The theory of rough paths was developed by Terry Lyons in the 1990s. Rough paths, along with their "branched" version, developed by Massimiliano Gubinelli a few years later, are the right tools to give meaning to certain (stochastic) differential equations of low regularity in affine space, and to solve them. It's natural to ask what happens if affine space is replaced by any differentiable manifold. Such a degree of generality seems out of reach at present. However, the case of a homogeneous space under the action of a Lie group (abelian or not) can be approached via a "planar" version of rough paths. Homogeneous spaces are non-linear spaces such as e.g. spheres and groups of rotations, which play a central role in geometric formulations of mechanics, control theory and mathematical physics. In the present project, we first propose to develop the theory of planar branched rough paths providing a satisfactory algebraic and analytical framework for the resolution of (stochastic) differential equations of low regularity over a homogeneous space. The central idea is to link branched rough paths with Hans Munthe-Kaas's theory of Lie-Butcher series from numerical analysis. A second, more ambitious part of the project aims at studying a large class of partial differential equations in the same framework. In affine space, the notion of a branched rough path must be replaced by the notion of regularity structure, introduced recently by Martin Hairer, and further developed together with Yvain Bruned, Ajay Chandra and Lorenzo Zambotti. It requires a nontrivial algebraic combinatorial framework, which involves Hopf algebras and bialgebras of rooted trees and forests. We believe that similar structures can be developed in the case of a homogeneous space, using planar trees and forests. The French and Norwegian teams consist of experts in stochastic and numerical analysis, and algebraic combinatorics, as well as strong PhD and postdoctoral students working in these areas.