Differential Geometry of Singular Spaces

黎曼流形的研究是数学的核心。它们是光滑的物体，可以使用几何和分析工具进行分析。在过去的15年里对黎曼流形某些极限的研究是很有成果的。一般来说，这些极限呈现出困难的奇点。极限空间的例子是具有下曲率界或上曲率界的度量空间。在这个项目中，我们想用几何测度理论来构造和研究其他类型的极限空间，这些极限空间更适合于标量曲率和里奇曲率的问题。微分几何、几何测度理论、凸几何和亚解析几何之间将会有有趣的相互作用。希望这一理论能应用于爱因斯坦度量的存在性，Yamable不变量的实现以及具有标量或里奇曲率边界的黎曼流形的几何。

The study of Riemannian manifolds is central in mathematics. They are smooth objects which can be analyzed using geometric and analytic tools. It turned out over the last fifteen years that the study of certain limits of Riemannian manifolds is very fruitful. In general, these limits present difficult singularities. Examples for limit spaces are metric spaces with a lower or an upper curvature bound. In this project, we want to use Geometric Measure Theory in order to construct and study other types of limit spaces which are better suited for questions about scalar curvature and Ricci curvature. There will be an interesting interplay between Differential Geometry, Geometric Measure Theory, Convex Geometry and Subanalytic Geometry. Hopefully, there will be applications of this theory to the existence of Einstein metrics, realization of Yamable invariants and to the geometry of Riemannian manifolds with scalar or Ricci curvature bounds.