Geometry and global analysis of Pailneve equations

从代数几何和动力系统理论的角度研究Painleve方程，特别是Painleve VI方程及其推广的Gamier系统，已取得了许多结果。Painleve动力学定律的建立主要包括基于代数几何的Painleve动力学定律的建立，以及Painleve动力学整体现象的阐明。更明确地说，Painleve动力学定律的结果包括Painleve动力学相空间的构造作为稳定抛物联络的模空间，Riemann-Hilbert对应的建立，Backlund变换的Riemann-Hilbert对应的刻画，Riccati解与奇性理论之间的密切关系的发现，Painleve方程的哈密顿结构的内在介绍等等这是最…的更值得注意的是，我们能够证明Painleve流的非线性单程在时间变量空间中几乎所有回路上都是混沌的。即证明了拓扑熵的正性，构造了鞍型最大熵双曲不变概率测度，建立了利用给定回路的约化字数计算熵的算法，给出了泛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，尽管以前只从可积系统的角度来研究它.上述成果是我们多次合作研究、出席各种会议和交流意见的结果，充分利用了这笔赠款。通过这笔赠款，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。关于Painleve方程的国际会议，其中首席研究员应邀发表演讲，包括法国昂格斯大学Painleve方程和方程的理论渐近；剑桥大学艾萨克·牛顿研究所的Painleve方程和单行问题。总而言之，这个项目的最初目标，即发展代数几何，以便为Painleve方程奠定坚实的基础，并探索Painleve动力学的全球现象，已经基本实现。在这些成就的基础上，可以预期这一项目将取得进一步的进展。较少

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