Spaces with sectional and Ricci curvature bounds

空间的曲率是其最重要的局部几何不变量。它描述了一个空间是如何从居住在该空间内的观察者的角度被局部塑造或弯曲的。 这一建议主要涉及具有各种下曲率下界的空间。具有较低截面界和Ricci曲率界的空间的概念在数学和物理的许多领域中发挥着重要作用：广义相对论、最优传输、大数据分析，当然还有黎曼几何本身。我计划研究具有较低截面曲率下界的广义空间(Alexandrov空间)和具有较低Ricci曲率下界的广义空间(限制曲率-维条件)的几个问题。我特别感兴趣的是这两种情况之间的相互作用，而这两种情况目前还没有得到充分的了解。

Curvature of a space is its most important local geometric invariant. It describes how a space is shaped or curved locally from the point of view of an observer living inside that space. This proposal is mostly concerned with spaces with various lower curvature bounds. The notions of spaces with lower sectional and Ricci curvature bounds play an important role in a number of fields in mathematics and physics: general relativity, optimal transport, big data analysis and, of course, Riemannian geometry itself. I plan to study several problems concerning generalized spaces with lower sectional curvature bounds (Alexandrov spaces) and generalized spaces with lower Ricci curvature bounds (Restricted Curvature-Dimension condition). I am particularly interested in the interplay between these two conditions which as yet is not sufficiently understood.