Pseudo-differential operators on homogeneous manifolds

Pseudo-differential operators can be considered as a natural extension of linear partial differential operators. The powerful modern theory of pseudo-differential operators grew out of research on singular integral operators with roots entwined deep down to solving differential equations. The main aim of this challenging and ambitious project is to develop the symbolic calculus and study regularity properties for pseudo-differential operators on homogeneous manifolds, with applications to partial differential equations (PDEs). We will concentrate on the analysis in two related settings: compact homogeneous manifolds and symmetric spaces of non-compact type. Many advantages of this project will include the development of very new research methodology applicable to the desired problems of symbolic calculus on homogeneous manifolds, to regularity of pseudo-differential operators, and their applications to linear and nonlinear evolution PDEs. The research program proposed here splits into three interacting work packages of varying scientific difficulty and risk: WP1: Pseudo-differential operators on compact homogeneous manifolds WP2: Pseudo-differential operators on non-compact homogeneous manifolds (symmetric spaces) WP3: Applications to partial-differential equations (PDEs)