By Jean Dolbeault, University of Paris-Dauphine

Nonlinear diffusion flows are very interesting tools for the study of sharp functional inequalities, the stability of the optimal functions and for improvements of the inequalities. The lecture will be devoted to a series of recent results which unify some rigidity methods in nonlinear elliptic equations, the Bakry-Emery method and various qualitative features related with functional inequalities in probability theory, information theory and nonlinear analysis.

By Jean-Claude Saut, University of Paris-Sud

We will survey recent results (both theoretical and numerical) on the effect of a "weak" dispersion on the dynamics of Burgers and hyperbolic equations: local Cauchy problem, existence or not of solitary waves, shock like structures, long time dynamics...).

By YanYan Li, Rutgers University

Let $(M,g)$ be an $n\ge 5$ dimensional smooth compact Riemannian manifold of positive Yamabe type, which is not conformally equivalent to the standard sphere. We prove compactness of conformal metrics of $g$ with positive constant Q-curvature provided that $(M,g)$ is locally conformally flat, or $n=5,6,7$. For $n\ge 8$, we prove the compactness result provided that the Weyl tensor of $g$ does not vanish anywhere. This is a joint work with Jingang Xiong.

By Gunther Uhlmann, University of Washington

Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc. including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last decade or so there have been several scientific proposals to achieve invisibility. We will introduce some of these in a non-technical fashion concentrating on the so-called "transformation optics" that has received the most attention in the scientific literature.