From Topology to Computation: Proceedings of the Smalefest

An extraordinary mathematical conference was held 5-9 August 1990 at the University of California at Berkeley: From Topology to Computation: Unity and Diversity in the Mathematical Sciences An International Research Conference in Honor of Stephen Smale's 60th Birthday The topics of the conference were some of the fields in which Smale has worked: • Differential Topology • Mathematical Economics • Dynamical Systems • Theory of Computation • Nonlinear Functional Analysis • Physical and Biological Applications This book comprises the proceedings of that conference. The goal of the conference was to gather in a single meeting mathemati­ cians working in the many fields to which Smale has made lasting con­ tributions. The theme "Unity and Diversity" is enlarged upon in the section entitled "Research Themes and Conference Schedule." The organizers hoped that illuminating connections between seemingly separate mathematical sub­ jects would emerge from the conference. Since such connections are not easily made in formal mathematical papers, the conference included discussions after each of the historical reviews of Smale's work in different fields. In addition, there was a final panel discussion at the end of the conference.

Table of Contents:
1: Autobiographical Material.- 1 Some Autobiographical Notes.- 2 On How I Got Started in Dynamical Systems, 1959–1962.- 3 The Story of the Higher Dimensional Poincaré Conjecture (What Actually Happened on the Beaches of Rio).- 4 On the Steps of Moscow University.- 5 Professional Biography, Bibliography, and Graduate Students.- 2: Informal Talks.- 6 Luncheon Talk and Nomination for Stephen Smale.- 7 Some Recollections of the Early Work of Steve Smale.- 8 Luncheon Talk.- 9 Banquet Address at the Smalefest.- 3: Differential Topology.- 10 The Work of Stephen Smale in Differential Topology.- Discussions.- 11 Discussion.- 12 Note on the History of Immersion Theory.- Contributed Papers.- 13 The Smale—Hirsch Principle in Catastrophe Theory.- 4: Economics.- 14 Stephen Smale and the Economic Theory of General Equilibrium.- Discussions.- 15 Topology and Economics: The Contribution of Stephen Smale.- 16 Comments.- 5: Dynamical Systems.- 17 On the Contribution of Smale to Dynamical Systems.- Discussions.- 18 Discussion.- Contributed Papers.- 19 Recurrent Sets for Planar Homeomorphisms.- 20 Convergence of Finite-Element Solutions for Nonlinear PDEs.- 21 Ergodic Theory of Chaotic Dynamical Systems.- 22 Beyond Hyperbolicity: Expansion Properties of One-Dimensional Mappings.- 23 Induced Hyperbolicity, Invariant Measures, and Rigidity.- 24 On the Enumerative Geometry of Geodesies.- 25 A Relation Between Newton’s Method and Successive Approximations for Quadratic Irrationals.- 26 On Dynamical Systems and the Minimal Surface Equation.- 27 A New Zeta Function, Natural for Links.- 6: Theory of Computation.- 28 On the Work of Steve Smale on the Theory of Computation.- Discussions.- 29 Smale’s Work on the Dynamics of Numerical Analysis.- 30 Steve Smale and the Geometry ofIll-Conditioning.- 31 On Smale’s Work in the Theory of Computation: From Polynomial Zeros to Continuous Complexity.- Contributed Papers.- 32 The Gödel Incompleteness Theorem and Decidability over a Ring.- 33 Ill-Posed Problem Instances.- 34 Cohomology of Braid Groups and Complexity.- 35 The Dynamics of Eigenvalue Computation.- 36 On the Role of Computable Error Estimates in the Analysis of Numerical Approximation Algorithms.- 37 Applications of Topology to Lower Bound Estimates in Computer Science.- 38 On the Distribution of Roots of Random Polynomials.- 39 A General NP-Completeness Theorem.- 40 Some Remarks on Bezout’s Theorem and Complexity Theory.- 41 Some Results Relevant to Smale’s Reports.- 42 Error Estimates of Ritz—Gelerkin Methods for Indefinite Elliptic Equations.- 7: Nonlinear Functional Analysis.- 43 Smale and Nonlinear Analysis: A Personal Perspective.- Discussion.- 44 Discussion.- 8: Applications.- 45 Steve Smale and Geometric Mechanics.- Discussion.- 46 Smale’s Topological Program in Mechanics and Convexity.- 47 Discussion.- Contributed Papers.- 48 On Paradigm and Method.- 49 Dynamical Systems and the Geometry of Singularly Perturbed Differential Equations.- 50 Cellular Dynamata.- 51 Rough Classification of Integrable Hamiltonians on Four-Dimensional Symplectic Manifolds.- 9: Final Panel.- 52 Final Panel.