Ricci flows for non-smooth spaces, monotonic quantities, and rigidity

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The focus of this project is to develop synthetic notions of Ricci flow for non-smooth spaces. In the general framework of time evolutions of metric measure spaces we will use ideas of geometric analysis and optimal transport to give several synthetic characterizations of Ricci flow and analyze the behavior of such flows.A second major goal will be to exhibit functionals that behave monotonically under this notion of Ricci flow and to establish rigidity results characterizing extremal evolutions. In particular, we will investigate natural non-smooth analogues of Perelman's W entropy and L length and we will explore their relation to generalized notions of Ricci solitons.Finally, we aim to establish sharp geometric and analytic comparison results for time-dependent metric measure spaces under Ricci flow which extend to a dynamic setting the powerful results developed recently for static spaces with synthetic lower Ricci bounds.