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
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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
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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
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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
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ISBN 978 - 0 - 8218 - 5067 - 1
9 780821 850671
CONM/64