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Selected Titles in This Series
60 David Jerison, I. M. Singer, and Daniel W . Stroock, Editors, The legacy of Norbert Wiener: A centennial symposium (Massachusetts Institute of Technology, Cambridge, October 1994)
59 Wil l iam Arveson, Thomas Branson, and Irving Segal, Editors, Quantization, nonlinear partial differential equations, and operator algebra (Massachusetts Institute of
Technology, Cambridge, June 1994)
58 Bill Jacob and Alex Rosenberg, Editors, K-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara, July 1992)
57 Michael C. Cranston and Mark A. Pinsky, Editors, Stochastic analysis (Cornell University, Ithaca, July 1993)
56 Wil l iam J. Haboush and Brian J. Parshall, Editors, Algebraic groups and their
generalizations (Pennsylvania State University, University Park, July 1991)
55 U w e Jannsen, Steven L. Kle iman, and Jean-Pierre Serre, Editors, Motives
(University of Washington, Seattle, July/August 1991)
54 Robert Greene and S. T. Yau, Editors, Differential geometry (University of
California, Los Angeles, July 1990)
53 James A. Carlson, C. Herbert Clemens , and David R. Morrison, Editors,
Complex geometry and Lie theory (Sundance, Utah, May 1989)
52 Eric Bedford, John P. D'Ange lo , Robert E. Greene, and Steven G. Krantz, Editors, Several complex variables and complex geometry (University of California, Santa
Cruz, July 1989)
51 Wil l iam B. Arveson and Ronald G. Douglas , Editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988)
50 James Gl imm, John Impagliazzo, and Isadore Singer, Editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988)
49 Robert C. Gunning and Leon Ehrenpreis, Editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987)
48 R. O. Wells, Jr., Editor, The mathematical heritage of Hermann Weyl (Duke University, Durham, May 1987)
47 Paul Fong, Editor, The Areata conference on representations of finite groups (Humboldt State University, Areata, California, July 1986)
46 Spencer J. Bloch, Editor, Algebraic geometry - Bowdoin 1985 (Bowdoin College,
Brunswick, Maine, July 1985)
45 Felix E. Browder, Editor, Nonlinear functional analysis and its applications (University
of California, Berkeley, July 1983)
44 Wil l iam K. Allard and Frederick J. Almgren, Jr., Editors, Geometric measure theory and the calculus of variations (Humboldt State University, Areata, California,
July/August 1984)
43 Frangois Treves, Editor, Pseudodifferential operators and applications (University of Notre Dame, Notre Dame, Indiana, April 1984)
42 Anil Nerode and Richard A. Shore, Editors, Recursion theory (Cornell University,
Ithaca, New York, June/July 1982)
41 Yum-Tong Siu, Editor, Complex analysis of several variables (Madison, Wisconsin, April 1982)
40 Peter Orlik, Editor, Singularities (Humboldt State University, Areata, California, July/August 1981)
39 Felix E. Browder, Editor, The mathematical heritage of Henri Poincare (Indiana
University, Bloomington, April 1980)
38 Richard V. Kadison, Editor, Operator algebras and applications (Queens University,
Kingston, Ontario, July/August 1980)
37 Bruce Cooperste in and Geoffrey Mason, Editors, The Santa Cruz conference on finite groups (University of California, Santa Cruz, June/July 1979)
(Continued in the back of this publication)
http://dx.doi.org/10.1090/pspum/060
Norbert Wiener (Photo courtesy of the MIT Museum.)
The Legacy of Norbert Wiener: A Centennial Symposium
Proceedings of Symposia in
PURE MATHEMATICS
Volume 60
The Legacy of Norbert Wiener: A Centennial Symposium
In Honor of the 100th Anniversary of Norbert Wiener's Birth October 8-14, 1994 Massachusetts Institute of Technology Cambridge, Massachusetts
David Jerison I. M. Singer Daniel W. Stroock Editors
American Mathematical Society Providence, Rhode Island
^ D E D
PROCEEDINGS OF A SYMPOSIUM ON THE LEGACY OF NORBERT WIENER
IN HONOR OF THE 100TH ANNIVERSARY OF HIS BIRTH HELD AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY
CAMBRIDGE, MASSACHUSETTS OCTOBER 8-14, 1994
with support from the Massachusetts Institute of Technology; the National Science Foundation, Grant DMS 9412279; Henry Singleton, and the Alfred P. Sloan Foundation
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
1991 Mathematics Subject Classification. Primary 01A06, 31-06, 34-06, 35-06, 40-06, 42-06, 45-06, 46-06, 60-06, 62-06, 68-06, 81-06, 82-06, 92-06, 94-06.
Library of Congress Cataloging-in-Publication Data
The legacy of Norbert Wiener : a centennial symposium in honor of the 100th anniversary of Norbert Wiener's birth, October 8-14, 1994, Massachusetts Institute of Technology, Cambridge, Massachusetts / David Jerison, I. M. Singer, Daniel W. Stroock, editors.
p. cm. — (Proceedings of symposia in pure mathematics, ISSN 0082-0717 ; v. 60) Includes bibliographical references. ISBN 0-8218-0415-4 1. Mathematics—Congresses. 2. Wiener, Norbert, 1894-1964—Congresses. I. Wiener,
Norbert, 1984 1964. II. Jerison, David, 1953- . III. Singer, I. M. (Isadore Manuel), 1924- . IV. Stroock, Daniel W. V. Series. QA1.L3786 1997 510—dc21 97-297
CIP
Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionOams.org.
Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.)
© 1997 by the American Mathematical Society. All rights reserved. Printed in the United States of America.
The American Mathematical Society retains all rights except those granted to the United States Government.
@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.
10 9 8 7 6 5 4 3 2 1 01 00 99 98 97
Contents
Preface ix
Program xi
Acknowledgments xix
Biographical and Historical Remarks 1
Norbert Wiener DAVID JERISON AND DANIEL W. STROOCK 3
The rise and fall of cybernetics in the US and the USSR P E T E R ELIAS 21
Reminiscences of Norbert Wiener
DIRK STRUIK 31
Speech by Fagi Levinson 33
Some memories of Norbert Wiener
PAUL A. SAMUELSON 37
Pictures 43
Scientific Articles 51
Gibbs measures, quasi-periodic solutions and nonlinear partial differential equations JEAN BOURGAIN 53
Wiener's Tauberian theorem LENNART CARLESON 65
The role of Paley-Wiener theory in partial differential equations LEON EHRENPREIS 71
Paley-Wiener on manifolds VICTOR GUILLEMIN 85
Norbert Wiener and the development of mathematical engineering THOMAS KAILATH 93
Oscillating networks of Neurons: Mathematics and function NANCY KOPELL 117
viii C O N T E N T S
Mapping heredity: Using probabilistic models and algorithms to map genes and genomes ERIC S. LANDER 137
A non-random walk down Wall Street ANDREW W. LO 149
Filtered Wiener space versus abstract Wiener space PAUL MALLIAVIN 185
Unsolved problems connected with the Wiener criterion VLADIMIR MAZ'YA 199
On the role of the Wiener process in finance theory and practice: The case of replicating portfolios ROBERT C. MERTON 209
Inference, learning, and recognition SANJOY K. MITTER 223
Issues in the mathematical modeling of cortical functioning and thought DAVID MUMFORD 235
Polymer braids and iterated Moire maps DAVID R. NELSON AND RANDALL D. KAMIEN 261
Networks that learn and how the brain works TOMASO POGGIO 275
The risk management revolution CHARLES S. SANFORD, J R . AND DAN BORGE 313
Complex noncommutative infinite dimensional analysis and Fermion-Boson interactions IRVING SEGAL 329
Complex integral transforms, diffraction, resonances and phase space tunneling JOHANNES SJOSTRAND 355
Scaling, the free field and statistical mechanics THOMAS SPENCER 373
Wiener-Hopf integral equations HAROLD WIDOM 391
Preface
Anybody looking at the list of topics of this symposium should have noticed that it is eclectic. And that is just what Wiener was. He sought out real-world phenomena that his mathematics might illuminate. He succeeded as often as he did because he searched far and wide.
From his forays into physics, engineering, and biology, Wiener brought back to mathematics new problems and insights that changed the course of mathematical research. Conversely, he used state-of-the-art techniques from mathematics to provide the first sound foundation for Einstein's formulation of Brownian motion and elucidated the fundamental role of electrostatic capacity in the analysis of solutions to Laplace's equation. Starting with basic ideas of signal processing, he developed his generalized harmonic analysis and his Tauberian theorems. These led, in turn, to a much broader understanding of spectral synthesis which has motivated generations of harmonic analysts. The Paley-Wiener theorems became a fundamental tool in the theory of linear partial differential equations. One could hardly have imagined the far-reaching consequences of Wiener's measure on path space, from quantum field theory to stochastic processes as used in economics or in the search for defective genes—not to mention Wiener-Hopf theory and prediction theory and their applications in electrical engineering.
The Wiener Symposium demonstrated the need for serious, sustained interactions between mathematics, broadly construed, and other sciences. A young Wiener today would bet his career on the fruitful outcome of such interactions.
David Jerison I. M. Singer Daniel W. Stroock Editors
Program
An * indicates that the speaker has contributed an article to this volume.
Saturday, October 8
I N T E G R A T I O N ON F U N C T I O N S P A C E AND P O T E N T I A L T H E O R Y
Morning
Welcome Charles M. Vest President, MIT
Irving E. Segal* MIT, emeritus Complex Infinite-Dimensional Noncommutative Analysis and Boson-Fermion Interactions
Paul Malliavin* Department of Mathematics, Unversite de Paris VI Cameron-Martin Theorems and Differential Space Structures on Path Spaces
Afternoon
Henry P. McKean Courant Institute for the Mathematical Sciences, New York University How to Build a Random Process from Independent Gaussian Variables: What Kolmogoroff and Wiener Did and What More Might Be Done
Peter W. Jones Department of Mathematics, Yale University Recent Advances in Potential Theory
Vladimir Maz'ya* Department of Mathematics, University of Linkoping Unsolved Problems Connected with the Wiener Criterion
XI
X l l P R O G R A M
Sunday, October 9
HARMONIC ANALYSIS
Morning
Lennart Carleson* Department of Mathematics, Royal Institute of Technology, Stockholm Tauberian Theorems
Charles Fefferman Department of Mathematics, Princeton University Electrons and Nuclei in Magnetic Fields
Afternoon
Stephane Mallat Courant Institute for the Mathematical Sciences, New York University Wavelets in Signal Processing
Elias M. Stein Department of Mathematics, Princeton University A Generalization of Wiener's Ergodic Theorem
Jean Bourgain* School of Mathematics, Institute for Advanced Study Nonlinear Schrodinger Equations and Gibbs Measures
P R O G R A M
Monday, October 10
WlENER-HOPF AND PALEY-WlENER THEORY
Morning
Harold Widom* Department of Mathematics, University of California, Santa Cruz Wiener-Hopf Integral Equations from the Classical and Operator-Theoretic Points of View
Lars Hormander Department of Mathematics, University of Lund General Mehler Formulas and the Weyl Calculus
Afternoon
Victor Guillemin* Department of Mathematics, MIT Paley-Wiener Theory on Manifolds
Johannes Sjostrand* Department of Mathematics, Universite de Paris, XI Complex Integral Transforms, Diffraction, Resonances and Phase Space Tunneling
Leon Ehrenpreis* Department of Mathematics, Temple University The Role of the Paley-Wiener Theorem in Partial Differential Equations
XIV P R O G R A M
Tuesday, October 11
STATISTICAL PHYSICS
Morning
Ek>ris Altshuler Department of Physics, MIT Universalities: from Anderson Localization to Quantum Chaos
Michael E. Fisher Institute for Physical Sciences and Technology, University of Maryland The Approach to the Limit in Statistical Mechanics
Afternoon
David Nelson* Department of Physics, Harvard University Path Integrals and Winding Numbers in High Temperature Superconductors
Yakov Sinai Department of Mathematics, Princeton University ID-Conservation Laws with Random Initial Data
Thomas Spencer* School of Mathematics, Institute for Advanced Study Quantum Dynamics in Random Media
CYBERNETICS
Evening
Peter Elias* Department of Electrical Engineering, MIT Cybernetics: Past and Present, East and West
Pesi R. Masani University of Pittsburgh, emeritus Wiener's Cybernetics and Its Societal Implications
P R O G R A M xv
Wednesday, October 12
E L E C T R I C A L E N G I N E E R I N G AND C O M P U T E R S C I E N C E
Morning
Robert Brayton Department of Electrical Engineering, University of California, Berkeley Synthesis and Verification of Finite State Systems
Robert G. Gallager Department of Electrical Engineering, MIT Universal Data Compression
Afternoon
Thomas Kailath* Department of Electrical Engineering, Stanford University The Wiener-Hopf Equation in Radiative Transfer and in Least-Squares Estimation
Sanjoy Mitter* Department of Electrical Engineering, MIT Pattern Recognition and Learning
Michael O. Rabin Department of Computer Science, Harvard University and Hebrew University Computability and Complexity from Turing to Feynman
NORBERT W I E N E R AT MIT
Evening
Walter A. Rosenblith MIT, emeritus Have I Had Lunch Yet?
Jerry Lettvin MIT, emeritus Pascal's Mother and Other Problems
Amar G. Bose Department of Electrical Engineering, MIT and Bose Corporation Ten Years with Norbert Wiener
xvi P R O G R A M
Thursday, October 13
FINANCIAL ECONOMICS
Morning
Paul A. Samuelson* M,IT, emeritus Some Memories of Norbert Wiener
Robert C. Merton* Business School, Harvard University On the Role of the Wiener Process in Finance Theory and Practice: The Case of Replicating Portfolios
Afternoon
Stephen A. Ross School of Organization and Management, Yale University Option Pricing
Andrew W. Lo* Sloan School of Management, MIT A Non-Random Walk Down Wall Street
Charles S. Sanford, Jr.* Chairman of the Board, Banker's Trust The Risk Management Revolution
P R O G R A M xvn
Friday, October 14
BIOLOGY
Morning
Nicholas Cozzarelli Department of Molecular and Cell Biology, University of California, Berkeley D N A Topology and Geometry
Eric Lander* Department of Biology and the Whitehead Institute, MIT Mathematics and Biology: New Frontiers
Afternoon
Nancy J. Kopell* Department of Mathematics, Boston University Networks and Neurons: Dynamics and Functions
David Mumford* Department of Mathematics, Harvard University Some Issues in the Mathematical Modeling of the Brain and Thought
Tomaso Poggio* Department of Brain and Cognitive Sciences, MIT Networks That Learn and How the Brain Works
Acknowledgments
The symposium was sponsored by the Massachusetts Institute of Technology and the American Mathematical Society, with financial support from Henry Singleton, the Massachusetts Institute of Technology, the Sloan Foundation, and the National Science Foundation.
The MIT Organizing Committee
Jonathan Allen, MIT David Benney, Chair, MIT Edward E. David, EED, Inc. Peter Elias, MIT Morris Halle, MIT David Jerison, ex-officio, MIT William Ted Martin, MIT Sanjoy Mitter, MIT Walter Rosenblith, Co-chair, MIT Jerome Wiesner, Co-chair, MIT
The Mathematics Organizing Committee
David Benney, ex-offlcio, MIT Bonnie A. Berger, MIT Roger Brockett, Harvard University Donald Burkholder, University of Illinois David Jerison, Chair, MIT Pesi R. Masani, University of Pittsburgh, emeritus Henry P. McKean, New York University Daniel Stroock, MIT I. M. Singer, MIT Elias M. Stein, Princeton University
Selected Titles in This Series (Continued from the front of this publication)
36 Robert Osserman and Alan Weinstein, Editors, Geometry of the Laplace operator (University of Hawaii, Honolulu, March 1979)
35 Guido Weiss and Stephen Wainger, Editors, Harmonic analysis in Euclidean spaces
(Williams College, Williamstown, Massachusetts, July 1978)
34 D . K. Ray-Chaudhuri , Editor, Relations between combinatorics and other parts of
mathematics (Ohio State University, Columbus, March 1978)
33 "A. Borel and W . Casselman, Editors, Automorphic forms, representations and
L-functions (Oregon State University, Corvallis, July/August 1977)
32 R. James Milgram, Editor, Algebraic and geometric topology (Stanford University,
Stanford, California, August 1976)
31 Joseph L. D o o b , Editor, Probability (University of Illinois at Urbana-Champaign,
Urbana, March 1976)
30 R. O. Wells, Jr., Editor, Several complex variables (Williams College, Williamstown,
Massachusetts, July/August 1975)
29 Robin Hartshorne, Editor, Algebraic geometry - Areata 1974 (Humboldt State
University, Areata, California, July/August 1974)
28 Felix E. Browder, Editor, Mathematical developments arising from Hilbert problems
(Northern Illinois University, Dekalb, May 1974)
27 S. S. Chern and R. Osserman, Editors, Differential geometry (Stanford University,
Stanford, California, July/August 1973)
26 Calvin C. Moore , Editor, Harmonic analysis on homogeneous spaces (Williams College,
Williamstown, Massachusetts, July/August 1972)
25 Leon Henkin, John Addison, C. C. Chang, Wil l iam Craig, Dana Scott , and Robert Vaught, Editors, Proceedings of the Tarski symposium (University of
California, Berkeley, June 1971)
24 Harold G. Diamond, Editor, Analytic number theory (St. Louis University, St. Louis,
Missouri, March 1972)
23 D . C. Spencer, Editor, Partial differential equations (University of California, Berkeley,
August 1971)
22 Arunas Liulevicius, Editor, Algebraic topology (University of Wisconsin, Madison,
June/July 1970)
21 Irving Reiner, Editor, Representation theory of finite groups and related topics
(University of Wisconsin, Madison, April 1970)
20 Donald J. Lewis, Editor, 1969 Number theory institute (State University of New York
at Stony Brook, Stony Brook, July 1969)
19 Theodore S. Motzkin, Editor, Combinatorics (University of California, Los Angeles,
March 1968)
18 Felix Browder, Editor, Nonlinear operators and nonlinear equations of evolution in
Banach spaces (Chicago, April 1968)
17 Alex Heller, Editor, Applications of categorical algebra (New York City, April 1968)
16 Shiing-Shen Chern and Stephen Smale, Editors, Global analysis, Part III
(University of California, Berkeley, July 1968)
15 Shiing-Shen Chern and Stephen Smale, Editors, Global analysis, Part II (University
of California, Berkeley, July 1968)
14 Shiing-Shen Chern and Stephen Smale, Editors, Global analysis, Part I (University
of California, Berkeley, July 1968)
13 Dana S. Scott (Part 1) and Thomas J. Jech (Part 2) , Editors, Axiomatic set
theory (University of California, Los Angeles, July/August 1967)
12 Wil l iam J. LeVeque and Ernst G. Straus, Editors, Number theory (Houston, Texas,
January 1967)
11 S. S. Chern, L. Ehrenpreis , J. Korevaar, W . H. J. Fuchs, and L. A. Rubel , Editors, Entire functions and related parts of analysis (University of California, San Diego, July 1966)