In the seminal paper of Jordan, Kinderlehrer and Otto the heat equation has been interpreted as the gradient flow, or deepest descent, of the entropy with respect to the  Wasserstein L^2 metric. In the same spirit the evolution associated to some partial differential equations has been characterized  in terms of an entropy dissipation inequality. Recently there have been some attempts to formulate the Fokker-Planck equation associated to continuous time reversible Markov chains, equivalently homogeneous linear kinetic equations, as gradient flow, and the approach has been extended to the case of homogeneous Boltzmann equations.
I will present some results on gradient flow formulation of linear, non-homogeneous kinetic equations. In particular, I will discuss  how the functional giving rise to the entropy dissipation inequality is related to large deviations and  I will derive the diffusive limit. I will finally discuss how the above approach could be extended to  non-linear Boltzmann equations.

Torus bundles over the circle are among the simplest and cutest examples of 3-dimensional manifolds. After presenting some of these examples, using in particular animations realized by Jos Leys, I will consider periodic orbits in these fiber bundles over the circle. We will see that their linking numbers --- that are rational numbers by definition --- can be computed as certain special values of Hecke L-functions.  Properly generalized this viewpoint makes it possible to give new topological proof of now classical rationality or integrality theorems of Klingen-Siegel and Deligne-Ribet. It also leads to interesting new "arithmetic lifts" that I will briefly explain. All this is extracted from an on going joint work with Pierre Charollois, Luis Garcia and Akshay Venkatesh.

A control system is a dynamical system on which one can act by using controls. For these systems a fundamental problem is the stabilization issue: Is it possible to stabilize a given unstable equilibrium by using suitable feedback laws? (Think to the classical experiment of an upturned broomstick on the tip of one's finger.) On this problem, we present some pioneer devices and works (Ctesibius, Watt, Maxwell, Lyapunov,...), some more recent results, and an application to the regulation of the rivers La Sambre and La Meuse in Belgium. A special emphasize is put on positive or negative effects of the nonlinearities.