May 28 - 31, 2015
In the first part of these lectures I will present certain, by now classic, results on local and global well-posedness for the NLS via Strichartz estimates.
In the second part I will show how one can use randomization of the initial data to prove well-posedness almost surely even when the problem lack enough regularity for a more deterministic approach. In this context I will also introduce certain Gibbs type measures and how their invariance can be used in certain cases to extend local
solutions to a global ones.
In the 1920's Besicovitch studied linearly measurable sets in the plane, that is sets with locally finite "length". The basic question he addressed was whether the infinitesimal properties of the "length" of a set E in the plane yield geometric information on E itself. This simple question marks the beginning of the study of the geometry of measures and the associated field known as Geometric Measure Theory (GMT).
In this series of lectures we will present some of the main results in the area concerning the regularity of the support of a measure in terms of the behavior of its density or in terms of its tangent structure. We will discuss applications to PDEs, free boundary regularity problem and harmonic analysis. The aim is that the mini-course will be self contained.