Geometric inequalities and Optimal Transport in Riemannian Geometry

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In this projects aims to investigate isoperimetric inequalities in Riemannian and subriemannian manifolds by variational methods, in particular by the mathematical theory of optimal transport. A particular question is the case of lower bounded Ricci curvature and indeed the case of an ambient Heisenberg group. In either case the analysis comes down to establishing control on certain types of geodesic decompositions of the ambient space, which allows for a reduction of the dimension of the problem. The project lies at the intersection of Riemannian geometry and Geometric analysis and requires deep understanding of wide areas in geometry and analysis.