Abstract: I will give an introduction to the renormalization theory of area-preserving maps, and present several recent results about the dynamics in a neighbourhood of the fixed point of the period doubling renormalization operator: $F\to\Lambda^{−1}\circ F\circ F\circ\Lambda$, where $\Lambda$ is a diagonal matrix. The results are based on a detailed study of the spectral properties of the derivative of the renormalization operator. The talk is based on joint work with Denis Gaidashev and Marco Martens.

Our starting point is a celebrated approximation theorem due to M.
Artin (1969) which roughly states that any formal solution of a system
of polynomial equations can be approximated (in the Krull topology)
by a sequence of algebraic solutions. Our goal in this talk is to explore
whether a similar conclusion holds when the system of polynomial is
coupled with certain (linear) pde’s. We will discuss the class of linear operators for which such a generalization is possible. These opera-
tors arise as tangential Cauchy-Riemann operators of real-algebraic CR
manifolds (and include, as a very special case, the standard Cauchy-
Riemann operator of the complex euclidean space).

Abstract: I will describe results obtained in collaboration with H. Jacobowitz
some years ago and recently with A. Bergamasco and S. Zani on topological con-
sequences of assumptions made on differential operators. This is a problem of a
global nature: the objects on which the operators act are functions (more generally
sections of a complex vector bundle) on a compact manifold without boundary.
The analytic conditions are either ellipticity, or global hypoellipticity. I will also
pose an open problem concerning solvability in the large.

Abstract: In our work we present a modification of the known Baouendi-Treves
Approximation Theorem. Instead of working with a general $N$ -dimensional smooth
manifold we will use a quadric manifold $M$.
While the original theorem deals with any locally integrable structure $\mathcal{L}$ of $\mathbb{C}T M$
we will focus on the CR-structure of $M$ and its solutions: the CR distributions.
With this restrictions we achieve convergence in any compact set instead of local
approximation. More precisely: our main theorem states that if $u\in$CR$(M)$ then one can find
smooth CR-polynomials $P_{n}(w, t)$ such that for every compact subset $K$ we can
approximate $u$ by $P_{n}$ in the distribution sense. In addition to that
(1) if $u\in C^{k}(M )$, $k = 0, 1,\ldots$, then the convergence occurs in the topology of $C^{k}(K)$;
(2) if $u\in h^{p}(M )$, $0 < p < \infty$, then the approximation occurs in the topology of $h^{p}(K)$.
If there is enough time, we will present an application.

Abstract: In this talk I will discuss recent results regarding regularity properties
of Green functions associated with elliptic differential operators of second and higher
order in irregular domains. This analysis includes the case of second and higher
order elliptic systems with constant coefficients, the bi-Laplacian, and the Stokes
system.

Abstract: After an introduction concerning the fractional Laplace operators and
the k-th Hessian operators, some relations between the k-Hessian energy and the
fractional Sobolev energy will be presented.

Abstract: We study the asymptotic behavior at infinity of solutions of a perturbed Dirac Operator and their relation to the outgoing and incoming solutions of
the Helmholtz equation and the Maxwell system in terms of their far-field patterns
(joint work with Salvador Perez-Esteva).

Abstract: A new proof of both analytic and $C^{\infty}$− hypoellipticity of Kohn’s
operator is given using FBI techniques introduced by J. Sjostrand. The same proof
allows us to obtain both kinds of hypoellipticity at the same time. We prove also
the hypoellipticity in the sense of germs, even though it fails to be hypoelliptic
in the strong sense, for an operator obtained by “slightly perturbing” the Kohn’s
operator.

Abstract: The plan of the talk is to describe some interesting trigonometric series
which we call “iterated Fourier series” and their connections to natural problems
of “physical reality”.

Abstract: The transmission eigenvalue problem is a new class of eigenvalue
problems that has recently appeared in inverse scattering theory for inhomogeneous media. Such eigenvalues provide information about material properties of
the scattering object and can be determined from scattering data, hence can play
an important role in a variety of problems in target identification. The transmission
eigenvalue problem is non-selfadjoint and nonlinear which make its mathematical
investigation very interesting.
In this lecture we will describe how the transmission eigenvalue problem arises
in scattering theory, how transmission eigenvalues can be computed from scattering
data and what is known mathematically about these eigenvalues. The investigation of transmission eigenvalue problem for anisotropic media will be discussed
and Faber-Krahn type inequalities for the first real transmission eigenvalue will be
presented.
We conclude our presentation with some recent preliminary results on transmission eigenvalues for absorbing and dispersive media, i.e. with complex valued index
of refraction, as well as for anisotropic media with contrast that changes sign.
Our presentation contains a collection of results obtained with several collaborators, in particular with David Colton, Drossos Gintides, Houssem Haddar and
Andreas Kirsch.