Geometric structures on low-dimensional manifolds

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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.