Rough paths on manifolds 3

Youness Boutaib

The general aim of this course is to present ways of translating

notions from classical analysis (understood to be built for and

adapted to the Euclidean setting) to manifolds without adding too

much structure on the manifolds trying to mimic the same

constructions. We use the theory of rough paths as an example to

show these ideas (but also for its own merit as the predecessor

for the ideas behind the currently much celebrated regularity

structures). During the first lecture, we will briefly review the

main ideas behind the theory of (geometric) rough paths (as

introduced by Terry Lyons) that will enable us to achieve our goal

in the following two lectures. Time allowing, these ideas will

include:

* The p-variation as a way of measuring the smoothness/regularity

of paths;

* Stieltjes and Young's integration theories and their limits;

* Signatures of moderately irregular paths;

* Geometric rough paths and rough differential equations.

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