Explorations des chemins rugueux – ECRU

After a decade of efforts, the basis of the rough path theory, initiated by T. Lyons, is now established on a solid ground for ordinary differential equations driven by an irregular signal. The time has thus come to explore new perspectives in the area, and we propose here a certain number of projects in this promising direction: definition of rough systems in infinite dimension, numerical approximations of rough differential equations and their concrete implementation, application of the rough path theory to dispersive deterministic PDEs, rough paths indexed by trees, to cite the most representative examples. We also wish to progress in the definition of concrete examples of rough paths, such as in the case of fractional Brownian motion with Hurst index lesser than 1/4 or other irregular processes. It should be noticed that all those projects lie at the edge of different areas of mathematics, ranging from PDE theory to Hopf algebras or complex analysis.