Geometric Methods in the Analytic Theory of Differential Equations

微分方程理论是物理学和工程学的基本数学工具。很少有微分方程可以显式求解，近似数值解并不总是能给出精确解的本质特征。几个世纪以来，数学家们一直在研究一类在应用中经常出现的简单微分方程；它们的解被称为数学物理的特殊函数，在科学上有广泛的应用。这个项目的总体目标是扩展这类很容易理解的方程。在物理学、控制理论和材料科学应用方面的早期工作的基础上，该项目旨在应用各种最近开发的方法来研究具有内在数学兴趣的长期问题。这项工作预计将进一步提高对数学物理和几何中出现的一些基本微分方程所定义的解析函数的定性特征的理解。数学物理中的大多数特殊函数都是由至多有三个奇点的线性微分方程定义的。Heun方程（有四个正则奇点）和painlevevi方程（非线性，有四个固定奇点，没有可动奇点）的解位于一类特殊函数的边界上。由于其固有的数学兴趣和在科学中的大量应用，自20世纪初以来，它们得到了广泛的研究。该研究项目旨在通过使用新的几何方法来促进对这些重要函数的理解。本课题主要研究了painlevevi方程实解的定性研究，具有二次奇异性的常正曲率黎曼度量的研究，重点研究了与Heun方程密切相关的具有四个奇异性的黎曼度量，最后研究了一些奇偶时对称非谐振子的特征值。

Theory of differential equations is a fundamental mathematical tool of physics and engineering. Few differential equations can be solved explicitly, and approximate numerical solutions do not always give essential features of the behavior of exact solutions. A class of simple differential equations that frequently occur in applications has been studied by mathematicians for centuries; their solutions are called special functions of mathematical physics, and they are widely used in science. The general goal of this project is to extend this class of well-understood equations. Building on earlier work relevant for applications to physics, control theory, and materials science, this project aims to apply a variety of recently-developed methods to study longstanding questions of intrinsic mathematical interest. The work is anticipated to further improve understanding of the qualitative features of analytic functions defined by some basic differential equations arising in mathematical physics and geometry.Most of the special functions of mathematical physics are defined by linear differential equations with at most three singularities. Solutions of the Heun equation (with four regular singularities) and the Painlevé VI equation (non-linear, with four fixed singularities and no movable singularities) lie on the boundary of the class of special functions. Because of their intrinsic mathematical interest and numerous applications in science, they have been extensively studied since the beginning of the 20th century. This research project aims to advance understanding of these important functions through the use of new geometric methods. The main topics of this project are the qualitative study of real solutions of the Painlevé VI equation, the study of Riemannian metrics of constant positive curvature with conic singularities, with the emphasis on the metrics with four singularities closely related to the Heun equation, and finally the study of the eigenvalues of some parity-time symmetric anharmonic oscillators.