Geometry, Topology, and Dynamics of Spaces of Non-Positive Curvature

几何学涉及空间的定量方面。负弯曲的空间局部看起来像一个马鞍的表面（在每对方向上）。虽然这可能很难想象，但具有这种性质的空间在数学和自然界中都很普遍。对于这些空间，在空间的几何和空间的拓扑（形状的定性方面）之间存在着紧密的联系。还有一个自然的关联动力系统，它记录了空间内的粒子如何运动，并表现出双曲线行为-对初始条件的极端敏感性。例如，如果你把一个弹珠放在马鞍上，看它是如何滚动的，那么附近的两个弹珠就可以朝不同的方向滚动。同样，双曲性是动力系统的一种非常常见的行为。本研究旨在研究这些空间的几何学、拓扑学和动力学如何相互作用和影响。主要研究者（PI）计划开展一系列以非正曲率概念为中心的项目。这些项目分为三大类。 (1)分支覆盖物：PI计划系统地研究非正弯曲（或负弯曲）局部对称流形的分支覆盖，分支于余维2的全测地子流形。主要目标是了解这些覆盖是否支持黎曼非正（或负）曲率度量。 (2)复几何：PI计划研究有关复流形的几个问题。 他计划创造新的封闭负弯曲凯勒流形。他想研究流形上的几乎复、复和Kaehler结构的空间的拓扑。 他还想给高维的例子几乎复杂的流形，不支持复杂的结构。 (3)几何/拓扑问题：PI希望证明双曲化总是产生基本群是线性的空间。他想研究的辛格猜想Gromov-Thurston流形。他有兴趣研究非算术格SO（n; 1），特别是结构的全测地子流形内nonarithmetic双曲流形。他还希望为流形找到新的障碍来支持Anosov同构。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Geometry is concerned with quantitative aspects of spaces. Negatively curved spaces locally look like the surface of a saddle (in every pair of directions). While this might be hard to visualize, spaces with this property are quite prevalent, both in mathematics, and in nature. For these spaces, there is a tight link between the geometry of the space and the topology (qualitative aspects of the shape) of the space. There is also a natural associated dynamical system, which records how a particle inside the space will move, and which exhibits hyperbolic behavior - extreme sensitivity to initial conditions. For example, if you put a marble on a saddle and see how it rolls, then two nearby marbles can roll in very different directions. Again, hyperbolicity is a very common type of behavior for dynamical systems. This research proposes to study how the geometry, topology, and dynamics of these spaces interact and influence each other.The Principal Investigator (PI) plans to work on a series of projects that are loosely centered around the notion of non-positive curvature. These projects fall into three broad categories. (1) Branched coverings: the PI plans to systematically study branched covers of non-positively curved (or negatively curved) locally symmetric manifolds, ramified over a codimension two totally geodesic submanifold. The main goal is to understand whether or not these covers support Riemannian non-positive (or negative) curvature metrics. (2) Complex geometry: the PI plans to study several questions concerning complex manifolds. He plans to create new closed negatively curved Kaehler manifolds. He wants to investigate the topology of the space of almost complex, complex, and Kaehler structures on a manifold. He also wants to give high dimensional examples of almost complex manifolds that do not support complex structures. (3) Geometry/topology problems: the PI wants to show that hyperbolizations always produce spaces whose fundamental groups are linear. He would like to study the Singer conjecture for Gromov-Thurston manifolds. He is interested in studying non-arithmetic lattices in SO(n; 1), particularly the structure of totally geodesic submanifolds inside nonarithmetic hyperbolic manifolds. He would also like to find new obstructions for manifolds to support Anosov diffeomorphisms.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.