Georgia Institute of Technology College of Sciences

We study the normally elliptic singular perturbation problems including both finite and infinite dimensional cases, which could also be nonautonomous. In particular, we establish the existence and smoothness of O(1) local invariant manifolds and provide various estimates which are independent of small singular parameters. We also use our results on local invariant manifolds to study the persistence of homoclinic solutions under weakly dissipative and conservative perturbations.

The aim of this talk is to present recent results obtained in collaboration with C. L\'eonard, C. Roberto and P.M Samson. In the first part, I will give a necessary and sufficient condition for Talagrand's inequality on the real line. In the second part, I will explain the links between Talagrand's inequality and the dimension-free Gaussian concentration phenomenon. This will lead us to a new proof of Otto-Villani Theorem. Finally, in the third part, we will show that Talagrand's inequality is equivalent to a variant of the log-Sobolev inequality, called the inf-convolution log-Sobolev inequality. This theorem will enable us to prove a general perturbation result for Talagrand's inequality.