Les groupes de cohomologie à coefficients tordus des groupes de difféotopie des surfaces et des corps en anses sont, de manière générale, très difficiles à calculer. Dans cet exposé, je présenterai des calculs de la cohomologie stable de ces groupes pour divers systèmes de coefficients, notamment liés à leur filtration de Johnson. Cet exposé présentera des résultats obtenus en collaboration avec Erik Lindell, ainsi que d’autres travaux en cours menés avec Andreas Stavrou.

We investigate the blow-up rate for the nonlinear wave equation with a superlinear power nonlinearity. While the blow-up rate for subconformal and conformal powers is well-known to follow the solution of the associated ODE, as established by Merle and Zaag (2003, 2005), the superconformal regime remains more complex. Although upper bounds have been established by Killip, Stovall, and Visan (2014) and in our previous work (2013), this talk presents a significant refinement of those results. Specifically, we demonstrate a logarithmic improvement of order |ln (T-t)|^q for an arbitrary q>0. This result is obtained by treating the superconformal case as a large perturbation of the conformal setting and introducing new analytical tools, most notably a Pohozaev-type identity involving a singular weight. This is a joint work with Hatem Zaag.

The Euler equation describes an incompressible and inviscid fluid. In the 2d case the vorticity is transported and hence bounded if it is bounded initially. In this joint work with Dimitri Cobb we consider solutions whose vorticity is bounded and whose velocity grows slower than linear at infinity. We construct global in time solutions which are unique up to the Galilei symmetry of the Euler equations. Key tools are local energy inequalities, a modification of Yudovich's uniqueness argument and the choice of several coordinate frames, using the Galilei symmetry.

The case of slower than square root growth is easier and allows to work with Leray type  projections.