Intrinsic Geometry, Topology, and Complexity of 3-Manifolds

一个被称为3流形的数学对象，如果你观察它的一小部分，就会发现它与我们的物理空间很相似。然而，在大范围内，3-流形的性质可能与我们习惯看到和经历的完全不同。这种流形是科学中常见的对象，例如出现在物理学、天文学或数据科学中，以及数学的各个领域。在这里，首席研究员将研究3流形的内在性质，以及它们与数学各个领域的关系。由于许多性质适合计算机研究并产生有趣的算法，PI也将研究包含“难”的3-流形计算的问题。该奖项为研究生提供资助。由于几何化（W. Thurston, Perelman），每个3流形都可以被正则分解成块，并且每个块都具有一定的几何结构。因此，在全局范围内，可以将流形的拓扑信息与相应的几何结构相匹配。然而，在局部尺度上，即本质上，3流形的几何和拓扑之间的联系并没有得到很好的理解。对于双曲3-流形尤其如此，不过其他课程也有问题。本项目第一部分的目标是深入了解有限双曲或简单体积的3-流形（因为不是所有的流形都是双曲的）。在3流形的嵌入表面和弧上有项目，尖顶或闭合，解决了该领域众所周知的猜想。虽然这些问题是先验的，但项目的第二部分关注的是已开发成果和技术的应用，并旨在利用获得的见解加深与其他领域的联系。特别是(a)通过研究3流形的表示变化，更好地理解3流形的几何拓扑与代数几何之间的相互作用；(b)通过与复杂性理论和可计算分析的重叠，解决理论计算机科学和低维拓扑学界面上的问题。该奖项反映了美国国家科学基金会的法定使命，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。

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.