Fractal boundary phenomena of harmonic functions

Abstract: The classical Herglotz-Poisson formula asserts that every positive harmonic function in a smooth domain is represented as the integral of the normal derivative of the (Dirichlet) Green function with respect a measure on the boundary. This formula was extended by Martin to a general (even fractal) domain by introducing an ideal boundary, referred to as the Martin boundary, given by the completion of the ratios of Green functions. The identification of the Martin boundary is an interesting problem and has attracted many attentions. The boundary behavior of the ratios of Green functions, or more generally, the ratios of two positive harmonic functions vanishing on a portion of boundary, is crucial. It is often called the boundary Harnack principle. In this talk, I will explain the background of the boundary Harnack principle and recent results about elliptic and parabolic boundary Harnack principles as well as a Harnack principle with exceptional sets.

Perelman, the Ricci flow and the Poincare' conjecture.

We will present an overview of the Ricci flow and of its geometric applications. This will be a panoramic talk aimed at non specialists.

An overview of the curve shortening flow.

Abstract: This talk concerns the curve shortening flow (CSF), which is a geometric flow that evolves planar curves according to its curvature. I will first introduce the necessary geometric background of plane curve geometry. Then I will introduce the CSF and its properties. We shall see some similarities between the CSF and the Ricci flow. At the end we will arrive at a great result, Grayson's theorem, which states that an initially embedded curve will shrink to a round point under the CSF.

On the growth of Sobolev norms for the non linear Schrodinger equation

Abstract: We will discuss the influence of the geometry of the spatial domain on the possible growth of the Sobolev norms under the flow of the cubic defocusing Schrodinger equation.

Existence and stability of stading waves to NLS with a partial confinement in the L^2 supercritical regime.

Abstract: I will present a joint work with Bellazzini, Boussaid,
Jeanjean. We show that the energy associated with NLS with a partial confinement has a local minima structure
when restricted on fixed charge constraint. By using a suitable adaptation of the concentration compactness argument we deduce the existence and stablitiy of standing waves.