Geometric Aspects of Painlevé Equations

Abstract

In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on ${\mathbb{P}}^1\times {\mathbb{P}}^1$ and classified according to the degeration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeomtric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

168 pages. Some errors found in the published version are corrected

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• ABJM Matrix Model and 2D Toda Lattice Hierarchy
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• Different Hamiltonians for differential Painlevé equations and their identification using a geometric approach
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• Discrete Painleve equation, Miwa variables, and string equation in 5d matrix models
• Full-parameter discrete Painlevé systems from non-translational Cremona isometries
• Duality Cascades and Parallelotopes
• Geometric approach for the identification of Hamiltonian systems of quasi-Painlevé type
• Elliptic Painlevé equations from next-nearest-neighbor translations on the $E_8^{(1)}$ lattice
• Geometric description of discrete power function associated with the sixth Painlevé equation
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• Discrete Painlevé equations from singularity patterns: the asymmetric trihomographic case
• Special solutions of a discrete Painlevé equation for quantum minimal surfaces
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• Fairing of Discrete Planar Curves by Integrable Discrete Analogue of Euler's Elasticae
• Hamiltonian reductions in Matrix Painlevé systems
• Involutions of Halphen Pencils of Index 2 and Discrete Integrable Systems
• Orthogonal Polynomials for the Gaussian Weight with a Jump and Discrete Painlevé Equations
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• Singularity confinement and proliferation of tau functions for a general differential-difference Sawada-Kotera equation
• Discrete equations from Bäcklund transformations of the fifth Painlevé equation
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• Solving the Quispel-Roberts-Thompson maps using Kajiwara-Noumi-Yamada's representation of elliptic curves