Regularity structures on manifolds

Over the past few years, significant progress has been made in understanding the behaviour and construction of solutions to very singular stochastic partial differential equations. This has lead to a much better understanding of the emergence of mesoscopic / macroscopic physical laws from a variety of microscopic systems and has shed light on the universality of these laws. Unfortunately, besides a few specific examples, current techniques are mostly restricted to the case in which the ambient space is flat. The goal of this project is to attempt to extend them to curved spaces. Such an extension would be a significant feat as it is not at all clear how to define even the most basic objects of the theory in this case. One particularly interesting case is when the underlying manifold is a symmetric space on which some Lie group acts in a transitive way. In this case, one would expect in principle to be able to get a natural theory of renormalisation provided that the equations under consideration satisfy a suitable invariance under the group action.

Research area: mathematical analysis