Oxford

Using a technique explored in unpublished work of Ball and Mizel I shall

show that already in 2 and 3 dimensions there are vectorfields which are

singular minimizers of integral functionals whose integrand is strictly

polyconvex and depends on the gradient of the map only. The analysis behind

these results gives rise to an interesting question about the relationship

between the regularity of a polyconvex function and that of its possible

convex representatives. I shall indicate why this question is interesting in

the context of the regularity results above and I shall answer it in certain

cases.

After a brief introduction to the basics of Rough Paths I'll

explain recent work by Peter Friz, Dan Stroock and myself proving that a

Brownian path conditioned to be uniformly close to a given smooth path

converges in distribution to that path in the Rough Path metric. The Stroock

Varadhan support theorem is an immediate consequence.

The novel part of the argument is to

obtain the estimate in a way that is independent of the particular norm used

in the Euclidean space when one defines the uniform norm on path space.