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Isomonodromic Deformations and Frobenius Manifolds

An Introduction

  • Textbook
  • © 2008

Overview

Authors:
  1. Claude Sabbah

    1. CNRS Centre de Mathématiques Laurent Schwartz, École Polytechnique, France

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  • First book to cover this material at a level that is accessible to graduate students and young researchers
  • Includes complete proofs, and applications to recent research questions

Part of the book series: Universitext (UTX)

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Table of contents (8 chapters)

  1. Front Matter

    Pages I-XIV
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  2. The language of fibre bundles

    • Claude Sabbah
    Pages 1-59

  3. Holomorphic vector bundles on the Riemann sphere

    • Claude Sabbah
    Pages 61-82

  4. The Riemann-Hilbert correspondence on a Riemann surface

    • Claude Sabbah
    Pages 83-120

  5. Lattices

    • Claude Sabbah
    Pages 121-143

  6. The Riemann-Hilbert problem and Birkhoff’s problem

    • Claude Sabbah
    Pages 145-166

  7. Fourier-Laplace duality

    • Claude Sabbah
    Pages 167-189

  8. Integrable deformations of bundles with connection on the Riemann sphere

    • Claude Sabbah
    Pages 191-221

  9. Saito structures and Frobenius structures on a complex analytic manifold

    • Claude Sabbah
    Pages 223-261

  10. Back Matter

    Pages 263-279
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Keywords

About this book

Based on a series of graduate lectures, this book provides an introduction to algebraic geometric methods in the theory of complex linear differential equations. Starting from basic notions in complex algebraic geometry, it develops some of the classical problems of linear differential equations and ends with applications to recent research questions related to mirror symmetry.

The fundamental tool used within the book is that of a vector bundle with connection. There is a detailed analysis of the singularities of such objects and of their deformations, and coverage of the techniques used in the resolution of the Riemann-Hilbert problem and Birkhoff’s problem. An approach to Frobenius manifolds using isomonodromic deformations of linear differential equations is also developed.

Aimed at graduate students and researchers, the book assumes some familiarity with basic complex algebraic geometry.

Authors and Affiliations

  • CNRS Centre de Mathématiques Laurent Schwartz, École Polytechnique, France

    Claude Sabbah

Bibliographic Information

  • Book Title: Isomonodromic Deformations and Frobenius Manifolds

  • Book Subtitle: An Introduction

  • Authors: Claude Sabbah

  • Series Title: Universitext

  • DOI: https://doi.org/10.1007/978-1-84800-054-4

  • Publisher: Springer London

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag London 2008

  • Softcover ISBN: 978-1-84800-053-7Published: 25 January 2008

  • eBook ISBN: 978-1-84800-054-4Published: 20 December 2007

  • Series ISSN: 0172-5939

  • Series E-ISSN: 2191-6675

  • Edition Number: 1

  • Number of Pages: XIV, 279

  • Number of Illustrations: 10 b/w illustrations

  • Additional Information: Original French edition published by EDP Sciences, CNRS Editions, France, 2007

  • Topics: Algebraic Geometry, Mathematics, general, Algebra

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