Analysis and PDEs on uniformly rectifiable sets

Over the past few decades uniformly rectifiability emerged as a natural geometric condition, necessary and sufficient for classical estimates in harmonic analysis, boundedness of singular integrals in L^2, and, in the presence of some background topological assumptions, for suitable scale invariant estimates on harmonic functions closely related to the solvability of the Dirichlet problem. In the first lecture will discuss the state of the art in the case of co-dimension one. The second one will concentrate on the new analogue of harmonic measure, recent results, and many remaining mysteries for sets of higher co-dimension