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Geometry and global analysis of Pailneve equations

Pailneve 方程的几何和全局分析

基本信息

批准号：
16340049
负责人：
IWASAKI Katsunori
金额：
$ 5.78万
依托单位：
Kyushu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

项目摘要

Many results have been obtained for Painleve equations, especially for Painleve VI equation and its generalization, Gamier systems, from the viewpoint of algebraic geometry and dynamical system theory. They consist of the establishments of laws of Painleve dynamics mainly based on algebraic geometry, and the elucidations of the global phenomena of Painleve dynamics mainly based on dynamical system theory.More explicitly, the results on the laws of Painleve dynamics include the construction of the phase spaces of Painleve dynamics as the moduli space of stable parabolic connections, the establishment of Riemann-Hilbert correspondence, a characterization of Backlund transformations in terms of the Riemann-Hilbert correspondence, the discovery of an initimate relation between Riccati solutions and singularity theory, an intrinsic introduction of the Hamiltonian structure of Painleve equations, and so on.On the other hand, among the results on the phenomena of Painleve dynamics, it is most … More remarkable that we were able to show that the nonlinear monodromy of the Painleve flow is chaotic along almost all loops in the space of time variable. Namely, the proof of the positivity of the topological entropy, the construction of a maximal-entropy hyperbolic invariant probability measure of saddle type, the establishment of an algorithm of calculating entropy in terms of the reduced word of a given loop and the geometric representation of a universal Coxeter group.These results clearly show that the Painleve equation is in fact a chaotic dynamical system, although it has previously been studied from the viewpoint of integrable systems only. So it is expected that our results would stimulate people to change minds in the future direction of research in the field of Painleve equations.The above-mentioned achievements are the results of many cooperative researches, attendances at various conferences and exchanges of ideas, making the best use of this grant. By virtue of this grant, we were also able to announce or describe the details of our results in various conferences, workshops and other academic meetings, either domestic or overseas.The international conferences on Painleve equations in which the head investigator were invited to give a lecture include Theories asymptotiques et equations de Painleve, Universite d'Angers, France; The Painleve equations and monodromy problems, Isaac Newton Institute, Cambridge University.In summary, the original aims of this project, i.e., to develop an algebraic geometry in order to lay a sound foundation of Painleve equations and to explore the global phenomena of Painleve dynamics, have largely been achieved. A further advances along the line of this project can be expected based on these achievements. Less

从代数几何和动力系统理论的观点出发，对Painleve方程，特别是Painleve VI方程及其推广的Gamier系统，已经取得了许多结果。这些结果包括主要基于代数几何的Painleve动力学定律的建立和主要基于动力系统理论的Painleve动力学全局现象的阐明，更具体地说，Painleve动力学定律的结果包括Painleve动力学相空间作为稳定抛物联络的模空间的构造，Riemann-Hilbert对应的建立，在Painleve动力学现象的研究中，它是最重要的结果之一，也是Painleve动力学现象研究中最重要的结果之一关于我们值得注意的是，我们能够表明，Painleve流的非线性monodromy是混乱的沿着几乎所有的时间变量的空间中的循环。即拓扑熵正性的证明，极大熵双曲不变鞍型概率测度的构造，用给定回路的约化字计算熵的算法的建立，泛Coxeter群的几何表示，这些结果清楚地表明Painleve方程实际上是一个混沌动力系统，尽管先前仅从可积系统的观点对其进行了研究。因此，我们的研究成果有望激发人们对Painleve方程领域未来研究方向的思考。上述成果是充分利用该基金，通过多次合作研究、参加各种会议和交流思想的结果。由于这项资助，我们还能够在国内外的各种会议、研讨会和其他学术会议上宣布或描述我们的结果的细节。Painleve方程和单值问题，艾萨克·牛顿研究所，剑桥大学。总之，这个项目的最初目的，即，发展一种代数几何，以便为Painleve方程打下坚实的基础，并探索Painleve动力学的全球现象，已在很大程度上取得了成就。在这些成果的基础上，可以期待该项目沿着该路线进一步沿着。少

项目成果

期刊论文数量（94）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Topological theory for Selberg type integrals coming from rigid local systems

来自刚性局部系统的 Selberg 型积分的拓扑理论

DOI：
发表时间：
2008
期刊：
影响因子：
0
作者：
原岡喜重;加藤満生;S. Tanabe,;S. Tanabe;田邊晋;原岡喜重;原岡喜重;横山利章;横山利章;S. Tanabe;原岡喜重;Y. Haraoka,;Yoshishige Haraoka;横山利章;横山利章;原岡喜重;下村俊;原岡喜重;下村俊;下村俊;下村俊;原岡喜重
通讯作者：
原岡喜重

Hierarchy of Backlund transformation groups of the Painleve systems

Painleve系统的Backlund变换群的层次结构