From Topology to Computation: Proceedings of the Smalefest

M.W. Hirsch J.E. Marsden Editors

From Topology to Computation:

M.Shub

Proceedings of the Smalefest

With 76 Illustrations

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest

Morris W. Hirsch Department of Mathematics University of California Berkeley, CA 94720 USA

Jerrold E. Marsden Department of Mathematics University of California Berkeley, CA 94720 USA

Michael Shub IBM Research T.J. Watson Research Center Yorktown Heights, NY 10598-0218 USA

Library of Congress Cataloging-in-Publication Data Smalefest (1990: Berkeley, Calif.)

From topology to computation: proceedings ofthe Smalefest / M. W. Hirsch, J.E. Marsden & M. Shub, editors.

p. em. Papers presented at an international research conference in honor

of Stephen Smale's 60th birthday, held Aug. 5-9, 1990 in Berkeley. Includes bibliographical references. ISBN-13: 978-1-4612-7648-7 e-ISBN-13: 978-1-4612-2740-3 DOl: 10.1007/978-1-4612-2740-3 1. Differential topology-Congresses. 2. Numerical calculations­

Congresses. 3. Differentiable dynamical systems-Congresses. I. Hirsch, Morris W., 1933- . II. Marsden, Jerrold E. III. Shub, Michael, 1943- . IV. Smale, Stephen, 1930- V. Title. QA613.6.S63 1990 514'.72-dc20 92-31991

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© 1993 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1993 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue. New York, NY 10010, USA), except for brief excepts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Acts, may accordingly be used freely by anyone.

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Preface

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 con­tributions. 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 sub­jects 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.

These discussions, with many contributions from the audience, turned out to be extremely useful in bringing out insights and patterns in Smale's work, and in bringing to light informative incidents and facts-some from decades ago-that are too often lost to the history of mathematics. We considered these discussions to be so valuable that we persuaded Springer-Verlag to

viii Preface

have them transcribed from the rather crude videotapes of the conference. The transcription was done with great patience and precision by Esther Zack. Some of the speakers had the opportunity to edit their transcriptions or submit manuscripts of their remarks; for mistakes in the other transcrip­tions, the editors are responsible. Whereas these discussions are not all of the utmost clarity, they present remarkable commentaries on important parts of contemporary mathematics by some of the people who developed it.

The main part of this volume consists of research papers contributed by speakers, colleagues, and students of Smale. All of these were refereed under the supervision of the editors. Also included are informal talks about Smale's life and career which were given at social events, and autobio­graphical writings by Smale, some of which are published here for the first time.

No discussion of Smale's work would be complete without reference to his political activities against oppression, some of which are discussed in his autobiographical articles. An interesting evening panel on Smale's politics was chaired by Serge Lang, but unfortunately we were not able to make a transcript.

The choice of topics and speakers inevitably reflects the interests of the organizing committee. Whereas we had intended to cover all aspects of Smale's mathematics, as it turned out some very interesting parts of his work were neglected: There were unfortunately no talks or contributed papers about mathematical biology, game theory, linear programming, or smooth solutions to partial differential equations. But to do justice to the full breadth of Smale's research would have required more than five days.

The organizing committee consisted of Gerard Debreu, Morris Hirsch, Nancy Kopell, Jerrold Marsden, Jacob Palis, Michael Shub, Anthony Tromba, and Alan Weinstein-all colleagues or former students of Smale.

The conference was generously supported by

IBM Research Mathematical Sciences Research Institute National Science Foundation Springer -Verlag University of California at Berkeley

Contents

Preface Contributors Research Themes Conference Schedule, Smalefest, August 5-9, 1990

Part 1: Autobiographical Material

vii xv

XIX

XXVll

Some Autobiographical Notes ............................. 3

2 On How I Got Started in Dynamical Systems, 1959-1962 ...... 22

3 The Story of the Higher Dimensional Poincare Conjecture (What Actually Happened on the Beaches of Rio) ............. 27

4 On the Steps of Moscow University ........................ 41

5 Professional Biography, Bibliography, and Graduate Students .. 53

Part 2: Informal Talks

6 Luncheon Talk and Nomination for Stephen Smale R. Bott

7 Some Recollections of the Early Work of

67

Steve Smale ............................................. 73 M.M. Peixoto

8 Luncheon Talk R. Thorn

76

x Contents

9 Banquet Address at the Smalefest E.c. Zeeman

Part 3: Differential Topology

10 The Work of Stephen Smale in Differential Topology Morris W. Hirsch

DISCUSSIONS

79

83

11 Discussion.............................................. 107 J. Stallings, A. Haejliger, and Audience Members

12 Note on the History of Immersion Theory David Spring

CONTRIBUTED PAPER

114

13 The Smale-Hirsch Principle in Catastrophe Theory ........... 117 V.A. Vassil'ev

Part 4: Economics

14 Stephen Smale and the Economic Theory of General Equilibrium ............................................. 131 Gerard Debreu

DISCUSSION

15 Topology and Economics: The Contribution of Stephen Smale .. 147 Graciella Chichilnisky

16 Comments 162 Y.-H. Wan

Part 5: Dynamical Systems

17 On the Contribution of Smale to Dynamical Systems .......... 165 J. Palis

DISCUSSION

18 Discussion .............................................. 179 S. Newhouse, R.F. Williams, and Audience Members

Contents xi

CONTRIBUTED PAPERS

19 Recurrent Sets for Planar Homeomorphisms ................. 186 Marcy Barge and John Franks

20 Convergence of Finite-Element Solutions for Nonlinear PDEs .. 196 X iaohua X uan

21 Ergodic Theory of Chaotic Dynamical Systems ............... 201 L.-S. Young

22 Beyond Hyperbolicity: Expansion Properties of One-Dimensional Mappings .............................................. 227 John Guckenheimer and Stewart Johnson

23 Induced Hyperbolicity, Invariant Measures, and Rigidity ....... 237 M.Jakobson

24 On the Enumerative Geometry of Geodesics ................. 243 Ivan A.K. Kupka and M.M. Peixoto

25 A Relation Between Newton's Method and Successive Approximations for Quadratic Irrationals Julian Palmore

254

26 On Dynamical Systems and the Minimal Surface Equation ..... 259 Per Tomter

27 A New Zeta Function, Natural for Links .................... 270 R.F. Williams

Part 6: Theory of Computation

28 On the Work of Steve Smale on the Theory of Computation .... 281 Michael Shub

DISCUSSIONS

29 Smale's Work on the Dynamics of Numerical Analysis ......... 302 Steve Batterson

30 Steve Smale and the Geometry of Ill-Conditioning ............ 305 James Demmel

Xli Contents

31 On Smale's Work in the Theory of Computation: From Polynomial Zeros to Continuous Complexity ................ 317 J.F. Traub

CONTRIBUTED PAPERS

32 The G6del Incompleteness Theorem and Decidability over a Ring ................................................. 321 Lenore Blum and Steve Smale

33 Ill-Posed Problem Instances ............................... 340 J ames Renegar

34 Cohomology of Braid Groups and Complexity ............... 359 V.A. Vassil'ev

35 The Dynamics of Eigenvalue Computation .................. 368 Steve Batterson

36 On the Role of Computable Error Estimates in the Analysis of Numerical Approximation Algorithms ...................... 387 Feng Gao

37 Applications of Topology to Lower Bound Estimates in Computer Science ....................................... 395 Michael D. Hirsch

38 On the Distribution of Roots of Random Polynomials ......... 419 E. Kostlan

39 A General NP-Completeness Theorem ...................... 432 Nimrod M egiddo

40 Some Remarks on Bezout's Theorem and Complexity Theory .. 443 Michael Shub

41 Some Results Relevant to Smale's Reports ................... 456 Xinghua Wang

42 Error Estimates of Ritz-Gelerkin Methods for Indefinite Elliptic Equations .............................................. 466 X iaohua X uan

Contents Xlll

Part 7: Nonlinear Functional Analysis

43 Smale and Nonlinear Analysis: A Personal Perspective ........ 481 Anthony J. Tromba

DISCUSSION

44 Discussion .............................................. 493 R. Abraham and F. Browder

Part 8: Applications

45 Steve Smale and Geometric Mechanics ...................... 499 Jerrold E. Marsden

DISCUSSIONS

46 Smale's Topological Program in Mechanics and Convexity ..... 517 Tudor S. Ratiu

47 Discussion .............................................. 530 A. Weinstein and Audience Members

CONTRIBUTED PAPERS

48 On Paradigm and Method ................................ 534 Philip Holmes

49 Dynamical Systems and the Geometry of Singularly Perturbed Differential Equations .................................... 545 N. Kopell

50 Cellular Dynamata ....................................... 557 R. Abraham

51 Rough Classification oflntegrable Hamiltonians on Four-Dimensional Symplectic Manifolds ......................... 561 A. T. Fomenko

Part 9: Final Panel

52 Final Panel ............................................. 589 R.F. Williams, D. Karp, D. Brown, o. Lanford, P. Holmes, R. Thom, E.C. Zeeman, M.M. Peixoto, and Audience Members

Contributors

R. Abraham Department of Mathematics University of California Santa Cruz, CA

M. Barge Department of Mathematics Montana State University Bozeman, MT

S. Batterson Department of Mathematics Emory University Atlanta, GA

L. Blum International Computer Science

Institute Berkeley, CA

R.Bott Department of Mathematics Harvard University Cambridge, MA

G. Chichilnisky Department of Economics Columbia University New York, NY

G.Debreu Economics Department

University of California Berkeley, CA

J.Demmel Mathematics Department and

Computer Science Division University of California Berkeley, CA

A.T. Fomenko Department of Geometry and

Topology Moscow State University Moscow, Russia

J. Franks Department of Mathematics Northwestern University Evanston, IL

F.Gao Department of Computer Science University of British Columbia Vancouver, Canada

J. Guckenheimer Department of Mathematics Cornell University Ithaca, NY

M.D. Hirsch Department of Mathematics

and Computer Science

xvi Contributors

Emory University Atlanta, GA

M.W. Hirsch Department of Mathematics University of California Berkeley, CA

P. Holmes Department of Theoretical and

Applied Mechanics and Mathematics

Cornell University Ithaca, NY

M. Jakobson Department of Mathematics University of Maryland College Park, MD

S. Johnson Department of Mathematics Williams College Williamstown, MA

N. Kopell Department of Mathematics Boston University Boston, MA

E. Kostlan Department of Mathematics Kapi Olani Community College Honolulu, HI

I.A.K. Kupka Department of Mathematics University of Toronto Ontario, Canada

J.E. Marsden Department of Mathematics University of California Berkeley, CA

N. Megiddo Almaden, IBM Research Center San Jose, CA

J. Palis Instituto de Matematica Pura e

Aplicada Rio de Janeiro, Brazil

J. Palmore Department of Mathematics University of Illinois

at Urbana Champaign Urbana,IL

M.M. Peixoto Instituto de Matematica Pura e

Aplicada Rio de Janeiro, Brazil

T. Ratiu Department of Mathematics University of California Santa Cruz, CA

J. Renegar School of Operations Research Cornell University Ithaca, NY

M. Shub IBM T.J. Watson Research Center Yorktown Heights, NY

S. Smale Department of Mathematics University of California Berkeley, CA

D. Spring Department of Mathematics Glendon College York University Toronto Ontario, Canada

R. Thorn Institut des Hautes

Etudes Scientifiques Bures-sur-Yvette, France

P. Tomter Matematisk Institutt Universitetet i Oslo Oslo, Norway

J.F. Traub Department of Computer Science Columbia University New York, NY

AJ. Tromba Mathematisches Institut Munich, Germany

V.A. Vassil'ev Institute of Systems Studies Academy of Science Moscow, Russia

Contributors xvii

X. Wang Department of Mathematics Hangzhou University Hangzhou, China

R.F. Williams Department of Mathematics University of Texas Austin, TX

X. Xuan Research and Development Hewlett Packard Company Santa Rosa, CA

L.-S. Young Department of Mathematics University of Arizana Tucson, AZ

E.c. Zeeman Hertford College Oxford University Oxford, England

Research Themes

Introduction

Many mathematicians have contributed powerful theorems in several fields. Smale is one of the very few whose work has opened up to research vast areas that were formerly inaccessible. From his early papers in differential topology to his current work in theory of computation, he has inspired and led the development of several fields of research: topology of nonlinear function spaces; structure of manifolds; structural stability and chaos in dynami­cal systems; applications of dynamical systems to mathematical biology, economics, electrical circuits; Hamiltonian mechanics; nonlinear functional analysis; complexity of real-variable computations. This rich and diverse body of work is outlined in the following subsection.

There are deep connections between Smale's work in apparently disparate fields, stemming from his unusual ability to use creatively ideas from one subject in other, seemingly distant areas. Thus, he used the homotopy theory of fibrations to study immersions of manifolds, and also the classification of differentiable structures. In another area, he applied handle body decomposi­tions of manifolds to structural stability of dynamical systems. Smale applied differential geometry and topology to the analysis of electrical circuits, and to several areas of classical mechanics. He showed how qualitative dynamical systems theory provides a natural framework for investigating complex phe­nomena in biology. A recent example is his application of algebraic topology to complexity of computation. In each case his innovative approach quickly became a standard research method. His ideas have been further developed by his more than 30 doctoral students, many of whom are now leading researchers in the fields he has pioneered.

This conference brought together mathematicians who are currently mak­ing important contributions to these fields. It had two purposes: First, to present recent developments in these fields; and second, to explore the con­nections between them. This was best done by examining the several areas of Smale's research in a single conference which crosses the traditional

XIX

xx Research Themes

boundary lines between mathematical subjects. In this way, a stimulating environment encouraged a fruitful exchange of ideas between mathemati­cians working in topics that are formally separate, but which, as Smale's work demonstrates, have strong intellectual connections.

Through this conference proceedings we hope that important new insights may be achieved into the extraordinary diversity and unity of mathematics.

Topics

Differential Topology Smale's first work in differential topology, on the classification of immersions of spheres, led to the general classification of immersions of manifolds. But it also presented, for the first time, the use of fibrations of function spaces in what is now called geometric topology. Through fibrations, the power­ful tools of algebraic topology were applied in new ways to a host of geo­metrical problems. This became, in the hands of Smale and many others, a standard approach to many areas: embeddings, diffeomorphisms, differential structures, piecewise linear theory, submersions, and other fields. The classi­fication of differential structures on topological manifolds due to R. Kirby and L. Siebernmann, and many of the profound geometrical theories of M. Gromov, are based on Smale's technique of function space fibrations.

In 1960, Smale startled the mathematical world with his proofs of the Generalized Poincare Conjecture and what is now called the H-Cobordism Theorem. Up to that time, the topological classification of manifolds was stuck at dimension three. John Milnor's exciting discovery in 1956 of exotic differential structures on the 7-sphere had pointed to the need for a theory of differential structures, but beyond his examples nothing was known about sufficient conditions for diffeomorphism. Smale had the audacity to attack the problem in dimensions five and above. His results opened the flood­gates of research in geometric topology. His techniques of handle cancella­tion and his constructive use of Morse theory proved enormously fruitful in a host of problems and have become standard approaches to the structural analysis of manifolds. Michael Freedman's recent topological classification of 4-dimensional manifolds is a far-reaching generalization of Smale's handle­canceling methods. It is closely related to exciting developments in Yang­Mills theory by Donaldson, Uhlenbeck, Taubes, and others. This work in­volves other areas of nonlinear functional analysis and mechanics that will be discussed below.

Dynamical Systems In the early 1960s, Smale embarked on the study of dynamical systems. Like topology, this subject was founded by Poincare, who called it the qualitative

Research Themes xxi

theory of differential equations. Intensively developed by G.D. Birkhoff, by 1960 it seemed played out as a source of new ideas. At this point, Smale introduced anew approach, based on geometrical assumptions about the dynamical process, rather than the standard method of examining specific equations coming from physics and engineering. The key notion was a hyper­bolic structure for the non wandering set, a far-reaching generalization of the standard notion of hyperbolic fixed point. Under this hypothesis, Smale proved that the nonwandering set (of points that are recurrent in a certain sense) breaks up into a finite number of compact invariant sets in a unique way; these he called basic sets. Each basic set was either a single periodic orbit or contained infinitely many periodic orbits that were tangled in a way that today would be called "chaotic." Moreover, he proved the dynamics in a basic set to be structurally stable.

These new ideas led to a host of conjectures, proofs, examples, and counterexamples by Smale, his many students, and collaborators. Above all, they led to new ways of looking at dynamical systems. These led to precise constructions and rigorous proofs for phenomena that, until then, were only vaguely describable, or only known in very special cases.

For example, Smale's famous Horseshoe is an easily described transforma­tion of the two-dimensional sphere that he proved to be both chaotic (in a precise sense) and structurally stable, and completely describable in com­binatorial terms. Moreover, this construction and analysis generalized to all manifolds of all dimensions. But it was more than merely an artificial class of examples, for Smale showed that any system satisfying a simple hypothesis going back to Poincare (existence of a transverse homoclinic orbit) must have a horseshoe system embedded in it. Such a system is, therefore, not only chaotic, but the chaos is stable in the sense that it cannot be eliminated by arbitrarily small perturbations. In this way, many standard models of natural dynamical processes have been proved to be chaotic.

Smale's new dynamical ideas were quickly applied, by himself and many others, to a variety of dynamical systems in many branches of science.

Nonlinear Functional Analysis Smale has made fundamental contributions to nonlinear analysis. His ap­plication (with R.S. Palais) of Morse's critical point theory to infinite­dimensional Hilbert space has been extensively used for nonlinear problems in both ordinary and partial differential equations. The "Palais-Smale" con­dition, proving the existence of a critical point for many variational problems, has been used to prove the existence of many periodic solutions for nonlinear Hamiltonian systems. Another application has been to prove the existence of minimal spheres and other surfaces in Riemannian manifolds.

Smale also was a pioneer in the development of the theory of manifolds of maps. The well-known notes of his lectures by Abraham and the related work

xxii Research Themes

of Eells has undergone active development ever since. For example, mani­folds of maps were used by Arnol'd, Ebin, Marsden, and others in their work on the Lagrangian representation of ideal incompressible fluids, in which the basic configuration space is the group of volume-preserving difTeomorphisms, and for which the Poisson reduced equations are the standard Euler equa­tions of fluid mechanics.

In 1965, Smale proved a generalization of the famous Morse-Sard theo­rem on the existence of regular values to a wide class of nonlinear mappings in infinite-dimensional Banach spaces. This permitted the use of transver­sality methods, so useful in finite-dimensional dynamics and topology, for many questions in infinite-dimensional dynamics. An important example is A. Tromba's proof that, generically (in a precise sense), a given simple closed curve in space bounds only a finite number of minimal surfaces of the topo­logical type of the disk. A similar result was proved by Foias and Temam for stationary solutions to the Navier-Stokes equations.

Physical and Biological Applications

Smale's first papers in mechanics are the famous ones on "Topology and Mechanics." These papers appeared in 1970 around the beginning of the geometric formulation of mechanics and its applications, when Mackey's book on the foundations of quantum mechanics and Abraham's book on the foundations of mechanics had just come out. Smale's work centered on the use of topological ideas, principally on the use of Morse theory and bifurca­tion theory to obtain new results in mechanics. Probably the best-known result in this work concerns relative equilibria in the planar n-body problem, which he obtained by exploiting the topological structure of the level sets of conserved quantities and the reduced phase space, so that Morse theory gave interesting results. For example, he showed that a result of Moulton in 1910, that there are !n! collinear relative equilibria, is a consequence of critical point theory. Smale went on to determine the global topology and the bifur­cation of the level sets of the conserved angular momentum and the energy for the problem. These papers were a great influence: for example, they led to further work of his former student Palmore on relative equilibria in the planar n-body problem and in vortex dynamics, as well as a number of studies by others on the topology of simple mechanical systems such as the rigid body. This work also was the beginning of the rich symplectic theory of reduction of Hamiltonian systems with symmetry. Smale investigated the case of the tangent bundle with a metric invariant under a group action, which was later generalized and exploited by Marsden, Weinstein, Guillemin, Sternberg, and others for a variety of purposes, ranging from fluids and plasmas to representation theory. The international influence of these papers on a worldwide generation of young workers in the now burgeoning area of geometric mechanics was tremendous.

Research Themes xxiii

Smale's work on dynamical systems also had a great influence in mechan­ics. In particular, the Poincare-Birkhoff-Smale horseshoe construction has led to studies by many authors with great benefit. For example, it was used by Holmes and Marsden to prove that the PDE for a forced beam has chaotic solutions, by Kopell and Howard to find chaotic solutions in reac­tion diffusion equations, by Kopell, Varaiya, Marsden, and others in circuit theory, by Levi in forced oscillations, and by Wiggins and Leonard to estab­lish connections between dynamical chaos and Lagrangian or particle mixing rates in fluid mechanics. This construction is regarded as a fundamental one in dynamical systems, and it is also one that is finding the most applications.

In 1972, Smale published his paper on the foundations of electric cir­cuit theory. This paper, highly influenced by an interaction between Smale, Desoer, and Oster, examines the dynamical system defined by the equations for an electric circuit, and gives a study of the invariant sets defined by Kirchhoff's laws and the dynamical systems on these sets. Smale was the first to deal with the implications of the topological complexities that this in­variant set might have. In particular, he raised the question of how to deal with the hysteresis or jump phenomena due to singularities in the constraint sets of the form f(x, dx/dt) = 0, and he discussed various regularizing devices. This had an influence on the electrical engineering community, such as the 1981 paper of Sastry and Desoer, "The Jump Behavior of Circuits and Sys­tems," which provided the answers to some of the questions raised by Smale's work. (This paper actually originated with Sastry's Masters thesis written in the Department of Mathematics at Berkeley.) This work also motivated the studies of Takens on constrained differential equations.

The best known of Smale's several papers in mathematical biology is the first, in which he constructed an explicit nonlinear example to illustrate the idea of Turing that biological cells can interact via diffusion to create new spatial and/or temporal structure. His deep influence on mathematical biol­ogy came less from the papers that he wrote in this field than from the impetus that his pure mathematical work gave to the study of qualitative dynamical systems. Because of the difficulty in measuring all of the relevant variables and the need for clarifying simplification, qualitative dynamical systems provides a natural framework for investigating dynamically com­plex phenomena in biology. These include "dynamical diseases" (Glass and Mackey), oscillatory phenomena such as neural "central pattern generators" (Ermentrout and Kopell), complexity in ecological equations (May) and im­mune systems (Perelson), and problems involving spontaneous pattern for­mation (Howard and Kopell, Murray, Oster).

Another important paper constructed a class of systems of classical com­peting species equations in [Rn with the property that the simplex N-1

spanned by the n coordinate unit vectors is invariant and the trajectories of the large system asymptotic to those of any dynamical system in Lln- 1 ;

this demonstrated the possible complexity in systems of competing species. Hirsch has shown that arbitrary systems of competing species decompose

XXIV Research Themes

into pieces which are virtually identical with Smale's construction. Thus, Smale's example, seemingly a very special case, turns out to be the basic building block for the general case.

Economics Smale's geometrical approach to dynamics proved fruitful not only in the physical and biological sciences, but also in economic theory. In 1973, he began a series of papers investigating the approach to equilibria in various economic models. In place of the then standard linear methods relying heav­ily on convexity, he used nonlinear differentiable dynamics with an emphasis on generic behavior. In a sense, this represented a return to an older tradition in mathematical economics, one that relied on calculus rather than algebra, but with intuitive arguments replaced by the rigorous and powerful methods of modern topology and dynamics.

Smale showed that under reasonable assumptions the number of equilibria in a large market economy is generically finite, generalizing work of Debreu. He gave a rigorous treatment of Pareto optimality. His interest in economic processes inspired his work on global Newton algorithms (see the section below). This led to an important paper on price-adjustment processes.

His interest in theoretical economics led Smale to work in the theory of games. In an original approach to the "Prisoner's Dilemma," which is closely related to economic competition, he showed that two players employing certain kinds of reasonable strategies will, in the long run, achieve optimal gains.

Smale's work in economics led to this research in the average stopping time for the simplex algorithm in linear programming, discussed below.

Theory of Computation Smale's work in economic equlibrium theory led him to consider questions about convergence of algorithms. The economists H. Scarf and C. Eaves had turned Sperner's classical existence proof for the Brouwer fixed-point theorem into a practical computational procedure for approximating a fixed point. Their methods were combinatorial; Smale transformed them into the realm of differentiable dynamics. His "global Newton" method was a simple­looking variant of the classical Newton-Raphson algorithm for solving f(x) = 0, where f is a nonlinear transformation of n-space. Unlike the classi­cal Newton's method, which guarantees convergence of the algorithm only if it is started near a solution, Smale proved that, under reasonable assump­tions, the global Newton algorithm will converge to a solution for almost every starting point which is sufficiently far from the origin. Because it is easy to find such starting points, this led to algorithms that are guaranteed to converge to a solution with probability one. The global Newton algorithm was the basis for an influential paper on the theory of price adjustment by Smale in mathematical economics.

Research Themes xxv

This geometrical approach to computation was developed further in a series of papers on Newton's method for polynomials, several of which were joint work with M. Shub. These broke new ground by applying to numer­ical analysis ideas from dynamical systems, differential topology and prob­ability, together with mathematical techniques from many fields: algebraic geometry, geometric measure theory, complex function theory, and differen­tial geometry.

Smale then turned to an algorithm that is of great practical importance, the simplex method for linear programming. It was known that this algo­rithm usually converged quickly, but that there are pathological examples requiring a number of steps that grows exponentially with the number of variables. Smale asked: What is the average number of steps required for m inequalities in n variables if the coefficients are bounded by 1 in absolute value? He translated this into a geometric problem which he then solved. The surprising answer is that the average number of steps is sublinear in n (or m)

if m (or n) is kept fixed. These new problems, methods, and results led to a great variety of papers

by Smale and many others, attacking many questions of computational com­plexity. Smale always emphasized that he looks at algorithms as mathe­maticians do, in terms of real numbers, and not as computer scientists do, in terms of a finite number of bits of information. Most recently this led Smale, L. Blum, and M. Shub to a new algebraic approach to the general theory of computability and some surprizing connections with G6de1's theorem.

Smalefest, August 5-9, 1990

Unity and Diversity in the Mathematical Sciences

Sunday, August 5, DIFFERENTIAL TOPOLOGY AND ECONOMICS

9:00-9: 15 9: 15-10:00

10:00-10:45

11: 15-12: 15

Dean G. Chew, Opening Remarks M. Hirsch, A Survey of Smale's Work in Differential Topology Discussion, J. Stallings and A. Haej/iger

M. Freedman, On Related Examples due to R.H. Bing and Ya. Zeldovich

Afternoon Session Chaired by Y-H. Wan 2:00-3:00 G. Debreu, Stephen Smale and the Economic Theory of General

Equilibrium 3: 00-3:30 Discussion

4:00-5:00

5:00-6:00

J. Geanakoplos, Solving Systems of Simultaneous Equations in Economics

A. M asCollel, Smale and First Order Conditions in Economics

Monday, August 6, DYNAMICAL SYSTEMS

Morning Sesstion Chaired by Jenny Harrison 9:00-9:45 J. Palis, A Survey of Smale's Work in Dynamical Systems 9: 45-10: 30 Discussion, S. Newhouse and R.F. Williams

11 : 00-12: 00 L. Young, Ergodic Theory of Chaotic Dynamical Systems

Afternoon Session Chaired by Charles Pugh 2:00-3:00 F. Takens, Homoclinic Bifurcations

3:30-4:30 D. Sullivan, Universal Geometric Structure of Quasi-Periodic Orbits

4:30-5:30 J. Franks, Rotation Numbers for Surface

6:15-7:30 7:30

Homeomorphisms

Reception Banquet, J. Kelley, R.F. Williams, M. Peixoto, C. Zeeman

XXVlll Conference Schedule

Tuesday, August 7, THEORY OF COMPUTATION

9:00-9:45 9:45-10: 30

11 :00-12:00

12: 15

2:00-3:00

3: 30-4: 30 4:30-5: 30

7:00-8:00

M. Shub, A Survey of Smale's Work in Computational Theory Discussion, J. Traub, S. Batterson, J. Demmel

L. Blum, Godel's Incompleteness Theorem and Decidability over a Ring

Film: Turning the Sphere Inside Out

J. Renegar, Approximating Solutions for Algebraic Formulae

V.A. Vasil'ev, Cohomology of Braid Groups and Complexity C. McMullen, Braids, Algorithms and Rigidity of Rational Maps

Evening Lecture, S. Lang, Smale's Political Work

Wednesday, August 8, NONLINEAR FUNCTIONAL ANALYSIS

Session Chair, Murray Protter 9: 00-9: 45 A.J. Tromba, A Survey of Smale's Work in Functional Analysis 9: 45-10: 30 Discussion, R. Abraham and F. Browder

11 :00-12:00 K. Uhlenbeck, The Topology of Moduli Spaces: A 90's Tribute to

12:30-2:30 2: 30-3 :30 3: 30-4: 30

5:00-6:00

Smale's 60's

Banquet Lunch, R. Bott and R. Thom R.S. Palais, Critical Point Theory when Condition C is Hiding P. Rabinowitz, Variational Methods and Hamiltonian Systems

N. Smale, Complete Conformally Flat Metrics with Constant Scalar Curvature

Thursday, August 9, PHYSICAL AND BIOLOGICAL APPLICATIONS

Morning Session Chaired by R. Abraham 9:00-9:45 J. Marsden, Steve Smale and Geometric Mechanics 9: 45-10: 30 Discussion, A. Weinstein and T. Ratiu

11 :00-12:00 N. Kopell, Dynamical Systems and the Geometry of Singularly Perturbed Equations

Afternoon Session Chaired by C. Desoer 2:00-3:00 P. Holmes, From Nonlinear Oscillations to Horseshoes and on to

3:30-4:30 4:30-6:00

6:00

Turbulence-Perhaps

S. Sastry, Smale's Work in Electrical Circuit Theory M. Shub and R. Williams (Organizers), Overview and Discussion,

D. Karp, D. Brown, O. Lanford, R. Bott, R. Thom, E.C. Zeeman, M. Peixoto

Close of Conference

From Topology to Computation: Proceedings of the Smalefest