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.

Hilbert's problem of the topology of algebraic curves and surfaces (the
16th problem from the famous list presented at the second International
Congress of Mathematicians in 1900) was difficult to formulate. The way it
was formulated made it difficult to anticipate that it has been solved. I
believe it has, and this happened more than thirty years ago, although the
World Mathematical Community missed to acknowledge this.