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

几何学涉及空间的量化方面。负弯曲的空间局部看起来像马鞍面(在每一对方向上)。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相当普遍。对于这些空间，空间的几何形状和空间的拓扑(形状的定性方面)之间存在着紧密的联系。还有一个自然的相关动力系统，它记录了空间中的粒子将如何运动，并表现出双曲线行为-对初始条件极端敏感。例如，如果你把一颗弹珠放在鞍子上，然后看它是如何滚动的，那么附近的两颗弹珠可以朝非常不同的方向滚动。同样，双曲性是动力系统的一种非常常见的行为类型。这项研究计划研究这些空间的几何、拓扑和动力学是如何相互作用和影响的。首席调查员(PI)计划开展一系列以非正曲率概念为松散中心的项目。这些项目分为三大类。(1)分支覆盖：PI计划系统地研究非正曲线(或负曲线)局部对称流形的分支覆盖，分支在余维两个全测地子流形上。主要目的是了解这些覆盖是否支持黎曼非正(或负)曲率度量。(2)复杂几何：PI计划研究与复杂流形有关的几个问题。他计划创造新的封闭式负曲线凯勒歧管。他想研究流形上几乎复数、复数和Kaehler结构的空间的拓扑。他还想给出几乎复杂的流形的高维例子，这些流形不支持复杂的结构。(3)几何/拓扑问题：PI想要证明双曲总是产生基本群是线性的空间。他想研究格罗莫夫-瑟斯顿流形的Singer猜想。他感兴趣的是研究SO(n；1)中的非算术格，特别是非算术双曲流形中的全测地子流形的结构。他还希望为多种形式支持Anosov差异同构找到新的障碍。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.