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

这个项目的重点是发展非光滑空间的里奇流的综合概念。在度量测度空间时间演化的一般框架下，利用几何分析和最优迁移的思想给出Ricci流的几个综合刻画，并分析这类流的行为;第二个主要目标是给出在Ricci流概念下单调行为的泛函，并建立刻画极值演化的刚性结果.特别是，我们将调查自然的非光滑类似物的Perelman的W熵和L长度，我们将探讨它们的关系，广义概念的Ricci solitons.Finally，我们的目标是建立尖锐的几何和解析比较结果的时间依赖性度量测度空间下的Ricci流扩展到一个动态设置的强大的结果最近开发的静态空间与合成的Ricci下界。

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.