OPTiMiSE · Optimal Transport and Metric Structures for Evolution Problems

Several evolution problems, such as gradient flows or rate-independent processes, are governed by variational principles which are extremely useful for studying the existence, stability, and structural properties of solutions by simple and general constructive approximation methods.Deep and beautiful ideas from the theory of Optimal Transport have contributed new insights and additional challenging questions to this scenario and have motivated flourishing and original developments. On the one hand, the applications to gradient flows in the Kantorovich-Wasserstein spaces of probability measures reveal the importance, the power, and the flexibility of the metric viewpoint. On the other hand, the interplay with evolutionary problems has in turn brought new ideas and perspectives to Optimal Transport, inspiring a powerful set of techniques for its applications, especially to the analysis and geometry in metric-measure spaces.In recent years, the PI and his collaborators have given relevant contributions to the general theory of gradient flows, in particular in Kantorovich-Wasserstein spaces, and they have obtained ground-breaking results for metric-measure spaces and Unbalanced Optimal Transport between positive measures with finite mass.The goal of the project is a wide-ranging analysis which aims to combine and broaden the above themes and perspectives, to address crucial and challenging open problems, and to open up novel research directions: - new generation results and metric-variational principles for evolution equations,- the interplay between curvature bounds and convergence of variational approximation schemes,- a new metric approach to dissipative evolution and saddle-point flows,- new methods and results for paradigmatic highly nonlinear and non-convex partial differential equations for probability measures,- the foundation of a mean-field theory for the rate-independent evolution of critical points.