Geometry and global analysis of Pailneve equations

从代数几何和动力系统理论的角度对Painleve方程，特别是Painleve VI方程及其推广Gamier系统得到了许多结果。它们包括主要基于代数几何的疼痛级动力学规律的建立，以及主要基于动力系统理论的疼痛级动力学全局现象的阐释。更明确地说，Painleve动力学定律的结果包括Painleve动力学相空间作为稳定抛物连接的模空间的构造，Riemann-Hilbert对应的建立，根据Riemann-Hilbert对应的Backlund变换的表征，Riccati解与奇点理论之间的初始关系的发现，Painleve方程的哈密顿结构的内在介绍，等等......。另一方面，在关于Painleve动力学现象的结果中，最值得注意的是，我们能够证明Painleve流的非线性单态在时间变量空间中几乎所有环路上都是混沌的。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，建立了用给定环的约化词计算熵的算法，并给出了普遍Coxeter群的几何表示。这些结果清楚地表明，尽管以前只从可积系统的角度对Painleve方程进行了研究，但它实际上是一个混沌动力系统。因此，我们的研究结果有望激发人们对Painleve方程未来研究方向的改变。上述成果是通过多次合作研究、参加各种会议和交流思想的结果，充分利用了这笔资助。凭借这笔资金，我们还能够在国内外各种会议、研讨会和其他学术会议上公布或描述我们的成果细节。邀请首席研究员在Painleve方程国际会议上发表演讲的会议包括：法国昂热大学的Painleve方程的渐近理论；Painleve方程与单一性问题，剑桥大学牛顿研究所。综上所述，该项目的最初目标，即发展代数几何，以奠定painlevel方程的良好基础，并探索painlevel动力学的全球现象，已基本实现。在这些成就的基础上，可以期望沿着这个项目的路线取得进一步的进展。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