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60 David Jerison, I. M. Singer, and Daniel W. Stroock, Editors, The legacy of : A centennial symposium ( Institute of Technology, Cambridge, October 1994) 59 William Arveson, Thomas Branson, and , 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 : 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 (, Ithaca, July 1993) 56 William J. Haboush and Brian J. Parshall, Editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991) 55 Uwe Jannsen, Steven L. Kleiman, 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'Angelo, Robert E. Greene, and Steven G. Krantz, Editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, Editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 , John Impagliazzo, and , Editors, The legacy of (Hofstra University, Hempstead, , 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, - Bowdoin 1985 (Bowdoin College, Brunswick, Maine, July 1985) 45 Felix E. Browder, Editor, Nonlinear and its applications (University of California, Berkeley, July 1983) 44 William K. Allard and Frederick J. Almgren, Jr., Editors, Geometric theory and the 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 Cooperstein 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) 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 Providence, Rhode Island

^DED 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 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 in the US and the USSR PETER 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 53 Wiener's Tauberian theorem 65 The role of Paley-Wiener theory in partial differential equations LEON EHRENPREIS 71 Paley-Wiener on VICTOR GUILLEMIN 85 Norbert Wiener and the development of mathematical 93 Oscillating networks of Neurons: Mathematics and function 117 viii CONTENTS

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 PAUL MALLIAVIN 185 Unsolved problems connected with the Wiener criterion VLADIMIR MAZ'YA 199 On the role of the 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 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, JR. 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 , engineering, and , 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 pro• vide the first sound foundation for Einstein's formulation of 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 and his Tauberian theorems. These led, in turn, to a much broader understanding of spectral synthesis which has motivated gen• erations 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 to stochastic processes as used in 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 interac• tions 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

INTEGRATION ON FUNCTION SPACE AND

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, 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, Recent Advances in Potential Theory

Vladimir Maz'ya* Department of Mathematics, University of Linkoping Unsolved Problems Connected with the Wiener Criterion

XI Xll PROGRAM

Sunday, October 9

HARMONIC ANALYSIS

Morning Lennart Carleson* Department of Mathematics, Royal Institute of Technology, Tauberian Theorems

Charles Fefferman Department of Mathematics, 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 PROGRAM

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 PROGRAM

Tuesday, October 11

STATISTICAL PHYSICS

Morning Ek>ris Altshuler Department of Physics, MIT Universalities: from Anderson Localization to Michael E. Fisher Institute for Physical Sciences and Technology, University of The Approach to the Limit in Statistical Mechanics

Afternoon David Nelson* Department of Physics, 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 PROGRAM xv

Wednesday, October 12

ELECTRICAL ENGINEERING AND

Morning Robert Brayton Department of Electrical Engineering, University of California, Berkeley Synthesis and Verification of Finite State

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 from Turing to Feynman

NORBERT WIENER 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 PROGRAM

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 PROGRAM xvn

Friday, October 14

BIOLOGY

Morning Nicholas Cozzarelli Department of Molecular and Cell Biology, University of California, Berkeley DNA and Geometry

Eric Lander* Department of Biology and the , MIT Mathematics and Biology: New Frontiers

Afternoon Nancy J. Kopell* Department of Mathematics, 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 , Co-chair, MIT

The Mathematics Organizing Committee

David Benney, ex-offlcio, MIT Bonnie A. Berger, MIT Roger Brockett, Harvard University Donald Burkholder, University of 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 and Stephen Wainger, Editors, Harmonic analysis in Euclidean spaces (Williams College, Williamstown, Massachusetts, July 1978) 34 D. K. Ray-Chaudhuri, Editor, Relations between 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. Doob, 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 , John Addison, C. C. Chang, William Craig, Dana Scott, and Robert Vaught, Editors, Proceedings of the Tarski symposium (University of California, Berkeley, June 1971) 24 Harold G. Diamond, Editor, Analytic (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, (University of Wisconsin, Madison, June/July 1970) 21 Irving Reiner, Editor, 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 , 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 , 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 (University of California, Los Angeles, July/August 1967) 12 William 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)