The Collected Papers of Stephen Smale

The following sections are included:

• Foreword

• Contents

• Permissions

The following sections are included:

• Introduction

• Topics
  • Differential Topology
  • Dynamical Systems
  • Nonlinear Functional Analysis
  • Physical and Biological Applications
  • Economics
  • Theory of Computation

The following sections are included:

• Nomination for Stephen Smale

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The following sections are included:

• Remarks on Part 1. Topology

• Remarks on Part 2. Economics

• Remarks on Part 3. Miscellaneous

• Remarks on Part 5. Dynamical Systems

• Remarks on Part 6. Mechanics

• Remarks on Part 7. Biology, Electric Circuits, Math. Programming

• Remarks on Part 8. Theory of Computation

The following sections are included:

• Background

• Immersions
  • Immersions of Circles
  • Immersions of Spheres in Euclidean Spaces
  • Further Development of Immersion Theory

• Diffeomorphisms of Spheres
  • Two-spheres
  • Three-spheres

• The Generalized Poincaré Conjecture and the h-Cobordism Theorem
  • Nice Functions, Handles, and Cell Structures
  • Eliminating Superfluous Handles
  • The h-Cobordism Theorem
  • The Structure of Manifolds

• The s-Cobordism Theorem

• References

The following sections are included:

• Definitions

• Proof of Theorem 1

• Applications

• BIBLIOGRAPHY

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The following sections are included:

• Introduction

• Fiber spaces

• Regular curves

• The reduction of the proof of Theorem B to 3.1 and 3.2.

• Proof of Proposition 3.1.

• Proof of Proposition 3.2.

• On the topology of the fiber Γ

• Classes of regular curves on a manifold

• Regular curves perpendicular to a submanifold

• Integral curves of a 1-form

• BIBLIOGRAPHY

The following sections are included:

• Gysin sequence of a Whitney sum

• Applications to immersed manifolds

• The intersection number of an immersion

• On the homology of Grassman manifolds

• The tangential map of an immersion

• BIBLIOGRAPHY

The following sections are included:

• Introduction

• Normal immersions

• The main theorem

• An orientability theorem

• REFERENCES

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The following sections are included:

• Introduction

• The covering homotopy theorem

• The weak homotopy equivalence theorem

• Applications

• Addenda

• REFERENCES

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The following sections are included:

• Introduction

• Proof of 1.2.

• Proof of 1.3.

• Proof of 1.1.

• REFERENCES

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The following sections are included:

• Proof of Theorem A.

• Proof of Theorem B.

• BIBLIOGRAPHY

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In this paper, we prove a number of theorems which give some insight into the structure of differentiable manifolds.

The methods, results and some notation of [13], hereafter referred to as GPC, and [12] will be used. These two papers and [14] can be considered as a starting point for this one. The main theorems in these papers are special cases of the theorems here…

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The following sections are included:

• General Equilibrium

• The Characterization of Pareto Optima

• The Existence of a General Equilibrium

• The Generic Local Uniqueness of General Equilibria

• The Global Newton Method

• Smale and the Theory of General Equilibrium

• Bibliography

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The following sections are included:

• Introduction

• References

• Appendix

Among other things a global version of the fundamental theorem of welfare economics is proved. One starts with a pure exchange economy with fixed total resources where hypotheses of differentiability, convexity, and monotonicity are made on the utility functions. Let A be the set of price equilibria where the initial allocation coincides with the final one. Then the map which assigns to such a price equilibrium, the corresponding allocation is a diffeomorphism (a complete correspondence) between A and the set of Pareto Optima.

In the spirit of Debreu's work on economies with a finite number of price equilibria, it is proved that under generic conditions, economies contain only a finite number of price equilibria when consumption sets are permitted to contain zero amounts of certain commodities. Also similar results are proved for economies with production (the last can be compared with works of Fuchs and Mas-Colell).

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The following sections are included:

• RESUME

• REFERENCES

• DISCUSSION

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A new way of looking at repeated games is introduced which incorporates a bounded memory and rationality. In these terms, a resolution of the prisoner's dilemma is given.

The following sections are included:

• The existence of equilibria

• Pure exchange economy: Existence of equilibria

• Pareto optimality

• Fundamental theorem of welfare economics

• Finiteness and stability of equilibria

• Appendix A. Existence of economic equilibrium with production

• Appendix B. A theorem on the approximation of multi-valued mappings

• References

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The following sections are included:

• Part I. Origins and Youth

• Part II. Vietnam Day

The following sections are included:

• Problem 1: The Riemann Hypothesis

• Problem 2: The Poincaré Conjecture

• Problem 3: Does P = NP?

• Problem 4: Integer Zeroes of a Polynomial

• Problem 5: Height Bounds for Diophantine Curves

• Problem 6: Finiteness of the Number of Relative Equilibrium in Celestial Mechanics

• Problem 7: Distribution of Points on the 2-Sphere

• Problem 8: Introduction of Dynamics into Economic Theory

• Problem 9: The Linear Programming Problem

• Problem 10: The Closing Lemma

• Problem 11: Is One-Dimensional Dynamics Generally Hyperbolic?

• Problem 12: Centralizers of Diffeomorphisms

• Problem 13: Hillbert's 16th Problem

• Problem 14: Lorenz Attractor

• Problem 15: Navier-Stokes Equations

• Problem 16: The Jacobian Conjecture

• Problem 17: Solving Polynomial Equations

• Problem 18: Limits of Intelligence

• REFERENCES