Aspects of non-positive curvature

摘要奖：DMS 1510640，主要研究者：Jean-Francois R.拉丰这个项目的主要目标是研究负（或更一般地说，非正）曲率的空间。负弯曲空间是指在每一点和每一对方向上看起来都像马鞍面的空间。虽然这种类型的空间可能听起来很奇怪，但它实际上在数学中很普遍。对于这些空间，几何（空间的定量特征）和拓扑（空间的定性特征）之间有着紧密的联系。此外，这些空间有一些非常有趣的动力学系统与它们相关联，在那里人们可以研究物体如何随时间移动。因此，对这些空间的研究提供了一个不同观点和方法的熔炉，使研究领域更加丰富。这个项目计划使用这三种不同的方法（几何学，拓扑学，动力学）来阐明这些空间。主要研究者（PI）计划围绕非正曲率的概念开展一系列项目。 PI的项目大致分为三大类：几何，拓扑和动力学。在几何方面，PI计划专注于流形的黎曼非正曲率和度量非正曲率之间的区别。PI还建议研究Gromov收缩不等式的离散版本。另一个项目涉及研究群和空间的几乎等距关系。在拓扑方面，PI计划给出3维双曲反射群的拓扑K理论的组合公式。PI还将对具有Sierpinski曲线边界的Gromov双曲群进行分类。另一个项目涉及双曲线化和欧拉特征线的符号。最后，PI计划研究Anosov分裂的拓扑。对于该提案的动态方面，PI希望为Fuchsian建筑建立显著的长度刚度。他还想证明某些非球面流形不能支持Anosov同构。另一个项目是证明局部CAT（-1）空间上的测地线流具有规范性质。

AbstractAward: DMS 1510640, Principal Investigator: Jean-Francois R. LafontThe main goal of this project is to study spaces of negative (or more generally, non-positive) curvature. Negatively curved spaces are ones which, at every point and every pair of directions, look like the surface of a saddle. While this type of space might sound odd, it is in fact pervasive in mathematics. For these spaces, there is a tight link between the geometry (quantitative features of the space) and the topology (qualitative features of the space). In addition, these spaces have some very interesting dynamical systems associated to them, where one can study how objects move over time. As a result, the study of these spaces provides a melting pot for different viewpoints and methods, making for a rich field of study. This project plans on using these three different approaches (geometry, topology, dynamics) to shed some light on these spaces.The Principal Investigator (PI) plans to work on a series of projects that are centered around the notion of non-positive curvature. The PI's projects loosely fall into three broad categories: geometry, topology, and dynamics. On the geometric side of things, the PI plans on focusing on the distinction, for manifolds, between Riemannian non-positive curvature and metric non-positive curvature. The PI also proposes to study discrete versions of Gromov's systolic inequality. Another project involves studying the relation of almost-isometry for groups and spaces. On the topological side of things, the PI plans on giving combinatorial formulas for the topological K-theory of 3-dimensional hyperbolic reflection groups. The PI will also classify Gromov hyperbolic groups with Sierpinski curve boundary. Another project concerns hyperbolization and the sign of the Euler characteristic. Finally, the PI plans on studying the topology of Anosov splittings. For the dynamical aspects of this proposal, the PI wants to establish marked length rigidity for Fuchsian buildings. He also wants to show that certain aspherical manifolds cannot support Anosov diffeomorphisms. Another project is to show the geodesic flow on locally CAT(-1) spaces has the specification property.