Ebook: 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.

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This volume is a Festschrift for Stephen Smale's 60th birthday; most of the chapters are based on papers given at a symposium in his honor at the University of California at Berkeley. The contributions reflect the breadth of Smale's interests, ranging over a wide variety of fields in pure and applied mathematics. In addition, Smale has provided an autobiographical essay, as well as a record of the lively panel discussions at the symposium. The research contributions include papers on differential topology, mathematical economics, dynamical systems, on the theory of computation, on nonlinear functional analysis, and on various applications in the physical and biological sciences. All are currently active areas of interdisciplinary research.

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This volume is a Festschrift for Stephen Smale's 60th birthday; most of the chapters are based on papers given at a symposium in his honor at the University of California at Berkeley. The contributions reflect the breadth of Smale's interests, ranging over a wide variety of fields in pure and applied mathematics. In addition, Smale has provided an autobiographical essay, as well as a record of the lively panel discussions at the symposium. The research contributions include papers on differential topology, mathematical economics, dynamical systems, on the theory of computation, on nonlinear functional analysis, and on various applications in the physical and biological sciences. All are currently active areas of interdisciplinary research.
Content:
Front Matter....Pages i-xxix
Front Matter....Pages 1-1
Some Autobiographical Notes....Pages 3-21
On How I Got Started In Dynamical Systems, 1959–1962....Pages 22-26
The Story of the Higher Dimensional Poincar? Conjecture (What Actually Happened on the Beaches of Rio)....Pages 27-40
On the Steps of Moscow University....Pages 41-52
Professional Biography, Bibliography, and Graduate Students....Pages 53-63
Front Matter....Pages 65-65
Luncheon Talk and Nomination for Stephen Smale....Pages 67-72
Some Recollections of the Early Work of Steve Smale....Pages 73-75
Luncheon Talk....Pages 76-78
Banquet Address at the Smalefest....Pages 79-80
Front Matter....Pages 81-81
The Work of Stephen Smale in Differential Topology....Pages 83-106
Discussion....Pages 107-113
Note on the History of Immersion Theory....Pages 114-116
The Smale—Hirsch Principle in Catastrophe Theory....Pages 117-128
Front Matter....Pages 129-129
Stephen Smale and the Economic Theory of General Equilibrium....Pages 131-146
Comments....Pages 162-162
Topology and Economics: The Contribution of Stephen Smale....Pages 147-161
Front Matter....Pages 1-1
On the Contribution of Smale to Dynamical Systems....Pages 165-178
Discussion....Pages 179-185
Recurrent Sets for Planar Homeomorphisms....Pages 186-195
Convergence of Finite-Element Solutions for Nonlinear PDEs....Pages 196-200
Front Matter....Pages 1-1
Ergodic Theory of Chaotic Dynamical Systems....Pages 201-226
Beyond Hyperbolicity: Expansion Properties of One-Dimensional Mappings....Pages 227-236
Induced Hyperbolicity, Invariant Measures, and Rigidity....Pages 237-242
On the Enumerative Geometry of Geodesics....Pages 243-253
A Relation Between Newton’s Method and Successive Approximations for Quadratic Irrationals....Pages 254-258
On Dynamical Systems and the Minimal Surface Equation....Pages 259-269
A New Zeta Function, Natural for Links....Pages 270-278
Front Matter....Pages 279-279
On the Work of Steve Smale on the Theory of Computation....Pages 281-301
Smale’s Work on the Dynamics of Numerical Analysis....Pages 302-304
Steve Smale and the Geometry of Ill-Conditioning....Pages 305-316
On Smale’s Work in the Theory of Computation: From Polynomial Zeros to Continuous Complexity....Pages 317-320
The G?del Incompleteness Theorem and Decidability over a Ring....Pages 321-339
Ill-Posed Problem Instances....Pages 340-358
Cohomology of Braid Groups and Complexity....Pages 359-367
The Dynamics of Eigenvalue Computation....Pages 368-386
On the Role of Computable Error Estimates in the Analysis of Numerical Approximation Algorithms....Pages 387-394
Applications of Topology to Lower Bound Estimates in Computer Science....Pages 395-418
On the Distribution of Roots of Random Polynomials....Pages 419-431
A General NP-Completeness Theorem....Pages 432-442
Some Remarks on Bezout’s Theorem and Complexity Theory....Pages 443-455
Front Matter....Pages 279-279
Some Results Relevant to Smale’s Reports....Pages 456-465
Error Estimates of Ritz—Gelerkin Methods for Indefinite Elliptic Equations....Pages 466-478
Front Matter....Pages 479-479
Smale and Nonlinear Analysis: A Personal Perspective....Pages 481-492
Discussion....Pages 493-496
Front Matter....Pages 497-497
Steve Smale and Geometric Mechanics....Pages 499-516
Smale’s Topological Program in Mechanics and Convexity....Pages 517-529
Discussion....Pages 530-533
On Paradigm and Method....Pages 534-544
Dynamical Systems and the Geometry of Singularly Perturbed Differential Equations....Pages 545-556
Cellular Dynamata....Pages 557-560
Rough Classification of Integrable Hamiltonians on Four-Dimensional Symplectic Manifolds....Pages 561-586
Front Matter....Pages 587-587
Final Panel....Pages 589-605

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This volume is a Festschrift for Stephen Smale's 60th birthday; most of the chapters are based on papers given at a symposium in his honor at the University of California at Berkeley. The contributions reflect the breadth of Smale's interests, ranging over a wide variety of fields in pure and applied mathematics. In addition, Smale has provided an autobiographical essay, as well as a record of the lively panel discussions at the symposium. The research contributions include papers on differential topology, mathematical economics, dynamical systems, on the theory of computation, on nonlinear functional analysis, and on various applications in the physical and biological sciences. All are currently active areas of interdisciplinary research.
Content:
Front Matter....Pages i-xxix
Front Matter....Pages 1-1
Some Autobiographical Notes....Pages 3-21
On How I Got Started In Dynamical Systems, 1959–1962....Pages 22-26
The Story of the Higher Dimensional Poincar? Conjecture (What Actually Happened on the Beaches of Rio)....Pages 27-40
On the Steps of Moscow University....Pages 41-52
Professional Biography, Bibliography, and Graduate Students....Pages 53-63
Front Matter....Pages 65-65
Luncheon Talk and Nomination for Stephen Smale....Pages 67-72
Some Recollections of the Early Work of Steve Smale....Pages 73-75
Luncheon Talk....Pages 76-78
Banquet Address at the Smalefest....Pages 79-80
Front Matter....Pages 81-81
The Work of Stephen Smale in Differential Topology....Pages 83-106
Discussion....Pages 107-113
Note on the History of Immersion Theory....Pages 114-116
The Smale—Hirsch Principle in Catastrophe Theory....Pages 117-128
Front Matter....Pages 129-129
Stephen Smale and the Economic Theory of General Equilibrium....Pages 131-146
Comments....Pages 162-162
Topology and Economics: The Contribution of Stephen Smale....Pages 147-161
Front Matter....Pages 1-1
On the Contribution of Smale to Dynamical Systems....Pages 165-178
Discussion....Pages 179-185
Recurrent Sets for Planar Homeomorphisms....Pages 186-195
Convergence of Finite-Element Solutions for Nonlinear PDEs....Pages 196-200
Front Matter....Pages 1-1
Ergodic Theory of Chaotic Dynamical Systems....Pages 201-226
Beyond Hyperbolicity: Expansion Properties of One-Dimensional Mappings....Pages 227-236
Induced Hyperbolicity, Invariant Measures, and Rigidity....Pages 237-242
On the Enumerative Geometry of Geodesics....Pages 243-253
A Relation Between Newton’s Method and Successive Approximations for Quadratic Irrationals....Pages 254-258
On Dynamical Systems and the Minimal Surface Equation....Pages 259-269
A New Zeta Function, Natural for Links....Pages 270-278
Front Matter....Pages 279-279
On the Work of Steve Smale on the Theory of Computation....Pages 281-301
Smale’s Work on the Dynamics of Numerical Analysis....Pages 302-304
Steve Smale and the Geometry of Ill-Conditioning....Pages 305-316
On Smale’s Work in the Theory of Computation: From Polynomial Zeros to Continuous Complexity....Pages 317-320
The G?del Incompleteness Theorem and Decidability over a Ring....Pages 321-339
Ill-Posed Problem Instances....Pages 340-358
Cohomology of Braid Groups and Complexity....Pages 359-367
The Dynamics of Eigenvalue Computation....Pages 368-386
On the Role of Computable Error Estimates in the Analysis of Numerical Approximation Algorithms....Pages 387-394
Applications of Topology to Lower Bound Estimates in Computer Science....Pages 395-418
On the Distribution of Roots of Random Polynomials....Pages 419-431
A General NP-Completeness Theorem....Pages 432-442
Some Remarks on Bezout’s Theorem and Complexity Theory....Pages 443-455
Front Matter....Pages 279-279
Some Results Relevant to Smale’s Reports....Pages 456-465
Error Estimates of Ritz—Gelerkin Methods for Indefinite Elliptic Equations....Pages 466-478
Front Matter....Pages 479-479
Smale and Nonlinear Analysis: A Personal Perspective....Pages 481-492
Discussion....Pages 493-496
Front Matter....Pages 497-497
Steve Smale and Geometric Mechanics....Pages 499-516
Smale’s Topological Program in Mechanics and Convexity....Pages 517-529
Discussion....Pages 530-533
On Paradigm and Method....Pages 534-544
Dynamical Systems and the Geometry of Singularly Perturbed Differential Equations....Pages 545-556
Cellular Dynamata....Pages 557-560
Rough Classification of Integrable Hamiltonians on Four-Dimensional Symplectic Manifolds....Pages 561-586
Front Matter....Pages 587-587
Final Panel....Pages 589-605
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