The Painlev Property One Century Later [Hardcover]

1 Singularities of Ordinary Linear Differential Equations and Integrability.- 1 Generalities.- 2 Structure of the Solutions of the Homogeneous Equation Around an Isolated Singular Point.- 3 Weakly Singular Equations (Fuchs).- 4 Thom?s Equations.- 5 Global Considerations.- 6 References.- 2 Introduction to the Theory of Isomonodromic Deformations of Linear Ordinary Differential Equations with Rational Coefficients.- 1 Introduction.- 2 Isomonodromic Deformations of Linear ODEs with Puchsian Singularities.- 3 The Isomonodromic Deformation Problem for Painlev? VI.- 4 Isomonodromic Deformations of Linear ODEs with Thom? Singularities.- 5 An Isomonodromic Deformation Problem for Painlev? I.- 6 Conclusion.- 7 Appendix A: Matrix Versus Scalar Formalisms and Fuchss Theorem.- 8 Appendix B: Asymptotic Power Series.- 9 References.- 3 The Painlev? Approach to Nonlinear Ordinary Differential Equations.- 1 Introduction.- 2 The Meromorphy Assumption.- 3 The True Problems.- 4 The Classical Results (L. Fuchs, Poincar?, Painlev?).- 5 Construction of Necessary Conditions. The Theory.- 6 Construction of Necessary Conditions. The Painlev? Test.- 7 Sufficiency: Explicit Integration Methods.- 8 Conclusion.- 9 References.- 4 Asymptotic Studies of the Painlev? Equations.- 1 Introduction.- 2 Linear and Nonlinear Asymptotic Models.- 3 The First and Second Painlev? Equations.- 4 Global Extensions.- 5 Conclusion.- 6 References.- 5 2-D Quantum and Topological Gravities, Matrix Models, and Integrable Differential Systems.- A 2-D Quantum Gravity.- 1 Introduction.- 2 The One-Matrix Model: Large N Limit.- 3 The One-Matrix Model: Exact Solution.- 4 The Double-Scaling Limit.- 5 Multimatrix Models.- 6 Conclusion.- B 2-D Topological Gravity.- 7 Introduction.- 8 Computing the Kontsevich Integral.- 9 The Kontsevich Integral as T-Function of the KdV Hierarchy.- 10 Main Equivalence Theorem Between Topological and Quantum Gravities.- 11 Conclusion.- 12 References.- 6 Painlev? Transcendents in Two-DimensionallÓÙ