The famous Lehmer problem asks whether there is a gap between 1 and the Mahler measure of algebraic integers which are not roots of unity. Asked in 1933, this deep question concerning number theory has since then been connected to several other subjects. After introducing the concepts involved, we will briefly describe a few of these connections with the theory of linear groups. Then, we will discuss the equivalence of a weak form of the Lehmer conjecture and the ``uniform discreteness" of cocompact lattices in semisimple Lie groups (conjectured by Margulis). (Joint work with Lam Pham.)

In this talk, we discuss the construction and invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree-nonlinearity.

In the first part of the talk, we construct the Gibbs measure and examine its properties. We discuss the
mutual singularity of the Gibbs measure and the so-called Gaussian free field. In contrast, the Gibbs
measure for one or two-dimensional wave equations is absolutely continuous with respect to the Gaussian free field.

In the second part of the talk, we discuss the probabilistic well-posedness of the corresponding nonlinear wave equation, which is needed in the proof of invariance. This was the first theorem proving the invariance of a singular Gibbs measure for any dispersive equation.

Free information theory is largely concerned with the following question: given a tuple of non-commutative random variables, what regularity properties of the algebra they generate can be inferred from assumptions about their joint distribution? This can include von Neumann algebraic properties, such as factoriality or absence of Cartan subalgebras, and free probabilistic properties, such as a lack of non-commutative rational relations.

After giving some background, I will talk on free Stein dimension, a quantity which measures the ease of defining derivations on a tuple of non-commutative variables which turns out to be a $*$-algebra invariant. I will mention some recent results on its theory, including behaviour in the presence of algebraic relations as well as under direct sum and amplification of algebras. I will also mention some recent attempts to
adapt its utility from polynomial algebras to W*-algebras, and time permitting, some cases where explicit estimates can be found on the Stein dimension of generating tuples of von Neumann algebras. This project is joint work with Brent Nelson.

I will present the dynamical formulation of optimal transport (a.k.a Benamou-Brenier formulation): it consists in writing the optimal transport problem as the minimization of a convex functional under a PDE constraint, and can handle a priori a vast class of cost functions and geometries. It is one of the oldest numerical method to solve the problem, and it is also the basis for a lot of extensions and generalizations of the optimal transport problem.

The optimization problem is then discretized to end up with a finite dimensional convex optimization problem. I will illustrate this method by presenting a discretization when the ground space is a surface. Although much effort has been devoted to solve efficently the discretized problem, the study of convergence under mesh refinement of the solution of the approximate problems has only been tackled recently. I will present an abstract framework guaranteeing convergence under mesh refinement, with no condition on the relative scale of the spacial and temporal mesh sizes, and even if the densities are very singular.

We provide a brief introduction to the theory of vector bundles and present a few useful and interesting constructions with a geometric flavor.