Different Faces of Geometry

429 Pages·2004·7.51 MB·English

by  Simon Donaldson

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
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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
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eBookISBN: 0-306-48658-X
Print ISBN: 0-306-48657-1
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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
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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
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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]
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