Abstract: The famous Furstenberg-Bergelson-Leibman conjecture says that the ergodic averages converge pointwise almost everywhere provided that the underlying measure preserving transformations generate a nilpotent group. The purpose of this talk is to discuss recent progress towards solving this conjecture. The talk will be based on joint work with Alexandru Ionescu, Akos Magyar and Mariusz Mirek

Abstract: In this talk we will discuss recent partial regularity results for a very weak notion of ‘solutions’ (u,d,p) to an extended (when the liquid crystal ‘director’ field d is non-trivial) Navier-Stokes system, based essentially only on the local energy inequality; in particular, the key vector-valued equations need not be satisfied even in a weak sense. This notion of `solution’ was introduced by V. Scheffer (1985) in the context of the Navier-Stokes partial regularity result of Caffarelli-Kohn-Nirenberg (1982), to identify the key properties required for such a proof. Similar results for the extended system were more recently achieved by F.H. Lin and C. Liu (1996) when d is bounded, which is a natural assumption (and such solutions are known to exist) when the vector equations themselves are satisfied. Our key focus is on treating the case when d is unbounded. We obtain partial regularity results in terms of both parabolic Hausdorff dimension (cf. [CKN82,LL96]) as well as the parabolic fractal dimension, the latter extending a recent result of Q. Liu. (2018).

Abstract: Most of the recent progress on the restriction problem in harmonic analysis has relied on Guth's polynomial partitioning method, which upgrades restriction estimates from low algebraic dimensions to higher ones. This talk presents a method for upgrading restriction estimates from low fractal dimensions to higher ones, and shows how the method can be used to prove estimates of the Mizohata-Takeuchi type in the plane.

Abstract: In this talk, we shall discuss recent efforts in finding a characterization of regularity of a set by the Green function. Our goal is to obtain a characterization with minimal background assumptions on the domain and the operator. We show that for an optimal class of elliptic operators with non-smooth coefficients on a domain that provides some access to its boundary, the boundary of the domain is uniformly rectifiable if and only if the Green function behaves like a distance function to the boundary. This talk is based on joint work with J. Feneuil and S. Mayboroda.