proceedings of the international congress of ...

PROCEEDINGS OF THE INTERNATIONAL CONGRESS OF MATHEMATICIANS

Volume 2

The Proceedings of the International Congress of Mathematicians held in Vancouver, August 21—29, 1974, is in two volumes. Volume 1 contains an account of the organization of the Congress, the list of members, the work of the Fields medalists, the expository addresses, and the addresses in Sections 1—7. Volume 2 contains the addresses in Sections 8—20 and the list of short communications.

Canadian Shared Cataloguing in Publication Data International Congress of Mathematicians, Vancouver, B.C., 1974. Proceedings of the International Congress of Mathematicians / [editor, Ralph D. James].— 1. Mathematics—Congresses. I. James, Ralph Duncan, ed. QA1.I8 1974 510.631 ISBN 0-919558-04-6 Library of Congress Catalog Card No. 74-34533 Copyright © 1975 by the Canadian Mathematical Congress All rights reserved Printed in the United States of America

Contents Section 8—Differential Geometry and Analysis on Manifolds Invariants of Flat Bundles

3

JEFF CHEEGER

Geometric Aspects of the Generalized Plateau Problem

7

H, BLAINE LAWSON, JR,

Problèmes de Géométrie Conforme

13

J. LELONG-FERRAND

ZjHCKpeTHbie rpynnu J\BmHOTnnoB Results and Independence Results in Set-Theoretical Topology

57 61

A. HAJNAL

Topological Structures

63

HORST HERRLICH

HeKOTopbie SiccTpeMajibHbie 3a#aHH TeopHH AnnpoKCHMaUHH H. n . KoPHEHHyK Quelques Problèmes de Factorisation d'Opérateurs Linéaires BERNARD MAUREY

iii

67 75

IV

CONTENTS

The Normality of Products

81

M A R Y ELLEN R U D I N

Section 10—Operator Algebras, Harmonic Analysis and Representation of Groups Structure Theory for Type III Factors A. CONNES Inversion Formula and Invariant Differential Operators on Solvable Lie Groups

87

93

MICHEL D U F L O

A Szegö Kernel for Discrete Series

99

A. W. K N A P P

Operator Algebras and Their Abelian Subalgebras

105

J. R. RINGROSE

Some Aspects of Ergodic Theory in Operator Algebras

Ill

ERLING ST0RMER

Homotopy Invariants for Banach Algebras

115

JOSEPH L. TAYLOR

Harmonic Analysis on Real Semisimple Lie Groups

121

V. S. VARADARAJAN

KoMnjieKCHbiH TapMOHHHecKHft AHajiH3 Ha riojiynpocTbix Tpynnax Jin R. n . ÄEJIOBEHKO

129

Section 11—Probability and Mathematical Statistics, Potential, Measure and Integration The Solution to the Buffon-Sylvester Problem and Stereology

137

R. V. AMBARTZUMIAN

The Gaussian Process and How to Approach It

143

R. M. D U D L E Y

Semigroups of Invariant Operators

147

J. FARAUT

Some Mathematical Problems Arising in Robust Statistics

153

PETER J. HUBER

Théorie du Potentiel Récurrent (Résultats Récents)

157

J. NEVEU

Functional Equations and Characterization of Probability Distributions

163

C. RADHAKRISHNA R A O

Random Time Evolution of Infinite Particle Systems FRANK

169

SPITZER

Limit Theorems for Dependent Random Variables Under Various Regularity Conditions V. STATULEVICIUS

173

CONTENTS

The Theory of Harmonic Spaces

V

183

BERTRAM WALSH

Stochastic Integrals in the Plane

189

JOHN WALSH

Section 12—Complex Analysis Theory of Factorization and Boundary Properties of Functions Meromorphic in the Disk 197 M, M, DZRBASJAN

Some Metric Properties of Quasi-Conformal Mappings

203

F. W. GEHRING

0 npeACTaBJieHHH AHajiHxmecKHx $yHKu;HH PHäUMH ^HpHXJie A- . JlEOHTbEB Classification of Kleinian Groups BERNARD

207 213

MASKIT

Intrinsic Metrics on Teichmüller Space

217

H. L. ROYDEN

On Quadratic Differentials and Extremal Quasi-Conformal Mappings

223

KURT STREBEL

Section 13—Partial Differential Equations Some Recent Advances in the Multidimensional Parametric Calculus of Variations

231

WILLIAM K. ALLARD

Free Boundary Problmes in the Theory of Fluid Flow Through Porous Media

237

CLAUDIO BAIOCCHI

On a Class of Fuchsian Type Partial Differential Operators

245

M. S. BAOUEKDI

Monotone Operators, Nonlinear Semigroups and Applications

249

HAIM BREZIS

Semigroups of Nonlinear Transformations and Evolution Equations

257

MICHAEL G. CRANDALL

Applications of Fourier Integral Operators

263

J, J. DUISTERMAAT

Elliptic Variational Inequalities

269

DAVID KINDERLEHRER

Monge-Ampère Equations and Some Associated Problems in Geometry . . , . 275 L. NIRENBERG AHajiHTHwecKHe PemeHHH ypaBHeHHft c BapnaijHOHHbiMH npoH3Bo^HHMH H HX npHJIO>KeHHH 281 M. H. BHIUHK

VI

CONTENTS

Section 14—Ordinary Differential Equations and Dynamic Systems Teojie3mecKm B OHHCJiepoBoft TeoMeTpHH

293

J\. B. AHOCOB

Symbolic Dynamics for Hyperbolic Flows

299

RUFUS BOWEN

On Generators in Ergodic Theory

303

W. KRIEGER

O rioBeAeHHH TaMHJibTOHOBbix CHCTeM, BJIH3KHX K HHTerpHpyeMbiM

309

H. H. HEXOPOIIIEB

On Bifurcations of Dynamical Systems

315

M. M. PEIXOTO

The Structure of Bernoulli Systems

321

BENJAMIN WEISS

Section 15—Control Theory and Related Optimization Problems Contrôle Impulsionnel et Inéquations Quasi Variationelles

329

A. BENSOUSSAN

Some Minimax Problems in Optimization Theory

335

V. F. DEMYANOV

Stochastic Differential Games with Stopping Times and Variational Inequalities

339

AVNER FRIEDMAN

Necessary and Sufficient Conditions for Local Controllability and Time Optimality

343

HENRY HERMES

A Finite Difference Method for Computing Optimal Stochastic Controls and Costs

349

HAROLD J. KUSHNER

Controllability in Topological Dynamics

355

LAWRENCE MARKUS ynpaBJieHne B VCJIOBKHX KoH(JMHKTa H

361

HeonpeÄejieHHOcra

A. H. CyBBOTHH Section 16—Mathematical Physics and Mechanics Spectral Deformation Techniques and Application to iV-Body Schrödinger Operators J.-M.

369

COMBES

Time Evolution of Infinite Classical Systems

377

OSCAR E. LANFORD III

Thomas-Fermi and Hartree-Fock Theory

383

ELLIOTT H. LIEB

Relations Between the Modulus and the Phase of Scattering Amplitudes ANDRé

MARTIN

387

CONTENTS

Markov Fields

Vil

395

EDWARD NELSON

Approximation of Feynman Integrals and Markov Fields by Spin Systems,,. 399 BARRY SIMON

Section 17—Numerical Mathematics Convergence in the Maximum Norm of Spline Approximations to Elliptic Boundary Value Problems

405

JAMES H. BRAMBLE

A Survey of Recent Progress in Approximation Theory

411

E. W. CHENEY

TeopHH ycTOHHHBOCTH Pa3HOCTHbix CxeM H HTepaiJHOHHbie MeTOßbl

417

A. A. CAMAPCKHö

Recent Progress in the Numerical Treatment of Ordinary Differential Equations

423

HANS J. STETTER

The Finite Element Method—Linear and Nonlinear Applications

429

GILBERT STRANG

MHCJieHHbie MeTOAbi B TeopHH #H(j)paKHHH

437

A. T. CBELUHHKOB

Invariant Subspaces

443

J. H. WILKINSON

Difficulty in Problems of Optimization

449

PHILIP WOLFE

Section 18—Discrete Mathematics and Theory of Computation HHAyKTHBHbift BbiBOß AßTOMaTOB, OyHKijHii H OporpaMM fl. M. BAP3£HHb Spectral Functions of Graphs

455 461

ALAN J. HOFFMAN

Extremal Properties on Partial Orders

465

D. J. KLEITMAN

On the Theory of Inference Operators

471

ROLF LINDNER

The Inherent Computational Complexity of Theories of Ordered Sets

477

ALBERT R. MEYER

Complexity of Product and Closure Algorithms for Matrices

483

M. S. PATERSON

Families of Sets

491

RICHARD RADO

Some Results in Algebraic Complexity Theory

497

VOLKER STRASSEN

3n Sets of Integers Containing No k Elements in Arithmetic Progression , , . 503 E. SZEMERÉDI

Vili

CONTENTS

Section 19—Applied Statistics, Mathematics in the Social and Biological Sciences CTOxacTHHecKHe ZjHHaMHHecKHe Mo/iejiH SKOHOMHnecKoro PaBHOBecnn — 509 E.B. ßbIHKHH The Future of Stochastic Modelling 517 P. A. P. MORAN

Mathematics and the Picturing of Data

523

JOHN W. TUKEY

Levels of Structure in Catastrophe Theory Illustrated by Applications in the Social and Biological Sciences 533 E. C. ZEEMAN

Section 20—History and Education 0 6 HccjieÄOBaHHHX no HcTopHH MaTeMaraKH, npOBOAflmnxcH B CoBeTCKOM Coirne

549

B. B. THEäEHKO

The Theory of Matrices in the 19th Century

561

THOMAS HAWKINS

Science as Handmaiden of Mathematics

571

GEOFFREY MATTHEWS

How to Understand and Teach the Logical Structures and the History of Classical Thermodynamics

577

C. TRUESDELL

Short Communications Index

587 599

Section 8 Differential Geometry and Analysis on Manifolds

Proceedings of the International Congress of Mathematicians Vancouver, 1974

Invariants of Flat Bundles Jeff Cheeger Let G be a Lie group with finitely many components. We are going to discuss some cohomology invariants of flat principal (/-bundles. Their properties were developed jointly with Simons [2] in an outgrowth of earlier work of Chern and Simons [3]; see also [4] for related ideas. We begin with some notation. Let /*(