Geometry and Topology of 3-Manifolds

这个项目的目的是基于主要与双曲几何有关的几个结构之间的联系，开展几何和拓扑相互作用的三维流形的跨学科研究。事实上，我们与拓扑学研究大会的活动一起推动了我们的项目。在这三个研究年中，我们取得了一些重大进展，结果比我们预期的要令人惊讶。本人与Mizushima和Tan在具有射影结构的曲面上的圆填充的研究，根据双曲三维流形的变形理论，阐明了他们的模空间的一个期望的整体结构。吉田在SU(2)共形场理论上的研究成功地建立了共形块基的显式描述，并探讨了三维流形的几何和拓扑之间的基本联系，例如体积猜想。此外，大树提出的三维流形拓扑不变世界中的全局图是最近由Habiro和Le以最普遍的方式完成的。此外，其他合作者也取得了扎实的进展，如森田关于曲面映射类群的研究，松本关于叶层的研究，Sakuma关于纽结和几何结构的研究，SOMA关于有界上同调的研究，总之，我们的研究澄清了我们现在应该关注的对象，以寻找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。注：此外，幸运的是，该专题作为第二部分得到了进一步研究的资金。

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.