Suite de l'exposé du 3 juin.

I will present a construction of universal mixers in all dimensions, i.e., incompressible flows that mix to arbitrarily small scales general solutions to the corresponding transport equation. While no universal mixer can have a uniform mixing rate for all measurable initial data, these flows are also almost-universal exponential mixers in the sense that they achieve exponential-in-time mixing (which is the optimal rate) for all initial data with at least some degree of regularity. The constructed flows are time-dependent with an alternating cellular structure, and exist on tori as well as on bounded domains in Euclidean spaces. I will also present numerical evidence of exponential mixing by a different class of flows, alternating shear flows on two-dimensional tori.

The theory of D-modules is an algebraic approach to linear partial differential equations. It was developed in the 1970s for complexes varieties using algebraic geometry objects and Grothentieck point of vue. It has many applications. For example Masaki Kashiwara and Zoghman Mebkhout proved a general version of the Riemann-Hibert correspondence in this context in the 1980s. D-modules are also very useful tools in representation theory. Today the theory of D-modules extends to the arithmetic case for arithmetic varieties. In my presentation I will expose some motivations of D-molules. I will explain why we introduce D-modules in order to solve partial differential equations. I will not talk about algebraic geometry. In a second time I will define the ring of differential equations with holomorphic coefficients and state some of its properties.

Deciding if a curve in a simplicial complex is contractible is known to be undecidable, even for curves in two dimensional complexes or 4-manifolds. It is decidable for a curve in a three manifold but the complexity status is unknown. We show that the problem is in NP when the curve lies on the boundary of a three manifold. To motivate this restricted problem we remark that the knot triviality problem reduces to it in polynomial time. Joint work with Erin Wolf Chambers , Arnaud de Mesmay and Salman Parsa

Stable commutator length (SCL) is a function one can define on a group that has been attracting interest over the past fifteen years for its links with the geometry of groups. After having introduced some ideas and questions of geometric group theory for motivation, we will give three points of view - algebraic, topological, and cohomological - on stable commutator length. We will present some recent results concerning the spectrum of SCL and, time permitting, we will explain some of the various ideas of proof behind those results.