Geometry and Topology of 3-Manifolds

该项目旨在发展三维流形的跨学科研究，该三维流形基于主要与双曲几何相关的几种结构之间的联系，将几何和拓扑相互作用。事实上，我们已经推动了我们的项目与活动的拓扑研究大会。我们取得了一定的重大进展，在这三年的研究结果是相当令人惊讶的，比我们预期的。我与Mizushima和Tan一起对具有射影结构的曲面上的圆填充的研究，根据双曲3-流形的变形理论，阐明了其模空间的预期整体结构。Yoshida对SU（2）共形场论的研究，成功地建立了共形块基的一个显式描述，并探讨了三维流形的几何与拓扑之间的基本联系，如体积猜想。此外，Ohtsuki提出的3-流形拓扑不变世界中的整体图最近由Habiro和Le以最普遍的方式完成。此外，其他合作者也取得了扎实的进展，如Morita对曲面映射类群的研究，松本对叶理的研究，Sakuma对结和几何结构的研究，索马对有界上同调的研究。总之，我们的研究澄清了我们现在应该关注的对象，以找到在三维流形理论中观察到的许多结构之间相互作用背后的数学原理。此外，幸运的是，该主题作为第二部分得到了进一步研究的资助。

This project was aimed to develop the interdisciplinary study of 3-manifolds which interacts geometry and topology based on the connection between several structures related mainly with hyperbolic geometry. We have in fact promoted our project in conjunction with the activity of the Topology Research Congress.We made certain significant progresses during these three research years which turned out to be rather surprising than what we had expected. The study of myself together with Mizushima and Tan on circle packings on surfaces with projective structures has clarified an expected global structure of their moduli space in the light of the deformation theory of hyperbolic 3-manifolds. The study of Yoshida on SU(2) conformal field theory has established successfully an explicit description of a basis of conformal blocks, and has approached to the fundamental connection between the geometry and topology of 3-manifolds suggested for instance by the volume con-jecture. Also, the global diagram in the 3-manifold topological invariant world suggested by Ohtsuki was completed quite recently in the most universal way by Habiro and Le. In addition, there have been down-to-earth progresses by other collaborators such as Morita's study on the mapping class group of surfaces, Matsumoto's on foliations, Sakuma's on knots and geometric structures, Soma's on bounded cohomology,In conclusion, our research has clarified the object on which we should now focus for finding the mathematical principle behind the interaction between many structures observed in the 3-manifold theory. Note in addition, the theme was fortunately funded for further study as the part II.