Sending quantum information over a distance is a challenging task and the challenge increases, if the task is to be performed several times, to send a message over a large distance. In general, this can only be done with a certain probability. On several quantum networks, a recently introduced procedure, which uses special quantum measurements, enhances this probability, as follows from the properties of related classical percolation models. ALL of the above terms will be explained in the talk, which is aimed at a general mathematical audience. While mathematically rigorous, the presented results were obtained in collaboration with a physics group and are of current physical interest, leading to a number of open problems, which will be mentioned.

A non-zero Abelian differential in a compact Riemann surface of genus $g \geq 1$ endows the surface with an atlas (outside the zeroes) whose coordinate changes are translations. There is a natural ``vertical flow'' (moving up with unit speed) associated with the translation structure, generalizing the genus $1$ case of irrational flows on the torus.

The Teichm\"uller flow in the moduli space of Abelian differentials can be seen as the renormalization operator of translation flows. In this talk, we will discuss how the chaoticity of the Teichm\"uller flow dynamics reflects on the (non-chaotic) dynamics of the associated vertical flows (for typical parameters), and the closely related interval exchange transformations.

After the seminal work of Paul Rabinowitz on periodic orbits of Hamiltonian Systems on starshaped surfaces in |R^n, Contact Structures have become a natural object of study for analysts. The search for invariants for these contact forms/structures benefited very much from the deeper understanding of the much more general associated variational problem used in the work of Paul Rabinowitz and of Conley-Zehnder. Contact Homology has then been defined using pseudo-holomorphic curves, but also via Legendrian curves. After broadly recalling the main steps in the formulation and the development of these tools, we present a more detailed account of the contact homology via Legendrian curves, including its definition, its compactness properties and the value of this homology for odd indexes.

There are no analogs of Navier-Stokes equations for solid mechanics. One reason is that information at the atomic scale seems to play a much more important role for solids than for fluids. A satisfactory mathematical theory for solids has to taken into account the behavior of solids at different scales, from electronic to atomic, to macroscopic scales.

I will discuss some of the fundamental problems that we have to resolve in order to build such a theory. I will start by reviewing the geometry of crystal lattices, the quantum as well as classical atomistic models of solids. I will then focus on a few selected problems:

(1) The crystallization problem -- why the ground states of solids are crystals and which crystal structure do they select?

(2) the microscopic foundation of elasticity theory;

(3) stability and instability of crystals;

(4) the electronic structure and density functional theory.

Holomorphic functions on a domain are the solutions to a set of homogeneous partial differential
equations called the Cauchy-Riemann equation, and CR functions are the solutions to the analogous
equations on the boundary. Many problems in complex analysis can be reduced to finding appropriate
solutions to the inhomogeneous versions of these equations. These solutions have been successfully
constructed when the geometry of the domain is sufficiently simple. I hope to show how these constructions
follow a pattern based on the singular integral operators of Calder\'on and Zygmund. I then plan to
discuss some more recent examples where this well-understood paradigm breaks down.