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The central theme of this volume is the contemporary mathematics of geometry and physics, but the work also discusses the problem of the secondary structure of proteins, and an overview of arc complexes with proposed applications to macromolecular folding is given. "Woods Hole has played such a vital role in both my mathematical and personal life that it is a great pleasure to see the mathematical tradition of the 1964 meeting resurrected forty years later and, as this volume shows, resurrected with new vigor and hopefully on a regular basis. I therefore consider it a signal honor to have been asked to introduce this volume with a few reminiscences of that meeting forty years ago." Introduction by R Bott (Wolf Prize Winner, 2000).
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Woods Hole Mathematics Perspectives in Mathematics and Physics K(XE Series on Knots and Everything — Vol. 34 Woods Hole Mathematics Perspectives in Mathematics and Physics editors Nils Tongring UJoods Hole Oceanographic Institution, USfl R C Penner University of Southern California, USfl \fc World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Woods Hole mathematics : perspectives in mathematics and physics / editors, Nils Tongring, R. C. Penner. p. cm. - (K & E series on knots and everything ; v. 34) Includes bibliographical references. ISBN 9812560211 (alk. paper) 1. Mathematical physics. 2. Operads. 3. Quantum algebra. 4. Differential equations. 5. Quantum physics. 6. Moduli theory. I. Tongring, Nils. II. Penner, R. C , 1956- III. Title. IV. Series. QC20 .W665 2005 53O'.15-dc22 2004059531 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore by World Scientific Printers (S) Pte Ltd V PREFACE The following eight papers are by mathematicians and physicists connected in some way to the three meetings held at the Woods Hole Oceanographic Institution (WHOI) during the past six years. This book is not a proceedings in the usual sense, but expositions of recent work contributed by those researchers interested in this form, and written with no pressure from editors. The topics of the meetings were suggested by the interests of the Sullivan Seminar at the City University of New York Graduate Center from diverse areas of mathematics and physics, but related certainly in concepts and techniques. One of the strands of this mix may have started in Woods Hole forty years ago (cf. the reminiscences of Raoul Bott in the Introduction to this book). The first meeting (1998) at Woods Hole was on fluid flow: the perennial problems of the Euler and Navier-Stokes equations and turbulence. A main theme was alternatives to the continuum model as a way of overcoming issues of regularity, the combinatorial difficulty of completing the evolution model when shock waves develop, and the limits