Covering Surfaces and Painleve Equations

建议：DMS-9970281首席调查员：Aimo Hinkkanen摘要：该项目有两个主要目标。第一个是发展一种更通用和更灵活的Ahlfors覆盖曲面理论，该理论适用于可变区域。这将特别强调在复杂动力学和共享价值问题上的应用。Ahlfors理论的思想是利用相对边界的长度，为不是真覆盖映射的映射找到Riemann-Hurwitz公式的一种修正形式。遗憾的是，目前该理论的结论只适用于固定的基域。欣卡宁计划考虑在适当意义上可变的领域，纳入从特定应用中产生的假设。该项目的第二个目标是解决Painleve微分方程式。其目的是创造新的方法，在适当的环境下建立这种解的单值性并研究它们的增长。复分析是与这一提议相关的数学的一般领域，它为数学的几个应用领域提供了技术和概念基础，包括许多在物理和工程中的应用。具体的例子包括电子工程，以及流体和气体运动的二维研究(由于对称性，许多三维问题都归结为这些问题)。作为一种非常特殊的情况，复杂动力学包含对由二次多项式所模拟的动力学行为的研究，二次多项式直接或通过逼近面作为研究物理、生物和经济学中众多动力系统的基本构件。这个项目所涉及的具体工作有望解释如何安排区域的几何形状的可能方法可以对某些复杂动力系统的行为产生重大影响。此外，这项研究还研究了一类著名的微分方程解的整体存在性和增长性，Painleve方程在许多领域都有应用，其中最突出的是微分几何和数学物理。在这样的联系中，特别值得注意的是最近发现了弦理论中的Painleve方程的用处，弦理论是物理学的一个分支，寻求在比当今量子物理学更基本的层面上解释空间和物质的结构。

Proposal: DMS-9970281Principal Investigator: Aimo Hinkkanen Abstract: This project has two main objectives. The first is to develop a more general and flexible form of the Ahlfors theory of covering surfaces, one that applies to variable domains. This will be done with particular emphasis on applications to complex dynamics and shared-value problems. The idea of the Ahlfors theory is to find a modified form of the Riemann-Hurwitz formula for a map which is not a proper covering map onto its image, using the length of the relative boundary. Unfortunately, the conclusions of this theory as presently formulated are only valid for fixed base domains. Hinkkanen plans to consider domains that are variable in a suitable sense, incorporating assumptions that arise from specific applications. The project's second objective deals with solutions to the Painleve differential equations. The aim is to create new methods to establish, in a proper setting, the single-valuedness of such solutions and to investigate their growth.Complex analysis, which is the general area of mathematics related to this proposal, provides technical and conceptual foundations for several fields of applications of mathematics, including many in physics and engineering. Specific examples include electrical engineering, and the two-dimensional study of the motion of fluids and gases (to which many three-dimensional problems reduce, due to symmetries). Complex dynamics contains, as a very special case, the study of the dynamical behavior modeled by quadratic polynomials, which both directly and via approximation surface as fundamental building blocks in the investigation of numerous dynamical systems in physics, biology, and economics. The specific work involved in this project is expected to explain how the possible ways of arranging the geometric shapes of regions can have a significant impact on the behavior of certain complex dynamical systems. In addition, this research addresses global existence and growth properties of solutions to a well-known class of differential equations, the Painleve equations, which have applications in a wide number of areas, differential geometry and mathematical physics prominent among them. Particularly notable among such connections is the recent discovery of the usefulness of Painleve equations in string theory, a branch of physics that seeks to explain the structure of space and matter at a more fundamental level than present-day quantum physics.