Different Faces of Geometry PDF
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Different Faces of Geometry INTERNATIONAL MATHEMATICAL SERIES Series Editor: Tamara Rozhkovskaya Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia NONLINEAR PROBLEMS IN MATHEMATICAL PHYSICS AND RELATED TOPICS I: In Honor of Professor O. A. Ladyzhenskaya Edited by M. Sh. Birman, S. Hildebrandt, V. A. Solonnikov, N. N. Uraltseva NONLINEAR PROBLEMS IN MATHEMATICAL PHYSICS AND RELATED TOPICS II: In Honor of Professor O. A. Ladyzhenskaya Edited by M. Sh. Birman, S. Hildebrandt, V. A. Solonnikov, N. N. Uraltseva DIFFERENT FACES OF GEOMETRY Edited by Simon Donaldson, Yakov Eliashberg, and Mikhael Gromov A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher. Different Faces of Geometry Edited by Simon Donaldson Imperial College of London London, UK Yakov Eliashberg Stanford University Stanford, California and Mikhael Gromov Institut des Haute Etudes Scientifiques Bures-sur-Yvette, France KLUWER ACADEMIC PUBLISHERS NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW eBookISBN: 0-306-48658-X Print ISBN: 0-306-48657-1 ©2004 Springer Science + Business Media, Inc. Print ©2004 Kluwer Academic/Plenum Publishers New York All rights reserved No part of this eBook maybe reproducedor transmitted inanyform or byanymeans,electronic, mechanical, recording, or otherwise,withoutwritten consent from the Publisher Createdin the UnitedStates of America Visit Springer's eBookstore at: http://www.ebooks.kluweronline.com and the Springer Global Website Online at: http://www.springeronline.com Preface Modern information technology allows most mathematicians unprece- dented access to the research literature. Published papers, reviews and preprints can be obtained in a few seconds. This mountain of informa- tion makes the need for articles which illuminate and survey important developments all the greater. Our volume brings together articles by leading experts on a variety of different topics which are the scene of exciting current activity. “Geometry” is famously hard to pin down, and means many different things to different mathematicians: it is probably best interpreted as a way of thought rather than a collection of specific subject areas-there is, perhaps, no branch of mathematics which cannot be considered a part of geometry, when approached in the right spirit. Certainly we have not set out to cover all the main topics which would normally be denoted as geometry (for example, there is relatively little in the volume on algebraic geometry, nor on the interface between partial differential equations and Riemannian geometry); but we hope and believe that these articles will give a valuable picture of some major areas. One can distinguish various themes running through the different contributions. There is some emphasis on invariants defined by elliptic equations and their applications in low-dimensional topology, symplectic and contact geometry (Bauer, Seidel, Ozsváth and Szabó). These ideas enter, more tangentially, in the articles of Joyce, Honda and LeBrun. Here and elsewhere, as well as explaining the rapid advances that have v vi been made, the articles convey a wonderful sense of the vast areas lying beyond our current understanding. Simpson’s article emphasizes the need for interesting new construc- tions (in that case of Kähler and algebraic manifolds), a point which is also made by Bauer in the context of 4-manifolds and the “11/8 conjec- ture.” LeBrun’s article gives another perspective on 4-manifold theory, via Riemannian geometry, and the challenging open questions involving the geometry of even “well-known” 4-manifolds. There are also striking contrasts between the articles. The authors have taken different approaches: for example, the thoughtful essay of Simpson, the new research results of LeBrun and the thorough exposi- tionswith homework problems of Honda. One can also ponder the differences in the style of mathematics. In the articles of Honda, Giannopoulos and Milman, and Mikhalkin, the “geometry” is present in a very vivid and tangible way; combining re- spectively with topology, analysis and algebra. The papers of Bauer and Seidel, on the other hand, makes the point that algebraic and algebro- topological abstraction (triangulated categories, spectra) can play an important role in very unexpected ways in concrete geometric problems. Finally, we wish to thank all the authors for their splendid contri- butions and express the hope that the reader will find as much interest and excitement in the articles as we ourselves have done. Simon Donaldson Yakov Eliashberg Mikhael Gromov London–Stanford–Paris March 2004 Main Topics Amoebas and Tropical Geometry Convex Geometry and Asymptotic Geometric Analysis Differential Topology of 4-Manifolds 3-Dimensional Contact Geometry Floer Homology and Low-Dimensional Topology Kähler Geometry Lagrangian and Special Lagrangian Submanifolds Refined Seiberg–Witten Invariants This page intentionally left blank Editors Simon Donaldson (UK) Department of Mathematics Imperial College of London Huxley Building, 180 Queen’s Gate London SW7 2BT, UK [email protected] Yakov Eliashberg (USA) Department of Mathematics Stanford University Stanford, CA 94305-2125, USA [email protected] Mikhael Gromov (France) Institut des Hautes Etudes Scientifiques 35, Route de Chartres F-91440 Bures-sur-Yvette, France [email protected] ix
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