The research focuses on certain aspects of functional and harmonic analysis together with their applications to differential-geometrical problems. In particular, several classes of differential and integral operators on Euclidean spaces and Lie groups are studied. Concerning Riemannian manifolds and Lie groups, we consider singular integral operators and a host of other related issues on spinor spaces. Another facet to be investigated concerns the existence and regularity of solutions to linear and non-linear differential equations, especially dispersive ones, and the motion of wave packets in phase space (microlocal analysis). Deformation issues on surfaces and integrable motions of curves are considered as well, with particular attention to the relationship with hierarchies of evolution equations. Finally, we examine multi-scale methods (wavelets, reproducing groups) and study time-frequency analysis (Gabor frames, time-frequency filters), with an eye to the characterization of functional spaces and the PDE discretization.