Skip to main content

Mathematical problems for the next century

This is a preview of subscription content, access via your institution.

References

  • Abraham, R. and Marsden, J. (1978).Foundations of Mechanics. Addison-Wesley Publishing Co., Reading, Mass.

    MATH  Google Scholar 

  • Babin, A.V. and Vishik, M.I. (1983). Attractors of partial differential evolution equations and their dimension.Russian Math. Surveys 38, 151- 213.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    MATH  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • BCSS: Blum, L, Cucker, F., Shub, M., and Smale, S. (1997).Complexity and Real Computation, Springer-Verlag.

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

  • Browder, F. (ed.), (1976).Mathematical Developments Arising from Hilbert Problems, American Mathematical Society, Providence, Rl.

    MATH  Google Scholar 

  • Brownawell, W. (1987). Bounds for the degrees in the Nullstellensatz.Annals of Math. 126, 577–591.

    Article  MATH  MathSciNet  Google Scholar 

  • Chern, S. and Smale, S. (eds.) (1970).Proceedings of the Symposium on Pure Mathematics, vol. XIV, American Mathematical Society, Providence, Rl.

    Google Scholar 

  • Chorin, A., and Marsden, J. (1993).A Mathematical Introduction to Fluid Mechanics, 3rd edition, Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • Chorin, A., Marsden, J., and Smale, S. (1977). Turbulence Seminar, Berkeley 1976-77,Lecture Notes in Math.615, Springer-Verlag, New York.

    Google Scholar 

  • Cucker, F., Koiran, P., and Smale, S. (1997). A polynomial time algorithm for Diophantine equations in one variable. To appear.

  • Debreu, G. (1959).Theory of Value, John Wiley & Sons, New York.

    MATH  Google Scholar 

  • Dulac, H. (1923). Sur les cycles limites.Bull. Soc. Math. France 51, 45–188.

    MATH  MathSciNet  Google Scholar 

  • Écalle, J. (1992).Introduction aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac. Hermann, Paris,

  • 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 

  • Freedman, M. (1982). The topology of 4-manifolds.J. Diff. Geom. 17, 357–454.

    MATH  Google Scholar 

  • Garey, M. and Johnson, D. (1979).Computers and Intractability, Freeman, San Francisco.

    MATH  Google Scholar 

  • Grötschel, M., Lovâsz, L., and Schrijver, A. (1993).Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • Guckenheimer, J. and Holmes, P. (1990).Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, third printing, Springer-Verlag, New York.

    Google Scholar 

  • Guckenheimer, J. and Williams, R.F. (1979). Structural stability of Lorenz attractors.Publ. Math. IHES 50, 59–72.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • Hirsch, M. and Smale, S. (1974).Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York,

    MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • llyashenko, Yu. (1991).Finiteness Theorems for Limit Cycles, American Mathematical Society, Providence, Rl.

    Google Scholar 

  • llyashenko, Yu. and Yakovenko, S. (1995). Concerning the Hilbert 16th problem.AMS Translations, series 2, vol.165, AMS, Providence, Rl.

    Google Scholar 

  • Jakobson, M. (1971). On smooth mappings of the circle onto itself.Math. USSR Sb. 14, 161–185.

    Article  Google Scholar 

  • Kuijlaars, A.B.J. and Saff, E.B. (1997). Asymptotics for minimal discrete energy on the sphere.Trans. Amer. Math. Soc, to appear.

  • 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.

    MATH  Google Scholar 

  • Lang, S. (1991).Number Theory III, vol. 60 ofEncyclopaedia of Mathematical Sciences, Springer-Verlag, New York.

    Google Scholar 

  • Linz, A., de Melo, W., and Pugh, C. (1977), in Geometry and Topology,Lecture Notes in Math.597, Springer-Verlag, New York.

    Google Scholar 

  • Liu, V. (1992). An example of instability for the Navier-Stokes equations on the 2-dimensional torus.Commun. PDE 17, 1995–2012.

    Article  MATH  Google Scholar 

  • Lloyd, N.G. and Lynch, S. (1988). Small amplitude limit cycles of certain Lienard systems.Proceedings Roy. Soc. London 418, 199–208.

    Article  MATH  MathSciNet  Google Scholar 

  • Lorenz, E. (1963). Deterministic non-periodic flow.J. Atmosph. Sci. 20,c 130–141.

    Article  Google Scholar 

  • Manders, K.L. and Adleman, L. (1978). NP-complete decision problems for binary quadratics.J. Comput. System Sci. 16, 168–184.

    Article  MATH  MathSciNet  Google Scholar 

  • McMullen, C. (1994). Frontiers in complex dynamics.Bull. Amer. Math. Soc. (2)31, 155–172.

    Article  MATH  MathSciNet  Google Scholar 

  • de Melo, W. and van Strien, S. (1993).One-Dimensional Dynamics. Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • 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.

    MATH  MathSciNet  Google Scholar 

  • Palmore, J. (1976). Measure of degenerative relative equilibria. I.Annals of Math. 104, 421–429.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • 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 

  • Penrose, R. (1991).The Emperor’s New Mind, Penguin Books.

  • Poincaré, H. (1953).Oeuvres, VI. Gauthier-Villars, Paris. Deuxième Complément à L’Analysis Situs.

  • Peixoto, M. (1962). Structural stability on two-dimensional manifolds.Topology 1, 101–120.

    Article  MATH  MathSciNet  Google Scholar 

  • Pugh, C. (1967). An improved closing lemma and a general density theorem.Amer. J. Math. 89, 1010–1022.

    Article  MATH  MathSciNet  Google Scholar 

  • Pugh, C. and Robinson, C. (1983). The C1 closing lemma including Hamiltonians.Ergod. Theory Dynam. Systems 3, 261–313.

    Article  MATH  MathSciNet  Google Scholar 

  • Rakhmanov, E.A., Saff, E.B., and Zhou, Y.M. (1994). Minimal discrete energy on the sphere.Math. Res. Lett. 1, 647–662.

    Article  MATH  MathSciNet  Google Scholar 

  • Robinson, C. (1989). Homoclinic bifurcation to a transitive attractor of Lorenz type.Non-linearity 2, 495–518.

    MATH  Google Scholar 

  • Rudin, W. (1995). Injective polynomial maps are automorphisms.Amer. Math. Monthly 102, 540–543.

    Article  MATH  MathSciNet  Google Scholar 

  • Ruelle, D. and Takens, F. (1971). On the nature of turbulence.Commun. Math. Phys. 20, 167–192.

    Article  MATH  MathSciNet  Google Scholar 

  • Samuelson, P. (1971).Foundations of Economic Analysis, Atheneum, New York.

    Google Scholar 

  • Saff, E. and Kuijlaars, A. (1997). Distributing many points on a sphere.Mathematical Intelligencer 10, 5–11.

    Article  MathSciNet  Google Scholar 

  • Schrijver, A. (1986).Theory of Linear and Integer Programming, John Wiley & Sons.

  • Shi, S. (1982). On limit cycles of plane quadratic systems.Sei. Sin. 25, 41–50.

    MATH  Google Scholar 

  • 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 

  • Shub, M. and Smale, S. (1993). Complexity of Bezout’s theorem, III: condition number and packing.J. of Complexity 9, 4–14.

    Article  MATH  MathSciNet  Google Scholar 

  • Shub, M. and Smale, S. (1994). Complexity of Bezout’s theorem, V: polynomial time.Theoret. Comp. Sci. 133, 141–164.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.)

  • Smale, S. (1963). A survey of some recent developments in differential topology.Bull. Amer. Math. Soc. 69, 131–146.

    Article  MathSciNet  Google Scholar 

  • Smale, S. (1967). Differentiable dynamical systems.Bull. Amer. Math. Soc. 73, 747–817.

    Article  MATH  MathSciNet  Google Scholar 

  • Smale, S. (1970). Topology and mechanics, I and II.Invent. Math. 10, 305–331 andInvent Math. 11, 45-64.

    Article  MATH  MathSciNet  Google Scholar 

  • Smale, S. (1976). Dynamics in general equilibrium theory.Amer. Economic Review 66, 288–294.

    Google Scholar 

  • Smale, S. (1980).Mathematics of Time, Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • 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 

  • 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).

    Article  MathSciNet  Google Scholar 

  • Smale, S. (1991). Dynamics retrospective: great problems, attempts that failed.Physica D 51, 267–273.

    Article  MATH  MathSciNet  Google Scholar 

  • Taubes, G. (July 1987). What happens when Hubris meets Nemesis?Discover.

  • Temam, R. (1979).Navier-Stokes Equations, revised edition, North- Holland, Amsterdam.

    MATH  Google Scholar 

  • Traub, J. and Wozniakowski, H. (1982). Complexity of linear programming.Oper. Res. Letts. 1, 59–62.

    Article  MATH  MathSciNet  Google Scholar 

  • Tsuji, M. (1959).Potential Theory in Modern Function Theory, Maruzen Co., Ltd., Tokyo.

    MATH  Google Scholar 

  • Wen, L. and Xia, Z. (1997). A simpler proof of the Cp1 connecting lemma. To appear.

  • Williams, R. (1979). The structure of Lorenz attractors.Publ. IHES 50, 101–152.

    Article  Google Scholar 

  • Wintner, A. (1941).The Analytical Foundations of Celestial Mechanics. Princeton University Press, Princeton, NJ.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Steve Smale.

Additional information

Lecture given on the occasion of Arnold’s 60th birthday at the Fields Institute, Toronto, June 1997.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Smale, S. Mathematical problems for the next century. The Mathematical Intelligencer 20, 7–15 (1998). https://doi.org/10.1007/BF03025291

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03025291

Keywords

  • Relative Equilibrium
  • Mathematical Intelligencer
  • Affirmative Answer
  • Riemann Hypothesis
  • Lorenz Attractor