Abstract: Lagrangian and totally real submanifolds are two objects deeply related because of their definitions. The tangent space of a totally real manifold meets its complex rotation in only one point, the origin; while the tangent space of a Lagrangian submanifold is orthogonal to its complex rotation. One should notice that there are totally real 3-spheres in $\mathbb C^3$, but these spheres cannot be Lagrangian.

Around 1995 Duval and Sibony introduced a new relation between totally real, Lagrangian, and rationally convex manifolds. They proved that, at least for compact totally real submanifolds, rational convexity is equivalent to be Lagrangian for some appropriate Kaehler form.

Moreover, in a recent paper Cieliebak and Eliashberg have proved that, for $n>2$, the closure of a bounded domain in $\mathbb C^n$ is isotopic to a rationally convex set if and only if it admits a defining Morse function with no critical points of index strictly larger than n. This result implies in particular that there are smooth 3-spheres in $\mathbb C^3$ with a compact and rationally convex tubular neighbourhood. These smooth 3-spheres cannot be Lagrangian.

Abstract: Techniques developed for transcendental number theory have had many surprising applications in the study of purely arithmetical questions. The aim of the talk will be to discuss this phenomenon.

Abstract: I will first explain the notion of ergodicity for classical dynamical systems and will go over some of the well known examples of such systems. I will then introduce the notion of quantum ergodicity for quantized Hamiltonian systems and discuss some open problems.