Geometric structures on low-dimensional manifolds

本课题旨在研究曲面和3流形上的几何结构及其在3流形拓扑研究中的应用。在这个项目中，PI提出了两种可能的方法来计算3流形上的双曲结构，以及一种量化某些几何结构空间的方法。该提议的部分智力价值来自于它位于其研究领域的前沿，并且还提供了数学不同海滩之间以及数学与理论物理之间的联系。近十年来，关于3-流形的主要开放问题已经得到了解决，其中包括几何化猜想和虚纤维猜想。然而，3流形还远未被分类。基于这些伟大的成就，PI计划在了解3流形的几何和拓扑方面取得进一步的进展。作为更广泛的影响，PI计划让研究生参与到他的工作中来。斯坦福大学有大量优秀的研究生，其中有相当一部分人对几何和拓扑学感兴趣。因此，PI正计划教授一系列关于这些主题的研究生课程，并应该在未来开始与一些研究生合作。这项工作还包括与大学以外的人合作。根据Thurston的几何化猜想和Perelman的证明，每个紧致3流形的内部都有正则分解成几何块，其中大多数具有唯一的双曲结构。反过来，明确地计算双曲结构对于理解3流形的几何和拓扑是必要的。PI的第一种方法是使用角度结构和体积优化来实现卡森的程序。这相当于找到一个理想的三角形的3-歧管，允许最大的体积角结构。一个很好的候选家族来自拉肯比的紧致理想三角；我们建议检验其中一些是否允许最大体积角结构。第二种方法是研究由PI和合作者最近开发的三角化3流形上的双曲锥度量。该方法的关键对象是给定三角化3-流形上所有双曲锥度量的组合曲率空间，该空间的零向量表明双曲结构的存在性。量子不变量及其与经典几何结构的关系为理解3流形的几何和拓扑结构提供了另一种可能的途径。这种关系的核心是Teichmuller空间和表面的特征变化，它们本质上包含了表面的所有几何信息，并且是量化的候选者。PI的方法是基于量子Teichmuller空间和由PI和合作者开发的圆弧和链路的串代数之间的关系。在这个构造中，关键的观察是由测地线弧的λ长度所满足的Penner's Ptolemy关系可以看作是一个skein关系。反过来，相关对象符合Bullock- Frohman- Kania-Bartoszynska和Przytycki- Sikora对SL(2,c)-字符多样性的串量化的描述。最终，PI希望利用束线量子化发展与非紧李群PSL（2，R）相关的拓扑量子场论。

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.