: Proceedings of the Smalefest, Springer-Verlag.Google Scholar
  8. Browder, F. (ed.), (1976).Mathematical Developments Arising from Hilbert Problems, American Mathematical Society, Providence, Rl.MATHGoogle Scholar
  9. Brownawell, W. (1987). Bounds for the degrees in the Nullstellensatz.Annals of Math. 126, 577–591.CrossRefMATHMathSciNetGoogle Scholar
  10. Chern, S. and Smale, S. (eds.) (1970).Proceedings of the Symposium on Pure Mathematics, vol. XIV, American Mathematical Society, Providence, Rl.Google Scholar
  11. Chorin, A., and Marsden, J. (1993).A Mathematical Introduction to Fluid Mechanics, 3rd edition, Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  12. Chorin, A., Marsden, J., and Smale, S. (1977). Turbulence Seminar, Berkeley 1976-77,Lecture Notes in Math.615, Springer-Verlag, New York.Google Scholar
  13. Cucker, F., Koiran, P., and Smale, S. (1997). A polynomial time algorithm for Diophantine equations in one variable. To appear.Google Scholar
  14. Debreu, G. (1959).Theory of Value, John Wiley & Sons, New York.MATHGoogle Scholar
  15. Dulac, H. (1923). Sur les cycles limites.Bull. Soc. Math. France 51, 45–188.MATHMathSciNetGoogle Scholar
  16. Écalle, J. (1992).Introduction aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac. Hermann, Paris,Google Scholar
  17. van den Essen, A. (1997). Polynomial automorphisms and the Jacobian conjecture, inAlgèbre non commutative, groupes quantiques et invariants, septième contact Franco-Belge, Reims, Juin 1995, eds. J. Alev and G. Cauchon, Société mathématique de France, Paris.Google Scholar
  18. Freedman, M. (1982). The topology of 4-manifolds.J. Diff. Geom. 17, 357–454.MATHGoogle Scholar
  19. Garey, M. and Johnson, D. (1979).Computers and Intractability, Freeman, San Francisco.MATHGoogle Scholar
  20. Grötschel, M., Lovâsz, L., and Schrijver, A. (1993).Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  21. Guckenheimer, J. and Holmes, P. (1990).Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, third printing, Springer-Verlag, New York.Google Scholar
  22. Guckenheimer, J. and Williams, R.F. (1979). Structural stability of Lorenz attractors.Publ. Math. IHES 50, 59–72.CrossRefMATHMathSciNetGoogle Scholar
  23. Hayashi, S. (1997). Connecting invariant manifolds and the solution of the C1-stability conjecture and ft-stability conjecture for flows.Annals of Math. 145, 81–137.CrossRefMATHGoogle Scholar
  24. Hirsch, M. and Smale, S. (1974).Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York,MATHGoogle Scholar
  25. llyashenko, J. (1985). Dulac’s memoir “On limit cycles” and related problems of the local theory of differential equations.Russian Math. Surveys VHO, 1–49.CrossRefGoogle Scholar
  26. llyashenko, Yu. (1991).Finiteness Theorems for Limit Cycles, American Mathematical Society, Providence, Rl.Google Scholar
  27. llyashenko, Yu. and Yakovenko, S. (1995). Concerning the Hilbert 16th problem.AMS Translations, series 2, vol.165, AMS, Providence, Rl.Google Scholar
  28. Jakobson, M. (1971). On smooth mappings of the circle onto itself.Math. USSR Sb. 14, 161–185.CrossRefGoogle Scholar
  29. Kuijlaars, A.B.J. and Saff, E.B. (1997). Asymptotics for minimal discrete energy on the sphere.Trans. Amer. Math. Soc, to appear.Google Scholar
  30. Kuz’mina, R., (1977). An upper bound for the number of central configurations in the plane n-body problem.Sov. Math. Dok. 18,818–821.MATHGoogle Scholar
  31. Lang, S. (1991).Number Theory III, vol. 60 ofEncyclopaedia of Mathematical Sciences, Springer-Verlag, New York.Google Scholar
  32. Linz, A., de Melo, W., and Pugh, C. (1977), in Geometry and Topology,Lecture Notes in Math.597, Springer-Verlag, New York.Google Scholar
  33. Liu, V. (1992). An example of instability for the Navier-Stokes equations on the 2-dimensional torus.Commun. PDE 17, 1995–2012.CrossRefMATHGoogle Scholar
  34. Lloyd, N.G. and Lynch, S. (1988). Small amplitude limit cycles of certain Lienard systems.Proceedings Roy. Soc. London 418, 199–208.CrossRefMATHMathSciNetGoogle Scholar
  35. Lorenz, E. (1963). Deterministic non-periodic flow.J. Atmosph. Sci. 20,c 130–141.CrossRefGoogle Scholar
  36. Manders, K.L. and Adleman, L. (1978). NP-complete decision problems for binary quadratics.J. Comput. System Sci. 16, 168–184.CrossRefMATHMathSciNetGoogle Scholar
  37. McMullen, C. (1994). Frontiers in complex dynamics.Bull. Amer. Math. Soc. (2)31, 155–172.CrossRefMATHMathSciNetGoogle Scholar
  38. de Melo, W. and van Strien, S. (1993).One-Dimensional Dynamics. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  39. Palis, J. and Yoccoz, J.C. (1989). (1) Rigidity of centralizers of diffeo- morphisms.Ann. Scient Ecole Normale Sup. 22, 81–98; (2) Centralizer of Anosov diffeomorphisms.Ann. Scient. Ecole Normale Sup. 22, 99-108.MATHMathSciNetGoogle Scholar
  40. Palmore, J. (1976). Measure of degenerative relative equilibria. I.Annals of Math. 104, 421–429.CrossRefMATHMathSciNetGoogle Scholar
  41. Petrovskπ, I.G. and Landis, E.M. (1957). On the number of limit cycles of the equationdy/dx = Pfc,y)/Q(x,y), whereP and Q are polynomials.Mat. Sb. N.S. 43 (85), 149–168 (in Russian), and (1960)Amer. Math. Soc. Transi. (2) 14, 181-200.MathSciNetGoogle Scholar
  42. Petrovskif, I.G. and Landis, E.M. (1959). Corrections to the articles “On the number of limit cycles of the equationdy/dx =P(x,y)/Q(>c,y), whereP and Q are polynomials.“Mat. Sb. N.S. 48 (90), 255–263 (in Russian)Google Scholar
  43. Penrose, R. (1991).The Emperor’s New Mind, Penguin Books.Google Scholar
  44. Poincaré, H. (1953).Oeuvres, VI. Gauthier-Villars, Paris. Deuxième Complément à L’Analysis Situs.Google Scholar
  45. Peixoto, M. (1962). Structural stability on two-dimensional manifolds.Topology 1, 101–120.CrossRefMATHMathSciNetGoogle Scholar
  46. Pugh, C. (1967). An improved closing lemma and a general density theorem.Amer. J. Math. 89, 1010–1022.CrossRefMATHMathSciNetGoogle Scholar
  47. Pugh, C. and Robinson, C. (1983). The C1 closing lemma including Hamiltonians.Ergod. Theory Dynam. Systems 3, 261–313.CrossRefMATHMathSciNetGoogle Scholar
  48. Rakhmanov, E.A., Saff, E.B., and Zhou, Y.M. (1994). Minimal discrete energy on the sphere.Math. Res. Lett. 1, 647–662.CrossRefMATHMathSciNetGoogle Scholar
  49. Robinson, C. (1989). Homoclinic bifurcation to a transitive attractor of Lorenz type.Non-linearity 2, 495–518.MATHGoogle Scholar
  50. Rudin, W. (1995). Injective polynomial maps are automorphisms.Amer. Math. Monthly 102, 540–543.CrossRefMATHMathSciNetGoogle Scholar
  51. Ruelle, D. and Takens, F. (1971). On the nature of turbulence.Commun. Math. Phys. 20, 167–192.CrossRefMATHMathSciNetGoogle Scholar
  52. Samuelson, P. (1971).Foundations of Economic Analysis, Atheneum, New York.Google Scholar
  53. Saff, E. and Kuijlaars, A. (1997). Distributing many points on a sphere.Mathematical Intelligencer 10, 5–11.CrossRefMathSciNetGoogle Scholar
  54. Schrijver, A. (1986).Theory of Linear and Integer Programming, John Wiley & Sons.Google Scholar
  55. Shi, S. (1982). On limit cycles of plane quadratic systems.Sei. Sin. 25, 41–50.MATHGoogle Scholar
  56. Shub, M. (1970). Appendix to Smale’s paper: Diagrams and relative equilibria in manifolds, Amsterdam, 1970.Lecture Notes in Math. 197, Springer-Verlag, New York.Google Scholar
  57. Shub, M. and Smale, S. (1993). Complexity of Bezout’s theorem, III: condition number and packing.J. of Complexity 9, 4–14.CrossRefMATHMathSciNetGoogle Scholar
  58. Shub, M. and Smale, S. (1994). Complexity of Bezout’s theorem, V: polynomial time.Theoret. Comp. Sci. 133, 141–164.CrossRefMATHMathSciNetGoogle Scholar
  59. Shub, M. and Smale, S. (1995). On the intractibility of Hubert’s Nullstellensatz and an algebraic version of “P = NP“,Duke Math. J. 81, 47–54.CrossRefMATHMathSciNetGoogle Scholar
  60. Smale, S. (1963). Dynamical systems and the topological conjugacy problem for diffeomorphisms, pages 490-496 in:Proceedings of the International Congress of Mathematicians, Inst. Mittag-Leffler, Sweden, 1962. (V. Stenström, ed.)Google Scholar
  61. Smale, S. (1963). A survey of some recent developments in differential topology.Bull. Amer. Math. Soc. 69, 131–146.CrossRefMathSciNetGoogle Scholar
  62. Smale, S. (1967). Differentiable dynamical systems.Bull. Amer. Math. Soc. 73, 747–817.CrossRefMATHMathSciNetGoogle Scholar
  63. Smale, S. (1970). Topology and mechanics, I and II.Invent. Math. 10, 305–331 andInvent Math. 11, 45-64.CrossRefMATHMathSciNetGoogle Scholar
  64. Smale, S. (1976). Dynamics in general equilibrium theory.Amer. Economic Review 66, 288–294.Google Scholar
  65. Smale, S. (1980).Mathematics of Time, Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  66. Smale, S. (1981). Global analysis and economics, pages 331-370 inHandbook of Mathematical Economics 1, editors K.J. Arrow and M.D. Intrilligator. North-Holland, Amsterdam.Google Scholar
  67. Smale, S. (1990). The story of the higher-dimensional Poincaré conjecture.Mathematical Intelligencer 12, no. 2, 40–51. Also in M. Hirsch, J. Marsden, and M. Shub, editors,From Topology to Computation: Proceedings of the Smalefest, 281-301 (1992).CrossRefMathSciNetGoogle Scholar
  68. Smale, S. (1991). Dynamics retrospective: great problems, attempts that failed.Physica D 51, 267–273.CrossRefMATHMathSciNetGoogle Scholar
  69. Taubes, G. (July 1987). What happens when Hubris meets Nemesis?Discover. Google Scholar
  70. Temam, R. (1979).Navier-Stokes Equations, revised edition, North- Holland, Amsterdam.MATHGoogle Scholar
  71. Traub, J. and Wozniakowski, H. (1982). Complexity of linear programming.Oper. Res. Letts. 1, 59–62.CrossRefMATHMathSciNetGoogle Scholar
  72. Tsuji, M. (1959).Potential Theory in Modern Function Theory, Maruzen Co., Ltd., Tokyo.MATHGoogle Scholar
  73. Wen, L. and Xia, Z. (1997). A simpler proof of the Cp1 connecting lemma. To appear.Google Scholar
  74. Williams, R. (1979). The structure of Lorenz attractors.Publ. IHES 50, 101–152.CrossRefGoogle Scholar
  75. Wintner, A. (1941).The Analytical Foundations of Celestial Mechanics. Princeton University Press, Princeton, NJ.Google Scholar

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© Springer-Verlag 1998

Authors and Affiliations

  1. 1.Mathematics DepartmentCity University of Hong KongKowtoon. Hong KongChina

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