Alexandrov Geometry in the light of symmetry and topology

Alexandrov空间是具有下截面曲率界的完备黎曼流形的推广。然而，它们可能表现出不同于黎曼流形的行为，因为它们携带的拓扑和度量奇点。因此，研究是否可以将黎曼环境中的给定性质扩展到亚历山德罗夫环境是至关重要的。在这个项目中，我们解决了其中的一些性质，并探索亚历山德罗夫空间如何表现出对每个性质的影响。在一个方向上，我们把注意力集中在拓扑特征上。这里的主要目标是了解亚历山德罗夫空间的拓扑与光滑流形的拓扑有多远。更具体地说，一方面，我们的目标是调查的基本和重要的问题是否亚历山德罗夫空间承认一个三角形，另一方面，我们要分析，使用理性同伦理论，上同调性质的亚历山德罗夫空间，如庞加莱对偶，在对称性的存在。在另一个方向上，我们研究了具有正曲率的Alexandrov空间的上齐性，目的是对它们进行分类。为此，我们需要找到障碍物和识别工具，特别是依赖于我们对前一方向研究的亚历山德罗夫空间的上齐性拓扑行为的理解。

Alexandrov spaces are a generalization of complete Riemannian manifolds with a lower sectional curvature bound. However, they may exhibit behaviors different from Riemannian manifolds due to the topological and metric singularities that they carry. It is then of fundamental importance to investigate whether one can extend a given property in the Riemannian setting to the Alexandrov setting.In this project we address some of these properties and explore how Alexandrov spaces behave with respect to each property. In the one direction, we focus our attention on the topological features. The primary objective here is to understand how far the topology of Alexandrov spaces are from that of the smooth manifolds. More specifically, on the one hand we aim to investigate the basic and important question of whether Alexandrov spaces admit a triangulation and on the other hand we want to analyze, using rational homotopy theory, the cohomological properties of Alexandrov spaces, such as Poincaré duality, in the presence of symmetry. In the other direction, we examine cohomogeneity one Alexandrov spaces with positive curvature and the goal is to classify them. To this end, we need to find obstructions and recognition tools, which, in particular, rely upon our understanding of topological behaviors of cohomogeneity one Alexandrov spaces investigated in the former direction.