POWRE: Applications of Recent Advances in Exponential Asymptotics

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

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 and sufficient 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).