The Painlevé Property: One Century Later

'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 Thomé'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 Painlevé VI.- 4 Isomonodromic Deformations of Linear ODEs with Thomé Singularities.- 5 An Isomonodromic Deformation Problem for Painlevé I.- 6 Conclusion.- 7 Appendix A: Matrix Versus Scalar Formalisms and Fuchs's Theorem.- 8 Appendix B: Asymptotic Power Series.- 9 References.- 3 The Painlevé Approach to Nonlinear Ordinary Differential Equations.- 1 Introduction.- 2 The Meromorphy Assumption.- 3 The True Problems.- 4 The Classical Results (L. Fuchs, Poincaré, Painlevé).- 5 Construction of Necessary Conditions. The Theory.- 6 Construction of Necessary Conditions. The Painlevé Test.- 7 Sufficiency: Explicit Integration Methods.- 8 Conclusion.- 9 References.- 4 Asymptotic Studies of the Painlevé Equations.- 1 Introduction.- 2 Linear and Nonlinear Asymptotic Models.- 3 The First and Second Painlevé 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 Painlevé Transcendents in Two-Dimensional Topological Field Theory.- 1 Algebraic Properties of Correlators in 2-D Topological Field Theories. Moduli of a 2-D TFT and WDW Equations of Associativity.- 2 Equations of Associativity and Probenius Manifolds. Deformed Flat Connection and Its Monodromy at the Origin.- 3 Semisimplicity and Canonical Coordinates.- 4 Stokes Matrices and Classification of Semisimple Probenius Manifolds.- 5 Monodromy Group and Mirror Construction for Semisimple Frobenius Manifolds.- 6 References.- 7 Discrete Painlevé Equations.- 1 Integrable Discrete Systems.- 2 Similarity Reduction and Direct Linearization.- 3 The Painlevé Property for Discrete Systems.- 4 Properties of the Discrete Painlevé Equations.- 5 Monodromy Problems and q-Difference Equations.- 6 References.- 8 Painlevé Analysis for Nonlinear Partial Differential Equations 517.- 1 Introduction.- 2 Integrable Equations.- 3 Painlevé Analysis for PDEs.- 4 Partially Integrable and Nonintegrable Equations.- 5 References.- 9 On Painlevé and Darboux-Halphen-Type Equations.- 1 Introduction.- 2 Painlevé Equations and 1ST.- 3 Darboux-Halphen Systems and Their Linear Problems as Reductions of SDYM.- 4 The Monodromy Evolving System and the Solution of the Generalized DH System.- 5 Discussion.- 6 References.- 10 Symmetry Reduction and Exact Solutions of Nonlinear Partial Differential Equations.- 1 Introduction.- 2 Algorithm for Calculating the Symmetry Group of a Differential System.- 3 Examples of Symmetry Groups.- 4 Symmetry Reduction, Group Invariant Solutions, Partially Invariant Solutions.- 5 Classification of the Subalgebras of a Finite-Dimensional Lie Algebra.- 6 Direct Reductions and Conditional Symmetries.- 7 Conclusions.- 8 References.- 11 Painlevé Equations in Terms of Entire Functions.- 1 Introduction.- 2 Hirota's Bilinear Method for Soliton Equations.- 3 Bilinear Forms and Similarity Reduction.- 4 Solutions in Terms of Entire Fu