Mathematical problems for the next century

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

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

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.

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.

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.

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.

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

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

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

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

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.

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

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.

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

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

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

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

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

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.

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

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.

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

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

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

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.

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

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

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

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

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

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

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

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

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.

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

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.

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)

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.

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

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

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

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

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

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

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

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

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.

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.

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

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

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.

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.

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

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

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

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

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.

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).

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

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

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

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

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

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.

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