Geometric structures on low-dimensional manifolds

低维流形上的几何结构

• 批准号: 1405066
• 负责人: Tian Yang
• 金额: $ 14.07万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405066&HistoricalAwards=false
• 关键词: Geometric; structures; low; dimensional; manifolds

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.

本计画旨在研究曲面与三维流形的几何结构，以及它们在三维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三维流形上的双曲结构，以及量化某些几何结构的空间的方法。这个提议的部分智力价值来自于它位于其研究领域的前沿，并且还提供了不同数学海滩之间以及数学和理论物理之间的联系。在过去的十年中，三维流形上的大多数重要的公开问题都得到了解决，包括几何化猜想和虚纤维猜想。然而，三维流形离分类还很远。基于这些伟大的成就，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。斯坦福大学有一大批优秀的研究生，其中相当一部分对几何和拓扑学感兴趣。因此，PI计划教授一系列关于这些主题的研究生课程，并在未来与一些研究生开始合作。由于Thurston的几何化猜想和Perelman的证明，每一个紧致三维流形的内部都有一个典型的几何分解，其中大多数都有一个独特的双曲结构。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的第一种方法是使用角结构和体积优化来实现Casson的程序。这相当于找到一个理想的三角形的3流形，承认最大体积角结构。一个很好的家庭的候选人来自Lackenby的紧张的理想三角形，我们建议检查，如果其中一些承认的最大体积角结构。第二种方法是研究PI和合作者最近开发的三角化3-流形上的双曲锥度量。在这种方法中的关键对象是一个给定的三角形3-流形上的所有双曲锥度量的组合曲率的空间，属于它的零向量意味着双曲结构的存在。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Teichmuller空间和表面的字符变体，它们在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和合作者开发的弧和链接的螺旋代数之间的关系。在这个构造中的关键观察是，由测地弧的长度所满足的Penner的托勒密关系可以被看作是一个绞链关系。反过来，相关的对象适合布洛克-弗罗曼-卡尼亚-巴托辛斯卡和普日蒂茨基-西科拉的SL（2，C）-字符簇的绞链量化。最终，PI希望使用skein量子化来发展与非紧李群PSL（2，R）相关的拓扑量子场论。