Intrinsic Geometry, Topology, and Complexity of 3-Manifolds

一个被称为三维流形的数学对象与我们的物理空间很相似，如果你只观察它的一小部分。然而，在大尺度上，三维流形的性质可能与我们习惯看到和体验的完全不同。这样的流形是科学中常见的对象，例如出现在物理学，天文学或数据科学，以及数学的各个领域。在这里，首席研究员将研究3-流形的内在属性，以及它们如何与数学的各个领域相关联。由于许多性质适合于计算机研究并产生有趣的算法，PI还将研究包括计算上如何“硬”3-流形的问题。该奖项为支持研究生提供资金。Thurston，Perelman），每个3-流形都可以正则地分解为若干块，并且每一块都有一定的几何结构。 因此，在全局尺度上，可以将流形的拓扑信息与相应的几何形状相匹配。然而，在局部尺度上，也就是说，本质上，三维流形的几何和拓扑之间的联系还没有得到很好的理解。这对于双曲三维流形尤其如此，尽管其他类也有问题。这个项目的第一部分的目标是深入了解这一点，对于具有有限双曲或单纯体积的3-流形（因为这里不是所有的流形都是双曲的）。有项目嵌入曲面和弧在3流形，尖或关闭，解决领域内的知名projectures。 虽然这些问题是有趣的先验，该项目的第二部分是关于开发的结果和技术的应用，并旨在利用所获得的洞察力加深与其他领域的联系。特别是，（a）通过研究三维流形的表示簇，更好地理解三维流形的几何拓扑和代数几何之间的相互作用;（B）解决理论计算机科学和低维拓扑学的接口问题，该奖项反映了NSF的法定使命，并被认为是值得支持的，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。

A mathematical object known as a 3-manifold resembles our physical space if one views a small piece of it. However on a large scale, 3-manifolds have properties that may be quite different from what we are used to seeing and experiencing. Such manifolds are common objects in science, appearing for example in physics, astronomy or data science, as well as in various fields of mathematics. Here the Principal Investigator will study intrinsic properties of 3-manifolds, and how they relate to various areas of mathematics. Since many properties lend themselves to computer study and yield interesting algorithms, the PI will also look at questions that encompass how "hard" 3-manifolds are computationally. The award provides funds for supporting a graduate student.Due to Geometrization (W. Thurston, Perelman), every 3-manifold can be canonically decomposed into pieces, and each piece has a certain geometric structure. Thus, on a global scale, one can match topological information for the manifold with the respective geometry. However, on a local scale, that is, intrinsically, the connection between the geometry and topology of a 3-manifold is not well understood. This is particularly so for hyperbolic 3-manifolds, though there are questions for other classes as well. The goal of the first part of this project is to obtain a deep insight into this, for 3-manifolds with finite hyperbolic or simplicial volume (since not all manifolds here are hyperbolic). There are projects on embedded surfaces and arcs in 3-manifolds, cusped or closed, addressing well-known conjectures in the field. While these questions are interesting a priori, the second part of the project is concerned with applications of the developed results and techniques, and aims to use the obtained insight for deepening connections with other areas. In particular, (a) to better understand the interplay between geometric topology and algebraic geometry of 3-manifolds, through the study representation variety of a 3-manifold; (b) to address the problems on the interface of theoretical computer science and low-dimensional topology, through an overlap with complexity theory and computable analysis.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.