Geometry and global analysis of Pailneve equations

点击翻译按钮获取中文摘要

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