Viscous incompressible fluids are described by the Navier-Stokes system. For fluids with small viscosity in the vicinity of a solid obstacle, Ludwig Prandtl predicted that a boundary layer (i.e. a zone of small width in which the velocity of the fluid changes abruptly) is created close to the boundary of the obstacle. He also derived a system of equations for the fluid within the boundary layer. This system has been a topic of intense research in the past decades. In this talk, we will focus on the stationary setting, and we will make a review of the existing results and of some open problems. I will also give some insight of the proofs.

Inverse problems are ubiquitous in nature, engineering and our daily lives. A prototypical nonlinear inverse problem is the Calderon problem which arises in electrical impedance tomography and in which one seeks to recover an unknown conductivity within a conducting body from voltage-to-current measurements at its surface. Major open questions concern the uniqueness, stability and reconstruction of the conductivity for irregular or anisotropic conductivities as well as settings in which only partial data are given. In this talk I will introduce this inverse problem and illustrate how nonlocality and associated uncertainty principles allow to solve some of these questions in the setting of the fractional Calderon problem, a nonlocal analogue of this problem.

I will describe several natural questions that arise in connection with the sequence of best approximation vectors. This sequence, which I will introduce in the talk, is a higher dimensional generalization of continued fraction convergents (which I will also discuss briefly). The questions involve the statistical behavior of certain observables, and the means to understand them involve a mix of number theory, ergodic theory, and elementary geometry. The talk will be based on joint work with Uri Shapira.