The Legacy of Sonya Kovalevskaya Proceedings of a Symposium Sponsored by The Association for Women in Mathematics and The Mary Ingraham Bunting Institute held October 25-28, 1985

AMERICAn MATHEMATICAL SOCIETY VOLUME 6G

Titles in This Series Volume

1 Markov random fields and their 18 Fixed points and nonexpansive applications. Ross Kindermann and mappings, Robert C. Sine. Editor J. Laurie Snell 19 Proceedings of the Northwestern

2 Proceedings of the conference homotopy theory conference. on Integration, topology, and Haynes R. Miller and Stewart B. geometry In linear spaces, Priddy, Editors William H. Graves. Editor 20 Low dimensional topology,

3 The closed graph and P-closed Samuel J. Lomonaco. Jr .. Editor graph properties In general

21 Topological methods in nonlinear topology, T. R. Hamlett and functional analysis. S. P. Singh, L. L. Herrington S. Thomeier. and B. Watson, Editors

4 Problems of elastic stability and 22 Factorizations of b" ± 1, b = vibrations. Vadim Komkov. Editor

2, 3, 5, 6, 7.10,11, 12 up to high 5 Rational constructions of powers, John Brillhart. D. H. Lehmer. modules for simple Lie algebras. J. L. Selfridge, Bryant Tuckerman. and George B. Seligman S. S. Wagstaff. Jr. 6 Umbral calculus and Hopf algebras,

23 Chapter 9 of Ramanujan's second Robert Morris. Editor notebook-Infinite series identities.

7 Complex contour integral transformations. and evaluations, representation of cardinal spline Bruce C. Berndt and Padmini T. Joshi functions. Walter Schempp

24 Central extensions. Galois groups, 8 Ordered fields and real algebraic and ideal class groups of number geometry, D. W. Dubois and fields. A. Frohlich

T. Recio. Editors 25 Value distribution theory and its 9 Papers In algebra, analysis and

applications. Chung-Chun Yang. statistics, R. Lidl. Editor Editor 10 Operator algebras and K-theory,

26 Conference in modern analysis Ronald G. Douglas and Claude and probability, Richard Beals. Schochet. Editors Anatole Beck. Alexandra Bellow. and 11 Plane ellipticity and related Arshag Hajian. Editors

problems, Robert P. Gilbert. Editor 27 Microlocal analysis, M. Salah 12 Symposium on algebraic topology in Baouendi. Richard Beals. and honor of Jose Adem, Samuel Gitler. Linda Preiss Rothschild. Editors Editor

Fluids and plasmas: geometry and 13 Algebraists' homage: Papers in 28 ring theory and related topics, dynamics, Jerrold E. Marsden. Editor S. A. Amitsur. D. J. Saltman. and 29 Automated theorem proving, G. B. Seligman. Editors W. W. Bledsoe and Donald Loveland.

14 Lectures on Nielsen fixed point Editors theory, Boju Jiang 30 Mathematical applications of

15 Advanced analytic number theory. category theory, J. W. Gray, Editor Part 1: Ramification theoretic 31 Axiomatic set theory, James E. methods. Carlos J. Moreno Baumgartner. Donald A. Martin. and

16 Complex representations of Saharon Shelah. Editors GL(2, K) for finite fields K. 32 Proceedings of the conference llya Piatetski-Shapiro on Banach algebras and several

17 Nonlinear partial differential complex variables. F. Greenleaf and equations. Joel A. Smoller. Editor D. Gulick. Editors

http://dx.doi.org/10.1090/conm/064

Titles in This Series Volume 33 Contributions to group theory,

Kenneth I. Appel. John G. Ratcliffe. and Paul E. Schupp, Editors

34 Combinatorics and algebra, Curtis Greene. Editor

35 Four-manifold theory, Cameron Gordon and Robion Kirby. Editors

36 Group actions on manifolds, Reinhard Schultz. Editor

37 Conference on algebraic topology in honor of Peter Hilton, Renzo Piccinini and Denis Sjerve. Editors

38 Topics in complex analysis. Dorothy Browne Shaffer. Editor

39 Errett Bishop: Reflections on him and his research. Murray Rosenblatt. Editor

40 Integral bases for affine Lie algebras and their universal enveloping algebras, David Mitzman

41 Particle systems. random media and large deviations. Richard Durrett. Editor

42 Classical real analysis, Daniel Waterman. Editor

43 Group actions on rings, Susan Montgomery. Editor

44 Combinatorial methods in topology and algebraic geometry, John R. Harper and Richard Mandelbaum. Editors

45 Finite groups-coming of age, John McKay, Editor

46 Structure of the standard modules for the affine Lie algebra A~ 1 ), James Lepowsky and Mirko Prime

47 Linear algebra and its role in systems theory, Richard A. Brualdi. David H. Carlson. Biswa Nath Datta. Charles R. Johnson. and Robert J. Plemmons. Editors

48 Analytic functions of one complex variable, Chung-chun Yang and Chi-tai Chuang, Editors

49 Complex differential geometry and nonlinear differential equations. Yum-Tong Siu. Editor

50 Random matrices and their applications, Joel E. Cohen. Harry Kesten. and Charles M. Newman. Editors

51 Nonlinear problems in geometry, Dennis M. DeTurck. Editor

52 Geometry of normed linear spaces, R. G. Bartle. N. T. Peck. A. L. Peressini. and J. J. Uhl. Editors

53 The Selberg trace formula and related topics. Dennis A. Hejhal. Peter Sarnak. and Audrey Anne Terras. Editors

54 Differential analysis and infinite dimensional spaces, Kondagunta Sundaresan and Srinivasa Swaminathan. Editors

55 Applications of algebraic K-theory to algebraic geometry and number theory, Spencer J. Bloch. R. Keith Dennis. Eric M. Friedlander. and Michael R. Stein. Editors

56 Multiparameter bifurcation theory, Martin Golubitsky and John Guckenheimer. Editors

57 Combinatorics and ordered sets. Ivan Rival. Editor

58 The Lefschetz centennial conference. Part 1: Proceedings on algebraic geometry, D. Sundararaman. Editor

59 Function estimates. J. S. Marron. Editor

60 Nonstrictly hyperbolic conservation laws. Barbara Lee Keyfitz and Herbert C. Kranzer. Editors

61 Residues and traces of differential forms via Hochschild homology, Joseph Lipman

62 Operator algebras and mathematical physics. Palle E. T. Jorgensen and Paul S. Muhly, Editors

63 Integral geometry, Robert L. Bryant. Victor Guillemin. Sigurdur Helgason. and R. 0. Wells. Jr .. Editors

64 The Legacy of Sonya Kovalevskaya, Linda Keen. Editor

The Legacy of Sonya Kovalevskaya

I Volume64

The Legacy of Sonya Kovalevskaya Proceedings of a Symposium Sponsored by The Association for Women in Mathematics and The Mary Ingraham Bunting Institute held October 25-28, 1985

Linda Keen1 Editor

AMERICAn MATHEMATICAL SOCIETY Prouldence • Rhode Island

EDITORIAL BOARD Irwin Kra.

managing editor Thomas F. Banchoff M. Salah Baouendi

W. H. Jaco Gerald J. Janusz Jan Mycielski Johannes C. C. Nitsche

Alan D. Weinstein

The Symposium in Honor of Sonya Kovalevskaya, held at Radcliffe College. October 27-28. 1985. sponsored by the Association for Women in Mathematics and the Mary Ingraham Bunting Institute. and three Special Sessions at the American Mathematical Society meeting in Amherst, October 25-27, 1985, have been combined in this book. Partial support was provided by the National Science Foundation. Grant number 405-669-3000.

1980 Mathematics Subject Classification (1985 Revision). Primary 01A70.10G05. 14H40. 35B40, 35K, 35L. 35L65, 35L70. 43A60. 49F22. 53A10, 53A55. 53B25. 53C42, 57S15. 58E20. 58F. 58G25. 65F. 73F. 82A60. 82A60, 82A65.

Library of Congress Cataloging-In-Publication Data

(Contemporary mathematics. ISSN 0271-4132: v. 64) 1. Mathematics-Congresses. 2. Kovalevskaia. S. V. (Sofia Vasil1evna).

185G-1891-Congresses. 3. Women in mathematics-Congresses. I. Kovalevskaia. S. V. (Sofia Vasil' evna). 185G-1891. II. Keen. Linda. Ill. Association for Women in Mathematics (U.S.) IV. Radcliffe College. Mary Ingraham Bunting Institute. V. Series: Contemporary mathematics (American Mathematical Society): v. 64. QA1.L38 1986 510 86-32032 ISBN 0-8218-5067-9

Copying and reprinting. Individual readers of this publication. and nonprofit libraries 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 pub-lication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director. American Mathematical Society. P.O. Box 6248, Providence. Rhode Island 02940.

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 .. 21 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 @1987 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted

to the United States Government. Printed in the United States of America.

This volume was printed directly from author-prepared copy. The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability.§

TABLE OF CONTENTS

Preface

List of Contributors

Section I. Kovalevskaya - her life and work

Sofia Kovaleskaia -- A biographical sketch by ANN HIBNER KOBLITZ

Sonya Kovalevskaya's place in nineteenth century mathematics

by ROGER COOKE

Changing views of Sofia Kovalevskaia by ANN HIBNER KOBLITZ

Section 2. Geometry and dynamical systems

Isospectral metrics and finite Riemannian coverings

ix

xiii

3

17

53

by DENNIS M. DETURCK and CAROLYN S. GORDON 79

Estimating small Eigenvalues of Riemann surfaces by JOZEF DODZIUK, THEA PIGNATARO,

BUR TON RANDOL, and DENNIS SULLIVAN 93

A conformal analog of Bernstein's Theorem for timelike surfaces in Minkowski 3-space

by TILLA KLOTZ MILNOR 123

Geometry of canonical forms by RICHARDS. PALAIS AND CHUU-LIAN TERNG 133

Flows on r-gonal Jacobians by EMMA PREVIA TO 153

Some linearly induced Morse-Smale systems, the QR algorithm and the Toda lattice

by MICHAEL SHUB and ALPHONSE THOMAS VASQUEZ 181

vii

viii TABLE OF CONTENTS

Some geometric developments related to Kowaleski's work by CHUU-LIAN TERNG 195

Section 3. Analysis

Large-time behavior of solutions to a scalar conservation Ia w in several space dimensions

by PATRICIA BAUMAN 209

Strong solutions for strongly damped quasilinear wave equations

by HANS ENGLER 219

Section 4. Applied mathematics

Quasicrystals, tilings, and algebraic number theory: some preliminary connections

by ENRICO BOMBIERI and JEAN E. TAYLOR 241

An introduction to quasicrystals by JOHN W. CAHN and JEAN E. TAYLOR 265

Geometric analysis in crystalline media by JEAN TAYLOR 287

PREFACE

The academic year 1985-86 marked the fifteenth anniversary of the Association for Women in Mathematics and the twenty-fifth anniversary of the Mary Ingraham Bunting Institute of Radcliffe College, Harvard University. Both these organizations were founded to support and encourage the contributions of women in scholarly and creative fields and both have enhanced the role of women in rna the rna tics.

The AWM was founded m 1970-71 by a group of women mathematicians who had begun to talk openly about the problems faced by women in the mathematical professions. In the period since its founding, the relative number of women in these professions has doubled. What is more important, is that women are now regularly invited to speak at conferences, to serve as editors of journals and to participate in the organizational structures of the various professional societies. The A WM has a membership of 1600 women and men, and has a European chapter.

The Bunting Institute was founded in 1960 as the Radcliffe Institute for Independent Study. It has provided opportunities for gifted women from a variety of fields to advance their careers by making available to them the time, space and support necessary to complete a major research project. Over the years, nineteen women mathematicians have been affiliated with the Institute.

This dual occasion provided an apt framework in which to celebrate the mathematical accomplishments of Sonya Kovalevskaya, who did her most important work just about a century ago in the mid-1880's and to present the work of mathematicians who are now working in related fields. Sonya Kovaleskaya was a distinguished mathematician who was considered among the best of her generation by her contemporaries. Her work, approach to mathematics, and ideas are still relevant

ix

X PREFACE

today. She was particularly interested in mathematical problems which originate from physical phenomena. She worked on questions involving the existence and uniqueness of solutions of differential equations and studied the quantitative and qualitative properties of these solutions. She won the prestigious Bordin Prize for her work on the rotation of a solid body about a fixed point --the Kovalevskaya top. She was the first woman in modern times to hold a professorship at a major university and to serve as an editor of an important mathematical journal.

Kovalevskaya was also an outspoken feminist. She was among the first group of Russian women who were determined to obtain the education unavailable to them in Russia and went abroad to study science. Later, she helped organize the women's university in St. Petersburg and sponsored the careers of young women in the sciences. This was a time of great social change, and Kovalevskaya led a very active intellectual and political life in addition to her mathematical life. Toward the end of her life, she published her childhood memoirs, a play, several essays on political and social themes and some poetry.

There were three parts to the Kovalevskaya celebration. On October 27-28, 1985, there was a symposium held at Radcliffe College to honor Kovalevskaya. There were nine speakers who spoke about Kovalevskaya, her work and their own work. At the meeting of the American Mathematical Society m Amherst, Massachusetts, on October 25-27, 1985 there were three special sessions organized in conjunction with the Kovalevskaya Symposium. At the Bunting Institute, on October 26, 1985, there was a program for high school students and teachers in the Boston area whose focus was the development of educational tools in mathematical education.

This volume is a collection of papers which were presented either at the Kovalevskaya Symposium at Radcliffe or at one of the special sessions of the AMS meeting. Some individuals gave more than one paper at these various sessions. The papers are grouped by subject, independently or where they were presented.

PREFACE xi

There is no general agreement on the spelling of Kovalevskaya's name in English; each author has used his/her own preference. The first group of papers contains background material. There is a biographical sketch and a discussion of Kovalevskaya as she has been perceived by the mathematical community over the last century (both by Ann Koblitz). The third paper in this group is a study of those works of Kovalevskaya which earned her her status as an eminent mathematician in her own time (by Roger Cooke). The rest of the papers contain new mathematics. They cover a wide variety of subjects which involve differential equations in one form or another, they reflect the interrelationships of various fields of mathematics and they study mathematical questions arising from physical phenomena. They are broadly grouped into categories of geometry, analysis and applied mathematics.

I would like to take this opportunity to thank the organizing committee for the symposium, and in particular, Bernice Auslander and Pamela Coxon, for making the Symposium at Radcliffe a success. I would also like to thank the organizers of the special sessions at the AMS meeting, Jane Cronin Scanlon, Lesley Sibner and Jean Taylor, the staff at the AMS in Providence for their help in the production of this volume and the National Science Foundation for financial support.

Linda Keen

LIST OF CONTRIBUTORS

Patricia Bauman, Purdue University Enrico Bombieri, Institute for Advanced Study John W. Cahn, Center for Materials Science, National Bureau of

Standards Roger Cooke, University of Vermont Dennis Deturck, University of Pennsylvania J ozef Dodzuik, Queens College, CUNY Hans Engler, Georgetown University Carolyn Gordon, University of Pennsylvania Ann Hibner Koblitz, University of Washington Tilla Klotz Milnor, Rutgers University Richard Palais, Brandeis University Thea Pignataro, Columbia University Emma Previato, Boston University Burton Randol, Graduate Center, CUNY Michael Shub, IBM T. J. Watson Research Labs. Dennis Sullivan, Queens College and Graduate Center, CUNY Jean Taylor, Rutgers University Chuu-lian Terng, Northeastern University Alphonse T. Vasquez, Graduate Center, CUNY

Editor: Linda Keen, Lehman College, CUNY

xiii

ISBN 978 - 0 - 8218 - 5067 - 1

9 780821 850671

CONM/64