Final Report Summary - GEMETHNES (Geometric Measure Theory in non-Euclidean spaces)
The main achievements of the project are:
-- A novel point of view for calculus in metric measure spaces, with applications to the theory of weakly differentiable functions, which allows to include in the theory finite and infinite dimensional spaces (80 pages paper, in collaboration with the team members Gigli and Savare', published on Inventiones)
-- Using the tools above, the PI, Gigli and Savare' provided in the paper published in Annals of Probability a key bridge between two theories developed in the recent past for synthetic Ricci lower bounds for metric measure spaces: the Lott-Sturm-Villani theory (of Lagrangian type, based on optimal transport) and the Bakry-Emery-Ledoux theory (of Eulerian type, based on the theory of Markov semigroups and Dirichlet forms)
-- On the geometric side, the mathematical tools introduced within this project have been used to study limits of Riemannian manifolds with Ricci lower bounds, providing to some extent a novel approach, compared to the techniques introduced by Cheeger-Colding
-- The Eulerian-Lagrangian duality mentioned above appears also in the theory of flows with respect to nonsmooth vector fields. In this direction a relevant progress has been made both in the Euclidean case and in the case of abstract metric measure spaces. In a joint work with D.Trevisan the PI proved the first well-posedness results in a metric measure setting, for general classes of vector fields not necessarily of gradient type
-- Most results have been inspired by ideas and techniques in Geometric Measure Theory. In some papers of the PI in collaboration with A.Figalli a novel point of view to study blow-up and rectifiability properties in infinite-dimensional spaces has been introduced: the main new idea is to use semigroup tools to bypass the difficulties due to infinite-dimensionality.
-- A novel point of view for calculus in metric measure spaces, with applications to the theory of weakly differentiable functions, which allows to include in the theory finite and infinite dimensional spaces (80 pages paper, in collaboration with the team members Gigli and Savare', published on Inventiones)
-- Using the tools above, the PI, Gigli and Savare' provided in the paper published in Annals of Probability a key bridge between two theories developed in the recent past for synthetic Ricci lower bounds for metric measure spaces: the Lott-Sturm-Villani theory (of Lagrangian type, based on optimal transport) and the Bakry-Emery-Ledoux theory (of Eulerian type, based on the theory of Markov semigroups and Dirichlet forms)
-- On the geometric side, the mathematical tools introduced within this project have been used to study limits of Riemannian manifolds with Ricci lower bounds, providing to some extent a novel approach, compared to the techniques introduced by Cheeger-Colding
-- The Eulerian-Lagrangian duality mentioned above appears also in the theory of flows with respect to nonsmooth vector fields. In this direction a relevant progress has been made both in the Euclidean case and in the case of abstract metric measure spaces. In a joint work with D.Trevisan the PI proved the first well-posedness results in a metric measure setting, for general classes of vector fields not necessarily of gradient type
-- Most results have been inspired by ideas and techniques in Geometric Measure Theory. In some papers of the PI in collaboration with A.Figalli a novel point of view to study blow-up and rectifiability properties in infinite-dimensional spaces has been introduced: the main new idea is to use semigroup tools to bypass the difficulties due to infinite-dimensionality.