The Degasperis-Procesi equation (DP) is a spatial one-dimensional model for nonlinear shallow waters phenomena and it is one of the few known Hamiltonian PDEs which is completely integrable, namely it possesses infinitely many constants of motion. Moreover this equation is quasi-linear, namely the nonlinear terms contain derivatives of the same order of the linear part. In this talk I will show a recent result of existence and stability of small amplitude quasi-periodic solutions for Hamiltonian perturbations of the DP equation on the circle. This result is based on a combination of Nash-Moser / KAM schemes and pseudo differential calculus techniques. There are several issues in dealing with this problem:

We consider the problem of inversion of weighted Radon transforms. This problem arises in different tomographies and, in particular, in emission tomographies. We present old and very recent results on this problem. This talk is based, in particular, on recent works [Goncharov, Novikov, 2016, 2018], [Goncharov, 2017].

Nonlinear Schrodinger equation (NLS) is in the following form: $$i\frac{du}{dt}=-\Delta_x u + |u|^2u,$$, where $x$ lies in the torus $\mathbb{T}^d$, and $t\in \mathbb{R}$. We are going to study the behavior of the solution $u(t,x)$ ( corresponding to initial value $u(0,x)$). By applying Birkhoff normal forms, we see that in the one dimensional context, all solutions are linear stable. However, in higher dimensions , the answer is not that simple. I will introduce some recent important results, and explain the main idea in each case.

I will describe an approach to studying the Kakeya maximal function in high dimensions via the Guth--Katz polynomial partitioning method. Although the approach does not currently produce better bounds than the record set by Katz--Tao, it is rather flexible, provides a lot of interesting structural information and gives rise to some interesting algebraic/geometric problems.

Motivated by some classical results of Meyer, Pichorides and Zygmund, we present a variant of Yano's extrapolation theorem for analytic Hardy spaces over the torus. Some related questions will also be discussed.

14h, Peter Topalov : On the group of almost periodic diffeomorphisms and its exponential map

We define the group of almost periodic diffeomorphisms on the Euclidean plane $\mathbb{R}^n$. We then study the properties of its Riemannian and Lie group exponential map and provide applications to fluid dynamics.

15h, Alexei Iantchenko : Semiclassical inverse problems for elastic surface waves in isotropic media

We carry out a semiclassical analysis of surface waves in Earth which is stratified near its boundary at some scale comparable to the wave length.

The Efimov effect is one of the interesting spectral properties of three-body systems. It asserts that if all the two-body subsystems do not have negative eigenvalues and have a resonance at zero energy, then the total system has an infinite number of negative eigenvalues accumulating at the origin. The effect holds only in dimension three. In recent physics papers, it has been reported to remain true even in dimension two or one under certain conditions. I talk about these results from a mathematical point of view.