"There are three different topics related to this project. The first one is an interplay between probability and Morse theory. The main idea is to exploit in the probabilitisc framework a suggestion of Witten and recover the Morse-Smale complex of a Morse function using asymptotics of Wiener functionals. The second topic is the study of transportation cost inequality on the path space of a Riemannian manifold and its connections with Log-Sobolev and HWI inequality of Otto-Villani. There were some attempts in the literature but it seems that the appropriate distance is yet to be settled. The third topic is in free probability. This regards first, second and higher order asymptotic freeness of Wigner ensembles and constant matrices. The results are to be applied to the study of random matrices with dependencies. Regarding functional inequalities in free probability, the standard tool is the random matrix approximation. The investigator used mass transportation thechiques to provide proofs of some of these inequalites in one dimensional free probability for measures on the line. Part of this proposal is an extension of these results to measures on the circle or in plane as well several noncommutative random variables."