POWRE: Applications of Recent Advances in Exponential Asymptotics

在过去的十年里，由于理论和实际的兴趣，人们开发了新的和强大的工具来理解奇异点附近微分算子的局部性质。 拟议的研究使用最近的技术和指数渐近的结果，以解决问题的分类常微分方程在奇异区域，以及偏微分算子的分析性质。 其中所讨论的问题有：（1a）找到一个常微分方程的线性部分具有多个正则奇点且是齐次的，与其线性部分解析等价的充分必要条件（在包含奇点的域中）;（1b）找到非线性方程与线性方程解析等价的必要条件（非齐次）部分在一个秩为1的不规则奇点的邻域中;（2）找到“平方和”型偏微分算子解析亚椭圆性的充要条件，特别关注Treves猜想。这个POWRE项目由MPS多学科活动办公室（OMA）和数学科学部（DMS）联合支持。

Motivated by theoretical as well as major practical interests, new and powerful tools have been developed in the last decade for understanding local properties of differential operators near singularities. The proposed research uses recent techniques and results of exponential asymptotics to address questions in the classification of ordinary differential equations in singular regions, as well as analytic properties of partial differential operators. Among the questions addressed are: (1a) finding necessary and sufficient criteria for an ordinary differential equation whose linear part has several regular singular points and is homogeneous, to be analytically equivalent to its linear part (in a domain which contains the singular points); (1b) finding necessary criteria for a nonlinear equation to be analytically equivalent to its linear (inhomogeneous) part in a neighborhood of one irregular singular point of rank 1; (2) finding necessary andsufficient conditions for analytic hypoellipticity of partial differential operators of type ``sum of squares'', with special focus on Treves' conjecture.This POWRE project is jointly supported by the MPS Office of Multidisciplinary Activities (OMA) and the Division of Mathematical Sciences (DMS).