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Birkhäuser

A Course on Holomorphic Discs

  • Textbook
  • © 2023

Overview

Authors:
  1. Hansjörg Geiges

    1. Mathematisches Institut, Universität zu Köln, Köln, Germany

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  2. Kai Zehmisch

    1. Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany

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    You can also search for this author in PubMed   Google Scholar

  • Provides a self-contained, comprehensive introduction to the theory of pseudoholomorphic curves
  • Utilizes the proof of a basic theorem to motivate study of advanced topics in symplectic geometry
  • Suitable for use in a graduate course or for independent study

Part of the book series: Birkhäuser Advanced Texts Basler Lehrbücher (BAT)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xviii
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  2. Gromov’s Nonsqueezing Theorem

    • Hansjörg Geiges, Kai Zehmisch
    Pages 1-39

  3. Compactness

    • Hansjörg Geiges, Kai Zehmisch
    Pages 41-64

  4. Bounds of Higher Order

    • Hansjörg Geiges, Kai Zehmisch
    Pages 65-88

  5. Elliptic Regularity

    • Hansjörg Geiges, Kai Zehmisch
    Pages 89-123

  6. Transversality

    • Hansjörg Geiges, Kai Zehmisch
    Pages 125-174

  7. Back Matter

    Pages 175-189
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Keywords

About this book

This textbook, based on a one-semester course taught several times by the authors, provides a self-contained, comprehensive yet concise introduction to the theory of pseudoholomorphic curves. Gromov’s nonsqueezing theorem in symplectic topology is taken as a motivating example, and a complete proof using pseudoholomorphic discs is presented. A sketch of the proof is discussed in the first chapter, with succeeding chapters guiding the reader through the details of the mathematical methods required to establish compactness, regularity, and transversality results. Concrete examples illustrate many of the more complicated concepts, and well over 100 exercises are distributed throughout the text. This approach helps the reader to gain a thorough understanding of the powerful analytical tools needed for the study of more advanced topics in symplectic topology.


This text can be used as the basis for a graduate course, and it is also immensely suitable for independentstudy. Prerequisites include complex analysis, differential topology, and basic linear functional analysis; no prior knowledge of symplectic geometry is assumed.


This book is also part of the Virtual Series on Symplectic Geometry.

Authors and Affiliations

  • Mathematisches Institut, Universität zu Köln, Köln, Germany

    Hansjörg Geiges

  • Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany

    Kai Zehmisch

About the authors

Hansjörg Geiges, born in Basel in 1966, is Professor of Mathematics at the Universität zu Köln. He received his PhD from the University of Cambridge in 1992. Before moving to his current position in 2002, he taught at Stanford University, ETH Zürich, and the Universiteit Leiden. His other published books are An Introduction to Contact Topology and The Geometry of Celestial Mechanics.

Kai Zehmisch, born in Leipzig in 1975, is Professor of Mathematics at the Ruhr-Universitat Bochum. He obtained his doctorate at the Universität Leipzig in 2009, after having lived through the failure of the second socialist experiment on German soil, and was rewarded nonetheless with a book prize on 100 Jahre Mathematisches Seminar der Karl-Marx-Universität Leipzig. Previous to his current position, he taught at the Westfälische Wilhems-Universität Münster (as it was then called) and at the Justus-Liebig-Universität Giessen.


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