http://dx.doi.org/10.1090/pspum/054.3

Recent Titles in This Series

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 James Glimm, 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 William 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 Francois 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) 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-f unctions (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) (Continued in the back of this publication) Differential Geometry: Riemannian Geometry Proceedings of Symposia in PURE MATHEMATICS

Volume 54, Part 3

Differential Geometry: Riemannian Geometry

Robert Greene S. T. Yau Editors

z((l*0S^iw American Mathematical Society Sll u h n n rj £ Mm * 3\\ ILliilil In Providence, Rhode Island PROCEEDINGS OF THE SUMMER RESEARCH INSTITUTE ON DIFFERENTIAL GEOMETRY HELD AT THE UNIVERSITY OF CALIFORNIA, LOS ANGELES LOS ANGELES, CALIFORNIA JULY 8-28, 1990 with the support from the National Science Foundation Grant DMS-8913610. 1991 Mathematics Subject Classification. Primary 53A10, 49F20, 58E20, 58E12, 58G11 (Part 1) 81E13, 53C80, 53B50, 32C10, 53C15, 58F05, 83C75 (Part 2) 53C20, 53C40, 58C40, 58G25, 58F07, 58F11, 58F17 (Part 3).

Library of Congress Cataloging-in-Publication Data Riemannian geometry/Robert Greene and S. T. Yau, editors. p. cm.—(Proceedings of symposia in pure mathematics; v. 54, pt. 3) (Differential ge• ometry; pt. 3) "Proceedings of a Summer Research Institute on Differential Geometry, held at the Uni• versity of California, Los Angeles, July 8-28, 1990"—T.p. verso. ISBN 0-8218-1494-X (Part 1) ISBN 0-8218-1495-8 (Part 2) ISBN 0-8218-1496-6 (Part 3) ISBN 0-8218-1493-1 (set) (alk. paper) 1. Geometry, Riemannian—Congresses. I. Greene, Robert Everist, 1943- . II. Yau, Shing- Tung, 1949- . III. Summer Research Institute on Differential Geometry (1990: University of California, Los Angeles) IV. Series. V. Series: Differential geometry; pt. 3. QA641.D3833 1993 pt. 3 516.3'6s—dc20 92-34056 [516.3'73] CIP COPYING AND REPRINTING. Individual readers of this publication, and nonprofit li• braries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publi• cation (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Services, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. The appearance of the code on the first page of an article in this book indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale.

Copyright ©1993 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. @ This publication was typeset using AMS-TgX, the American Mathematical Society's TgX macro system. 10 9 8 7 6 5 4 3 2 1 98 97 96 95 94 93 Dedication.

To Professor S.-S. Chern, in appreciation of his formative influence on

modern differential geometry. Contents (* denotes one-hour survey lectures) Preface xxi

PART 1

Open Problems in Geometry* SHING TUNG YAU 1

Questions and Answers about Area-Minimizing Surfaces and Geometric Measure Theory* FRED ALMGREN 29

On the Geometrical Origin and the Solutions of a Degenerate Monge-Ampere Equation DAVID BAO AND TUDOR RATIU 55

Differential Geometry and the Design of Gradient Algorithms* ROGER W. BROCKETT 69

Spectral Geometry of F-Manifolds and its Application to Harmonic Maps YUAN-JEN CHIANG 93

Constructing Harmonic Maps into the Hyperbolic Space HYEONG IN CHOI AND ANDREJS TREIBERGS 101

Spherical Minimal Immersions of Spherical Space Forms DENNIS DETURCK AND WOLFGANG ZILLER 111

Banach Manifolds of Solutions to Nonlinear Partial Differential Equa• tions, and Relations with Finite-Dimensional Manifolds JOSEF DORFMEISTER 121

Some New Harmonic Maps ROBERT HARDT 141 x CONTENTS

Existence of Unstable Minimal Surfaces in Manifolds with Homology and Applications to Triply Periodic Minimal Surfaces JOEL HASS, JON T. PITTS, AND J. H. RUBINSTEIN 147

Closed Minimal Submanifolds in the Spheres* WU-YIHSIANG 163

Local and Global Behaviour of Hypersurfaces Moving by Mean Curvature* GERHARD HUISKEN 17 5

The Level-Set Flow on a Manifold TOM ILMANEN 193

Unstable Solutions of Two-Dimensional Geometric Variational Problems* JURGEN JOST 205

Harmonic Maps and Superrigidity JURGEN JOST AND SHING TUNG YAU 245

Harmonic Functions of Polynomial Growth on Complete Manifolds ATSUSHI KASUE 281

The Structure of Constant Mean Curvature Embeddings in Euclidean Three Space NICK KOREVAAR AND ROB KUSNER 291

Uniformization of Spherical CR Manifolds and the CR Yamabe Problem ZHONGYUAN LI 299

The Theory of Harmonic Functions and Its Relation to Geometry* PETER LI 307

Harmonic Maps with Prescribed Singularities YAN YAN LI AND GANG TIAN 317

Some Recent Results on Harmonic Maps to Spheres* FANG HUA LIN 327

The Geometry, Topology, and Existence of Periodic Minimal Surfaces* WILLIAM H. MEEKS III 333 CONTENTS xi

Soap Films and Mathematics FRANK MORGAN 375

Uniform Boundary Regularity Estimates for Minima of Certain Quadratic Functional LIBINH. Mou 381

Self-similar Solutions and Asymptotic Behavior of Flows of Nonpara- metric Surfaces Driven by Gauss or Mean Curvature VLADIMIR OLIKER 389

A Report on Geometric Quantization P. L. ROBINSON 403

Motion of Curves by Crystalline Curvature, Including Triple Junctions and Boundary Points JEAN E. TAYLOR 417

Recent Progress in Submanifold Geometry* CHUU-LIAN TERNG 439

Constant Mean Curvature Surfaces in the Heisenberg Group PER TOMTER 485

Complete Immersions of Constant Mean Curvature* HENRY C. WENTE 497

Banach Manifolds of Minimal Surfaces in the 4-Sphere HONGYOU Wu 513

On the Isolatedness for the Solutions of Plateau's Problem STEPHEN ZHENG 541

PART 2

Invariants of Gauss Maps of Theta Divisors MALCOLM ADAMS, CLINT MCCRORY, TED SHIFRIN, AND ROBERT VARLEY 1

On Characteristics of Hypersurfaces in Symplectic Manifolds AUGUSTIN BANYAGA 9 xii CONTENTS

Disprisoning and Pseudoconvex Manifolds JOHN K. BEEM 19

On the Holonomy of Lorentzian Manifolds L. BERARD BERGERY AND A. IKEMAKHEN 27

Spinors, Dirac Operators, and Changes of Metrics JEAN-PIERRE BOURGUIGNON 41

The Hyperkahler Geometry of the ADHM Construction and Quaternionic Geometric Invariant Theory CHARLES P. BOYER AND BENJAMIN M. MANN 45

Non-Abelian Vortices and a New Yang-Mills-Higgs Energy STEVEN B. BRADLOW 85

What Are the Best Almost-Complex Structures on the 6-Sphere? EUGENIO CALABI AND HERMAN GLUCK 99

The Failure of Complex and Symplectic Manifolds to be Kahlerian Luis A. CORDERO, MARISA FERNANDEZ, AND ALFRED GRAY 107

Nonabelian Hodge Theory* KEVIN CORLETTE 12 5

Geometric Invariants and Their Adiabatic Limits XIANZHE DAI 145

Geometry of Elementary Particles ANDRZEJ DERDZINSKI 15 7

Existence of Connections with Prescribed Yang-Mills Currents DENNIS DETURCK, HUBERT GOLDSCHMIDT, AND JANET TALVACCHIA 17 3

Vector Bundles over Homogeneous Spaces and Complete, Locally Symmetric Spaces LANCE D. DRAGER AND ROBERT L. FOOTE 183

Margulis Space-Times TODDA. DRUMM 191

Incomplete Flat Homogeneous Geometries D. DUNCAN AND E. IHRIG 197 CONTENTS xiii

Geodesic and Causal Behavior of Gravitational Plane Waves: Astigmatic Conjugacy PAUL E. EHRLICH AND GERARD G. EMCH 203

Curvature of Singular Spaces via the Normal Cycle JOSEPH H. G. Fu 211

The Nonintegrable Phase Factor and Gauge Theory RONALD O. FULP 223

The Lorentzian Version of the Cheeger-Gromoll Splitting Theorem and its Application to General Relativity GREGORY J. GALLOWAY 249

Weyl Structures on Self-Dual Conformal Manifolds PAUL GAUDUCHON 259

Chiral Anomalies and Dirac Families in Riemannian Foliations JAMES F. GLAZEBROOK AND FRANZ W. KAMBER 271

What is the Shape of Space in a Spacetime? STEVEN G. HARRIS 287

The Kinematics of the Gravitational Field ADAM D. HELFER 297

Support Theorems in Integral Geometry and Their Applications SIGURDUR HELGASON 317

Killing Spinors and Eigenvalues of the Dirac Operator OUSSAMA HIJAZI 325

Einstein Metrics on Circle Bundles GARY R. JENSEN AND MARCO RIGOLI 331

Topological and Differentiable Structures of the Complement of an Arrangement of Hyperplanes TAN JIANG AND STEPHEN S.-T. YAU 337

The Relationship between the Moduli Spaces of Vector Bundles on K3 Surfaces and Enriques Surfaces HOIL KIM 359 xiv CONTENTS Self-Dual Manifolds with Symmetry C. LEBRUN AND Y. S. POON 365

Remarks on the Skyrme Model ELLIOTT H. LIEB 379

Geometric Settings for Quantum Systems with Isospin E. B. LIN 385

Heat Kernels on Covering Spaces and Topological Invariants JOHN LOTT 391

The Heat Kernels of Symmetric Spaces QI-KENG LU 401

CR-Geometry and Deformations of Isolated Singularities* JOHN J. MILLSON 411

Generalized Symplectic Geometry on the Frame Bundle of a Manifold L. K. NORRIS 435

Complex Geometry and String Theory* D. H. PHONG 467

Constructing Non-Self-Dual Yang-Mills Connections on S4 with Arbitrary Chern Number LORENZO SADUN AND JAN SEGERT 529

The Yang-Mills Measure for the Two-Sphere AMBAR SENGUPTA 539

Examples of Nonminimal Critical Points in Gauge Theory* L. M. SIBNER 547

Spectral Invariants of Pseudoconformal Manifolds NANCY K. STANTON 551

L2-Cohomology and Index Theory of Noncompact Manifolds* MARK STERN 559

On Calabi-Yau Three-Folds Fibered Over Smooth Complex Surfaces WING-WAH SUNG 577 CONTENTS xv

Degeneration of Kahler-Einstein Manifolds. I GANG TIAN 595

Flat Connections on Products of Determinant Bundles YUELINL. TONG 611

Surfaces Riemanniennes a Singularites Simples MARC TROYANOV 619

Remarks on Certain Higher-Dimensional Quasi-Fuchsian Domains S.-T. YAU AND F. ZHENG 629

ZACohomology: Banach Spaces and Homological Methods on Riemannian Manifolds* STEVEN ZUCKER 637

PART 3

Some Concepts and Methods in Riemannian Geometry* ROBERT E. GREENE 1

Isometric Immersions of Hn into Hn+l KINETSU ABE AND ANDREW HAAS 23

The Distance-Geometry of Riemannian Manifolds with Boundary STEPHANIE B. ALEXANDER, I. DAVID BERG, AND RICHARD L. BISHOP 31

Hypersurfaces and Nonnegative Curvature STEPHANIE B. ALEXANDER AND ROBERT J. CURRIER 37

Shortening Space Curves STEVEN J. ALTSCHULER 45

Degeneration of Metrics with Bounded Curvature and Applications to Critical Metrics of Riemannian Functional* MICHAEL T. ANDERSON 53

The Geometry of Totally Geodesic Hypersurfaces in Hyperbolic Manifolds ARA BASMAJIAN 81

Finiteness of Diffeomorphism Types of Isospectral Manifolds ROBERT BROOKS, PETER PERRY, AND PETER PETERSEN V 89 xvi CONTENTS

Small Eigenvalues of the Laplacian on Negatively Curved Manifolds P. BUSER, B. COLBOIS, AND J. DODZIUK 95

Geometrically Tame Hyperbolic 3-Manifolds RICHARD D. CANARY 99

Isoperimetric Constants and Large Time Heat Diffusion in Riemannian Manifolds ISAAC CHAVEL AND EDGAR A. FELDMAN 111

Large Time Behavior of Solutions of the Heat Equation ISAAC CHAVEL AND LEON KARP 123

Manifolds with 2-Nonnegative Curvature Operator HAIWENCHEN 129

The Geometry of Isospectral Deformations DENNIS DETURCK, HERMAN GLUCK, CAROLYN GORDON, AND DAVID WEBB 135

Quadric Representation of a Submanifold and Spectral Geometry IVKO DIMITRIC 155

Embedded Eigenvalues for Asymptotically Flat Surfaces HAROLD DONNELLY 169

Manifolds of Nonpositive Curvature* PATRICK EBERLEIN, URSULA HAMENSTADT, AND VIKTOR SCHROEDER 179

Topological Rigidity for Compact Nonpositively Curved Manifolds* F. T. FARRELL AND L. E. JONES 229

Almost Nonnegatively Curved Manifolds* KENJI FUKAYA AND TAKAO YAMAGUCHI 275

Local Structure of Framed Manifolds PATRICK GHANAAT 283

Constant Affine Mean Curvature Hypersurfaces of Decomposable Type SALVADOR GIGENA 289

Heat Equation Asymptotics PETER B. GILKEY 317 CONTENTS xvii

Nonnegatively Curved Manifolds Which Are Flat Outside a Compact Set R. E. GREENE AND H. WU 327

Spaces of Nonnegative Curvature* DETLEF GROMOLL 337

Critical Point Theory for Distance Functions* KARSTEN GROVE 357

Spectral Geometry in Higher Ranks: Closed Geodesies and Flat Tori DAVID GURARIE 387

Ellipticity of Local Isometric Embeddings CHONG-KYU HAN 409

Curve Flows on Surfaces and Intersections of Curves JOEL HASS AND PETER SCOTT 415

Curve Shortening, Equivariant Morse Theory, and Closed Geodesies on the 2-Sphere NANCY HINGSTON 423

Morse Theory for Geodesies DIANE KALISH 431

On Some Metrics on S2 x S2* MASATAKE KURANISHI 439

Curve-straightening ANDERS LINNER 451

Ball Covering Property and Nonnegative Ricci Curvature Outside a Compact Set ZHONG-DONG LIU 459

General Conjugate Loci are not Closed CHRISTOPHE M. MARGERIN 465

Collapsing of Riemannian Manifolds and Their Excesses YUKIO OTSU 479

Gromov-Hausdorff Convergence of Metric Spaces* PETER PETERSEN V 489 xviii CONTENTS

Cartan Connections* ERNST A. RUH 505

Eigenvalue Estimates for the Laplacian with Lower Order Terms on a Compact Riemannian Manifold ALBERTO G. SETTI 521

Positive Ricci Curvature on Compact Simply Connected 4-Manifolds JI-PING SHA AND DAGANG YANG 529

Volume Growth and Finite Topological Type ZHONGMIN SHEN AND GUOFANG WEI 539

Recent Developments in Sphere Theorems* KATSUHIRO SHIOHAMA 5 51

On the Excess of Open Manifolds TAKASHI SHIOYA 577

Local Theory of Affine 2-Spheres U. SIMON AND C. P. WANG 585

Riemannian Manifolds with Completely Integrable Geodesic Flows R. J. SPATZIER 599

Differentiate Structure on Spheres and Curvature YOSHIHIKO SUYAMA 609

Spectral Theory for Operator Families on Riemannian Manifolds* Z. I. SZABO 615

Riemannian Foliations and Tautness PHILIPPE TONDEUR 667

Eigenvalue Problems for Manifolds with Singularities JOHAN TYSK 673

Some Rigidity Aspects of Riemannian Fibrations GERARD WALSCHAP 679 CONTENTS xix

Hausdorff Convergence and Sphere Theorems JYH-YANG WU 685

Automorphism Groups and Fundamental Groups of Geometric Manifolds ROBERT J. ZIMMER 693 Preface

The 1990 American Mathematical Society Summer Institute on Differen• tial Geometry took place at University of California, Los Angeles from July 9 to July 27, 1990. This was the largest AMS Summer Institute ever. There were 426 registered participants and 270 lectures. The organizing commit• tee for the Institute consisted of Robert Bryant, Duke University; Eugenio Calabi, University of Pennsylvania; S. Y. Cheng, University of California, Los Angeles; H. Blaine Lawson, State University of New York, Stony Brook; H. Wu, University of California, Berkeley; and as co-chairmen, Robert E. Greene, University of California, Los Angeles, and S. T. Yau, Harvard Uni• versity. In the years since the previous AMS Summer Institute on Differential Geometry in 1973, the field has undergone a remarkable expansion, both in the number of people working in the area and in the number and scope of the topics under investigation. Even in the context of the rapid growth of mathematics as a whole during this period, the growth of geometry is striking. It is our hope that the three volumes of these proceedings, taken as a whole, will provide a broad overview of geometry and its relationship to mathematics in toto, with one obvious exception; the geometry of complex manifolds and the relationship of complex geometry to complex analysis were the subject of one week of the (three-week) 1989 AMS Summer Institute on Several Complex Variables and Complex Geometry. While some topics in complex geometry arose naturally in the 1990 Summer Institute and are covered in these Proceedings, the coverage of this subject in 1989 justified a reduced emphasis in 1990. Thus the reader seeking a complete view of geometry would do well to add the second volume on complex geometry from the 1989 Proceedings to the present three volumes. Each week of the 1990 Summer Institute was given a general emphasis as to subject and the Proceedings volumes are organized in the same way. While overlap is natural, and indeed inevitable, the subjects of the volumes are as follows:

xxi XX11 PREFACE

I. Partial differential equations on manifolds: harmonic functions and mapping, Monge-Ampre equation, differential systems, minimal submani- folds; II. Geometry in mathematical physics and related topics: gauge theory, symplectic geometry, complex geometry, L cohomology, Lorentzian geom• etry; III. Riemannian geometry: curvature and topology, groups and manifolds, dynamical systems in geometry, spectral theory of Riemannian manifolds. The articles in these Proceedings are also of several types. We requested broad-ranging surveys from the people who had given one-hour survey lec• tures at the Institute. These articles are marked with an asterisk in the table of contents. Such surveys were also encouraged from other participants. The remaining articles are either research papers in the usual sense or relatively brief announcements of results. But in these categories as well, we encouraged authors to provide more background information and references to related work than usual. Thus we hope that readers will find the volumes a source of broad perspectives on the rapidly expanding literature of geometry. The editors themselves essayed two efforts in this direction. Volume I be• gins with a problem list by S. T. Yau, successor to his 1980 list [Seminar in Differential Geometry, Annals of Math. Studies, no. 102, Princeton Univer• sity Press]. Volume III begins with an overview by R. E. Greene of some recent trends in Riemannian geometry, in the interests of identifying themes common to the remaining papers in that volume and in outlining certain topics that, as it happened, would not otherwise have been included. An event such as the 1990 AMS Summer Institute involves the efforts of a great many people. We would like to thank all the participants, both for their participation in the Institute and for their prompt and abundant re• sponse to our call for papers for these Proceedings. We are indebted to the staff of the UCLA Mathematics Department for their cooperation during the Summer Institute and to the American Mathematical Society in general and in particular to Mr. Wayne Drady and Ms. Susan Blyth for their invaluable on-the-spot assistance during the Institute. Without the financial support of the National Science Foundation, the Summer Institute could not have had anything like the scope it in fact attained; we are particularly grateful for their willingness to support the participation of advanced graduate students, a willingness that we believe was a valuable investment in the future of ge• ometry. Finally, we are indebted to the publication staff of the American Mathematical Society and particularly to Ms. Alison Buckser, Ms. Donna Harmon, and Ms. Christine Thivierge, whose unfailing patience and helpful• ness made the preparation of these volumes a far easier task than it would have been otherwise. Robert E. Greene S. T. Yau Recent Titles in This Series {Continued from the front of this publication)

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, 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 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 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) 10 Alberto P. Calderdn, editor, Singular integrals (University of Chicago, April 1966) 9 Armand Borel and George D. Mostow, editors, Algebraic groups and discontinuous subgroups (University of Colorado, Boulder, July 1965) 8 Albert Leon Whiteman, editor, Theory of numbers (California Institute of Technology, Pasadena, November 1963) 7 Victor L. Klee, editor, Convexity (University of Washington, Seattle, June 1961) (See the AMS catalog for earlier titles)