Abstract: The theory of curvature-dimension bounds for metric measure structure has several motivations: the study of functional and geometric inequalities in structures which are very far from being Euclidean, therefore with new non-Riemannian tools, the description of the “closure” of classes of Riemannian manifolds under suitable geometric constraints, the stability of analytic and geometric properties of spaces. In the last few years, with crucial inputs coming from the theory of optimal mass transportation, we have seen a spectacular progress in all these directions, which also stimulated the development of new calculus tools in metric measure spaces. The lecture is meant both as a survey and as an introduction to this quickly developing research field
https://eta.impa.br/dl/PL011.pdf
http://cvgmt.sns.it/media/doc/paper/3779/ICM_Ambrosio.pdf
本文讨论非光滑空间的曲率-维数的有界性理论。作者首先介绍了三个基本的等价性结果：Cheeger能量与弱可微函数，向量场的流与叠加原理，度量与能量结构。然后介绍最优输运的背景。最后给出处理曲率-维数条件的两种成功理论：BE(Bakry-Émery)理论与CD理论。
相关附件
7-PL011 Ambrosio
