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Painlevé III: A Case Study in the Geometry of Meromorphic Connections

Martin Guest
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      The purpose of this monograph is two-fold:  it introduces a conceptual language for the geometrical objects underlying Painlevé equations,  and it offers new results on a particular Painlevé III equation of type  PIII (D6), called PIII (0, 0, 4, −4), describing its relation to isomonodromic families of vector bundles on P1  with meromorphic connections.  This equation is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears widely in geometry and physics.   It is used here as a very concrete and classical illustration of the modern theory of vector bundles with meromorphic connections. Complex multi-valued solutions on C* are the natural context for most of the monograph, but in the last four chapters real solutions on R>0 (with or without singularities) are addressed.  These provide examples of variations of TERP structures, which are related to  tt∗ geometry and harmonic bundles.    As an application, a new global picture o0 is given.
      Format: Paperback / softback CONTRIBUTORS: Martin Guest EAN: 9783319665252 COUNTRY: Switzerland PAGES: WEIGHT: 3343 g HEIGHT: 235 cm
      PUBLISHED BY: Springer International Publishing AG DATE PUBLISHED: 2017-10-15 CITY: GENRE: MATHEMATICS / Differential Equations / General, MATHEMATICS / Geometry / Algebraic, MATHEMATICS / Mathematical Analysis WIDTH: 155 cm SPINE:

      Book Themes:

      Complex analysis, complex variables, Functional analysis and transforms, Differential calculus and equations, Algebraic geometry

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      The purpose of this monograph is two-fold:  it introduces a conceptual language for the geometrical objects underlying Painlevé equations,  and it offers new results on a particular Painlevé III equation of type  PIII (D6), called PIII (0, 0, 4, −4), describing its relation to isomonodromic families of vector bundles on P1  with meromorphic connections.  This equation is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears widely in geometry and physics.   It is used here as a very concrete and classical illustration of the modern theory of vector bundles with meromorphic connections. Complex multi-valued solutions on C* are the natural context for most of the monograph, but in the last four chapters real solutions on R>0 (with or without singularities) are addressed.  These provide examples of variations of TERP structures, which are related to  tt∗ geometry and harmonic bundles.    As an application, a new global picture o0 is given.
      Format: Paperback / softback CONTRIBUTORS: Martin Guest EAN: 9783319665252 COUNTRY: Switzerland PAGES: WEIGHT: 3343 g HEIGHT: 235 cm
      PUBLISHED BY: Springer International Publishing AG DATE PUBLISHED: 2017-10-15 CITY: GENRE: MATHEMATICS / Differential Equations / General, MATHEMATICS / Geometry / Algebraic, MATHEMATICS / Mathematical Analysis WIDTH: 155 cm SPINE:

      Book Themes:

      Complex analysis, complex variables, Functional analysis and transforms, Differential calculus and equations, Algebraic geometry

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