Lie groups and manifolds of non-smooth mappings

We propose to construct and study Lie groups of paths of bounded variation with values in a Lie group. The driving motivation for this construction comes from the theory of rough paths. Rough path theory was invented by T. Lyons in the 90s and provides a fresh look at integration and differential equations driven by rough signals. A rough path can be completely characterised by its signature, which is a series of tensors satisfying certain algebraic constraints. Associating rough paths of a fixed p-variation to their signatures, one ends up with a subgroup of the group of all signatures. To understand the geometry of these groups, we propose to construct (infinite-dimensional) Lie groups of p-variation paths with values in a Lie group (in this setting, the pointwise product of the paths yields the group structure). First studies indicate that these groups provide a proper framework to exploit geometric properties of the signature. Further, the groups envisaged here are also of interest in infinite-dimensional Lie theory and are related to the open problem of J. Milnor whether each infinite-dimensional Lie group is regular. Our second (related aim) is the development of exponential laws for mappings of mixed differentiability. In recent years functions on a product U x V of manifolds which are up to m times differentiable in the second variable and up to n times differentiable in the first variable. For suitable topological spaces X, Y, and Z, an exponential law C(X x Y,Z) ~ C(X, C(Y,Z)) is well known for continuous functions, with analogous results for smooth mappings (under suitable hypotheses). Our goal is to establish a natural smooth manifold structure on manifolds of mappings of mixed differentiability together with a version of the exponential law. When the target is a Lie group, we strive to obtain a Lie group structure. The new constructions will enhance the toolbox of infinite-dimensional calculus for non-smooth objects, and the scope of its applications.