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Isomonodromic Deformations and Frobenius Manifolds : an Introduction.

Author: Claude Sabbah
Publisher: London : Springer London, 2008.
Series: Universitext.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
The notion of a Frobenius structure on a complex analytic manifold appeared at the end of the seventies in the theory of singularities of holomorphic functions. Motivated by physical considerations, further development of the theory has opened new perspectives on, and revealed new links between, many apparently unrelated areas of mathematics and physics. Based on a series of graduate lectures, this book provides an  Read more...
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Details

Genre/Form: Electronic books
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Claude Sabbah
ISBN: 1848000537 9781848000537
OCLC Number: 990412618
Language Note: French.
Description: 1 online resource (289 pages).
Contents: The language of fibre bundles --
Holomorphic vector bundles on the Riemann sphere --
The Riemann-Hilbert correspondence on a Riemann surface --
Lattices --
The Riemann-Hilbert problem and Birkhoff's problem --
Fourier-Laplace duality --
Integrable deformations of bundles with connection on the Riemann sphere --
Saito structures and Frobenius structures on a complex analytic manifold.
Series Title: Universitext.
Other Titles: Universitext

Abstract:

The notion of a Frobenius structure on a complex analytic manifold appeared at the end of the seventies in the theory of singularities of holomorphic functions. Motivated by physical considerations, further development of the theory has opened new perspectives on, and revealed new links between, many apparently unrelated areas of mathematics and physics. 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.

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