Table Of ContentFrom Topology to
Computation:
Proceedings of the Smalefest
M.W. Hirsch J.E. Marsden M.Shub
Editors
From Topology to
Computation:
Proceedings of the Smalefest
With 76 Illustrations
Springer-Verlag
New York Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
Morris W. Hirsch Michael Shub
Department of Mathematics IBM Research
University of California T.J. Watson Research Center
Berkeley, CA 94720 USA Yorktown Heights, NY 10598-0218 USA
Jerrold E. Marsden
Department of Mathematics
University of California
Berkeley, CA 94720 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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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 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.
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 transcrip
tions, 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 autobio
graphical 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 vii
Contributors xv
Research Themes XIX
Conference Schedule, Smalefest, August 5-9, 1990 XXVll
Part 1: Autobiographical Material
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 67
R. Bott
7 Some Recollections of the Early Work of
Steve Smale ............................................. 73
M.M. Peixoto
8 Luncheon Talk 76
R. Thorn
x Contents
9 Banquet Address at the Smalefest 79
E.c. Zeeman
Part 3: Differential Topology
10 The Work of Stephen Smale in Differential Topology 83
Morris W. Hirsch
DISCUSSIONS
11 Discussion.............................................. 107
J. Stallings, A. Haejliger, and Audience Members
12 Note on the History of Immersion Theory 114
David Spring
CONTRIBUTED PAPER
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 254
Julian Palmore
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
