I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close together (super-exponentially in their lengths). Applications include resolution of some conjectures of Furstenberg on the dimension of sumsets and, together with work of Shmerkin, progress on the absolute continuity of Bernoulli convolutions. The main new ingredient is a statement in additive combinatorics concerning the structure of measures whose entropy does not grow very much under convolution. If time permits I will discuss the analogous results in higher dimensions.[-]

Left-handed flows are 3-dimensional flows which have a particular topological property, namely that every pair of periodic orbits is negatively linked. This property (introduced by Ghys in 2007) implies the existence of as many Bikrhoff sections as possible, and therefore allows to reduce the flow to a suspension in many different ways. It then becomes natural to look for examples. A construction of Birkhoff (1917) suggests that geodesic flows are good candidates. In this conference we determine on which hyperbolic orbifolds is the geodesic flow left-handed: the answer is that yes if the surface is a sphere with three cone points, and no otherwise.
dynamical system - geodesic flow - knot - periodic orbit - global section - linking number - fibered knot[-]

I will discuss some recent results with Aaron Brown and Zhiren Wang on actions by higher rank lattices on nilmanifolds. I will present the result in the simplest case possible, $SL(n,Z)$ acting on $Tn$, and try to present the ideas of the proof. The result imply existence of invariant measures for $SL(n,Z)$ actions on $Tn$ with standard homotopy data as well as global rigidity of Ansosov actions on infranilmanifolds and existence of semiconjugacies without assumption on existence of invariant measure.[-]