Kunihiko Kodaira
Complex Analysis
Description
All three volumes of Kodaira’s classic text on complex analysis are collected together in English for the first time. The author develops the classical theory of functions of a complex variable in a clear and logical manner. Starting from the basics, students are led on to the study of conformal mappings, Riemann’s mapping theorem and analytic functions on a Riemann surface. Caucy’s integral formula is first proved in a topologically simple case from which the author deduces the basic properties of holomorphic functions. In general, the approach taken here emphasises geometrical aspects of the theory in order to avoid some of the topological pitfalls associated with this subject. Profusely illustrated and with many problems and examples, this book should be an ideal text for a course in complex analysis.
Chapter Contents
1. Holomorphic functions; 2. Cauchy’s theorem; 3. Conformal mappings; 4. Analytic continuation; 5. Riemann’s mapping theorem; 6. Riemann surfaces; 7. The structure of Riemann surfaces; 8. Analytic functions on a closed Riemann surface.
ISBN: 0-521-00398-9
Binding: Paperback
ISBN: 0-521-80937-1
Binding: Hardback
Pages: 400
Figures: 160 line diagrams
available from January 2002
I. R. Porteous
Geometric Differentiation, 2nd edition
Description
This is a revised and extended version of the popular first edition. Inspired by the work of Thom and Arnold's on singularity theory, such topics as umbilics, ridges and subparabolic lines, all robust features of a smooth surface, which are rarely treated in elementary courses on differential geometry, are considered here in detail. These features are of immediate relevance in modern areas of application such as interpretation of range data from curved surfaces and the processing of magnetic resonance and cat-scan images. The text is based on extensive teaching at Liverpool University to audiences of advanced undergraduate and beginning postgraduate students in mathematics. However, the wide applicability of this material means that it will also appeal to scientists and engineers from a variety of other disciplines. The author has included many exercises and examples to illustrate the results proved.
Chapter Contents
1. Plane curves; 2. Some elementary geometry; 3. Plane kinetics; 4. The derivatives of a map; 5. Curves on the unit sphere; 6. Space curves; 7. k-times linear forms; 8. Probes; 9. Contact; 10. Surfaces in R3; 11. Ridges and ribs; 12. Umbilics; 13. The parabolic line; 14. Involutes of geodesic foliations; 15. The circles of a surface; 16. Examples of surfaces; 17. Flexicords of surfaces; 18. Duality.
ISBN: 0-521-00264-8
Binding: Paperback
0-521-81040-X Hardback
Pages: 350
Figures: 39 line diagrams 26 colour plates 6 figures
available from January 2002
S. K. Donaldson
Floer Homology Groups in Yang-Mills Theory
Description
The concept of Floer homology has been one of the most striking developments in differential geometry over the past 20 years. It yields rigorously defined invariants which can be viewed as homology groups of infinite-dimensional cycles. The ideas have led to great advances in the areas of low-dimensional topology and symplectic geometry and are intimately related to developments in Quantum Field Theory. The first half of this book gives a thorough account of Floer's construction in the context of gauge theory over 3 and 4-dimensional manifolds. The second half works out some further technical developments of the theory, and the final chapter outlines some research developments for the future - including a discussion of the appearance of modular forms in the theory. The scope of the material in this book means that it will appeal to graduate students as well as those on the frontiers of the subject.
Chapter Contents
1. Introduction; 2. Basic material; 3. Linear analysis; 4. Gauge theory and tubular ends; 5. The Floer homology groups; 6. Floer homology and 4-manifold invariants; 7. Reducible connections and cup products; 8. Further directions.
ISBN: 0-521-80803-0
Binding: Hardback
Pages: 235
available from February 2002
Francois Berteloot, Universite Paul Sabatier (Toulouse III), France,
and Volker Mayer, Universite de Lille I, France
Rudiments de Dynamique Holomorphe
A publication of the Societe Mathematique de France.
Description
This book is an introduction to rational iteration theory. In the first four chapters, the authors deal with the classical theory. The basic properties of the Julia set and its complement, the Fatou set, are presented; the highest points of the treatment are the classification of the components of the Fatou set and Sullivan's non-wandering theorem.
The second part of the book studies several topics in more detail. The authors begin by considering at length two classes of rational maps: the chaotic maps and the hyperbolic maps. In the closing chapters, they include respectively a study of holomorphic families of rational maps with a view to discussing Fatou's famous problem concerning the density of hyperbolic maps and an exposition of the methods of potential theory, touching on questions of ergodicity, which may serve as a preparation for generalizations in higher dimensions.
A number of the developments treated here appear for the first time in book form. Several original proofs are presented.
Contents
Introduction
La dichotomie dynamique de Fatou et Julia
Dynamiques locales et composantes de Fatou
Ensemble de Julia
Classification des composantes de Fatou
Fractions rationnelles chaotiques
Fractions rationnelles hyperboliques
Familles holomorphes de fractions rationnelles
Le point de vu potentialiste
Mesure et dimension de Hausdorff
Applications quasiconformes et structures conformes
Quelques points de theorie du potentiel
Bibliographie
Index
Details:
Publisher: Societe Mathematique de France
Distributor: American Mathematical Society
Series: Cours Specialises--Collection SMF Number: 7
Publication Year: 2001
ISBN: 2-86883-521-X
Paging: 160 pp.
Binding: Softcover
Michele Audin, Universite Louis Pasteur et CNRS, Strasbourg, France
Les Systemes Hamiltoniens et Leur Integrabilite
A publication of the Societe Mathematique de France.
Description
This book presents some modern techniques in the theory of integrable systems viewed as variations on the theme of action-angle coordinates. These techniques include analytical methods coming from the Galois theory of differential equations, as well as more classical algebro-geometric methods related to Lax equations. Many examples are given.
Contents
Introduction
Introduction aux systemes integrables
Variables action-angles
Integrabilite et groupes de Galois
Une introduction aux equations de Lax
Appendix
Ce qu'il faut savoir en theorie de Galois differentielle
Ce qu'il faut savoir sur les courbes algebriques
Bibliographie
Index
Details:
Publisher: Societe Mathematique de France
Distributor: American Mathematical Society
Series: Cours Specialises--Collection SMF, Number: 8
Publication Year: 2001
ISBN: 2-86883-522-8
Paging: 160 pp.
Binding: Softcover