Geometric structures on low-dimensional manifolds

本课题旨在研究曲面和三维流形上的几何结构及其在研究三维流形拓扑中的应用。在这个项目中，PI提出了两种可能的方法来计算三维流形上的双曲结构，以及一种量化某些几何结构的空间的方法。该提议的部分学术价值来自于它处于其研究领域的前沿，也提供了数学不同领域之间以及数学和理论物理之间的联系。在过去的十年里，三维流形上的大多数主要公开问题都得到了解决，包括几何猜想和虚拟Fibered猜想。然而，三维流形仍远未被归类。在这些巨大成就的基础上，PI计划在理解三维流形的几何和拓扑方面取得进一步的进展。作为一个更广泛的影响，PI计划让研究生参与他的工作。斯坦福大学拥有一大批优秀的研究生，其中相当一部分人对几何和拓扑学感兴趣。因此，PI正计划就这些主题教授一系列研究生课程，并应在未来开始与一些研究生合作。这项工作还涉及到与大学外的人的合作。由于瑟斯顿的几何化猜想和佩雷尔曼的证明，每个紧致三维流形的内部都有一个几何碎片的典范分解，其中大多数几何碎片具有独特的双曲线结构。反过来，显式计算双曲结构对于理解三维流形的几何和拓扑是必要的。PI的第一种方法是利用角度结构和体积优化来实现卡森程序。这相当于找到了允许最大体积角结构的3-流形的理想三角剖分。一个很好的候选者家族来自Lackenby的严格理想三角剖分；我们建议检查他们中的一些人是否承认最大体积角结构。第二种方法是研究PI和他的合作者最近发展起来的三角剖分三维流形上的双曲锥度量。该方法的关键目标是给定的三角剖分三维流形上所有双曲锥度量的组合曲率空间，其中的零向量蕴含着双曲结构的存在。量子不变量及其与经典几何结构的关系为理解三维流形的几何和拓扑提供了另一种可能的途径。这种关系的核心是曲面的Teichmuller空间和特征变量，它们的本质包含了曲面的所有几何信息，并且是量化的候选者。PI的方法是基于量子Teichmuller空间和由PI和合作者开发的弧和链接的skein代数之间的关系。在这一结构中的关键观察是，由测地弧的波长长度所满足的Penner托勒密关系可以看作是一种斜交关系。反过来，相关的对象适合布洛克-弗罗曼-卡尼亚-巴托辛斯卡和Przytycki-Sikora的画面，用于SL(2，C)字符种类的倾斜量化。最终，PI想要发展一个与非紧致李群PSL(2，R)有关的拓扑量子场论，使用斜率量子化。

This project aims at studying the geometric structures on surfaces and 3-manifolds, and their applications in studying the topology of 3-manifolds. In this project, the PI proposes to develop two possible approaches to calculate the hyperbolic structure on 3-manifolds, and an approach of quantizing the spaces of certain geometric structures. Part of the intellectual merit of the proposal comes from the fact that it lies at the frontier of its research area, and also provides a link between different beaches of mathematics, and between mathematics and theoretical physics. During the last ten years, most major open problems on 3-manifolds have been solved, including the Geometrization Conjecture and the Virtually Fibered Conjecture. However, 3-manifolds are still far from being classified. Based on those great achievements, the PI plans on making a further progress in understanding the geometry and topology of 3-manifolds. As a broader impact, the PI plans on involving graduate students in his work. Stanford University has a large number of excellent graduate students, a good amount of which are interested in geometry and topology. As such, the PI is planning to teach a series of graduate courses on the topics, and should start collaborations with some of the graduate students in the future. This work also involves collaborations with people from outside the University.Due to Thurston's Geometrization Conjecture and Perelman's proof, the interior of every compact 3-manifold has a canonical decomposition into geometric pieces, most of which have a unique hyperbolic structure. In turns, explicitly calculating the hyperbolic structure becomes necessary to understand the geometry and topology of 3-manifolds. The PI's first approach is to realize Casson's program using angle structures and volume optimization. This amounts to finding an ideal triangulation of the 3-manifold that admits the maximum volume angle structure. A good family of candidates come from Lackenby's taut ideal triangulations; and we propose to examine if some of them admit the maximum volume angle structure. The second approach consists in studying the hyperbolic cone metrics on triangulated 3-manifolds developed recently by the PI and a collaborator. The key object in this approach is the space of combinatorial curvatures of all hyperbolic cone metrics on a given triangulated 3-manifold, the zero vector belonging to which implies the existence of the hyperbolic structure. The quantum invariants and their relationship with classical geometric structures provides another possible approach to understand the geometry and topology of 3-manifolds. At the heart of this relationship is the Teichmuller space and character varieties of surfaces, which in their nature contain all the geometric information of the surfaces and are candidates for quantization. The PI's approach is based on a relationship between the quantum Teichmuller space and the skein algebra of arcs and links developed by the PI and a collaborator. The key observation in this construction is that Penner's Ptolemy relation satisfied by the lambda-lengths of geodesic arcs could be viewed as a skein relation. In turns, the related objects fit into the picture of Bullock--Frohman--Kania-Bartoszynska and Przytycki--Sikora for the skein quantization of the SL(2,C)-character variety. Ultimately, the PI wants to develop a topological quantum field theory associated to the non-compact Lie group PSL(2,R) using the skein quantization.