In this talk, I will discuss results obtained with Yvan Dynnikov, Paul Mercat, Olga Paris-Romaskevich and Sasha Skripchenko. Novikov's conjecture for foliations states that the restriction of a linear foliation to a triply periodic surface is generically periodic or integrable (leaves stay at bounded distance from a line). I will explain some results on families of interval exchange transformations with flips. This approach gives a partial solution to the conjecture in a non-trivial case.[-]

This talk is focused on the Radon-Carleman Problem, dealing with computing and/or estimating the essential norm and/or the Fredholm radius of singular integral operators of double layer type associated with elliptic partial dfferential operators, on function spaces naturally intervening in the formulation of boundary value problems for the said operator in a given domain. The main goal is to monitor how the geometry of the domain affects the complexity of this type of study and to present a series of results in increasingly more irregular settings, culminating with that of uniformly rectifiable domains.
This is based on joint work with Dorina Mitrea and Marius Mitrea from Baylor University, which has recently appeared in volume V of our Geometric Harmonic Analysis research monograph series in Developments in Mathematics, Springer.[-]