POWRE: Applications of Recent Advances in Exponential Asymptotics

在理论和主要实践兴趣的推动下，在过去十年中已经开发出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。本研究利用指数渐近的最新技术和结果来解决奇异区域中常微分方程的分类问题，以及偏微分算子的解析性质。所讨论的问题包括：（1a）寻找一个常微分方程的线性部分有几个正则奇点且齐次的必要和充分准则，以解析等价于它的线性部分（在包含奇异点的域内）；（1b）在一个秩为1的不规则奇点的邻域中，找到一个非线性方程解析等价于它的线性（非齐次）部分的必要准则；(2)找到“平方和”型偏微分算子解析亚椭圆性的充分必要条件，重点研究了Treves猜想。该power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).