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摘要翻译：
这基本上是在Hitchin的60岁生日会议上发表的一份调查报告（题为“Painleve,Klein和二十面体”）的文本。讨论了第六类Painleve微分方程的代数解的寻找和构造，该方程可以看作是Gauss超几何方程的非线性类比。代数和超越方法都被使用，故事涉及仿射Weyl群，编织群和三次曲面。重点讨论了第六个Painleve方程作为最简单的非阿贝尔Gauss-Manin联系的显式形式的解释，即作为一个“来自几何”的非线性微分方程，就像Picard-Fuchs方程在具有阿贝尔系数的上同调情况下所产生的一样。
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英文标题：
《Towards a nonlinear Schwarz's list》
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作者：
Philip Boalch
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Classical Analysis and ODEs 经典分析与颂歌
分类描述：Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics
特殊函数、正交多项式、调和分析、Ode、微分关系、变分法、逼近、展开、渐近
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Physics 物理学
二级分类：Exactly Solvable and Integrable Systems 精确可解可积系统
分类描述：Exactly solvable systems, integrable PDEs, integrable ODEs, Painleve analysis, integrable discrete maps, solvable lattice models, integrable quantum systems
精确可解系统，可积偏微分方程，可积偏微分方程，Painleve分析，可积离散映射，可解格模型，可积量子系统
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英文摘要：
This is basically the text of a survey talk (entitled 'Painleve, Klein and the icosahedron') given at Hitchin's 60th birthday conference. It discusses the search for and construction of algebraic solutions of the sixth Painleve differential equation, which may be viewed as a nonlinear analogue of the Gauss hypergeometric equation. Both algebraic and transcendental methods are used and the story involves affine Weyl groups, braid groups and cubic surfaces. Some emphasis is given to the interpretation of the sixth Painleve equation as the explicit form of the simplest nonabelian Gauss-Manin connection, i.e. as a nonlinear differential equation 'coming from geometry', much as Picard-Fuchs equations arise in the case of cohomology with abelian coefficients.
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PDF链接：
https://arxiv.org/pdf/0707.3375
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