Geometry and topology of 3-manifolds II

该研究项目的目的是看到三维流形上各种几何结构的相互作用，找到它们背后的主要思想，如物理定律，并促进三维流形相互作用几何和拓扑学的跨学科研究。目前的项目是在2003年3月终止的主要项目的基础上建立的，为期4年，到目前为止我们已经花了3年时间。然后，我们决定通过将重点更多地转移到不变量上来预先更新该项目，因为根据当前工作主题的最新进展，似乎有必要这样做。幸运的是，新项目被授予了持续性项目，因此我们在这里报告我们过去3年的活动。在过去的3年里，我们作为研究小组的成员和合作者，通过进行研究交流，组织会议和发表研究报告来推动项目。综上所述，我们作为课题组的一员，取得了许多重大的进展，同时也为这一领域的发展做出了贡献。研究成果在包括国际会议在内的学术会议上发表。详细情况，包括合作者的情况，见所附报告。综上所述，根据课题的目的，我们对研究三维流形拓扑学的几何方法的要求越来越高，回顾近三年来的研究，我们必须注意到历史性的进展，如Minsky-Brock-Canary对终结层猜想的解决，Agol和Calegari-Gabai对马尔登猜想的解决，以及汉密尔顿和Perelman对几何化的探讨。其中至少包括庞加莱猜想的解。在此基础上，我们更新了课题“三维流形的几何与不变量”，建立了三维流形的后几何化。

The purpose of this research project is to see the interaction of various geometric structures on 3-manifolds, to find principal idea behind them such as laws in physics and to promote the study of 3-manifolds interacting geometry and topology interdisciplinary. The current project based on the primary one that terminated on March 2003 was set up for 4 years and we have spent 3 years so far. Then we have decided to pre-renew the project by shifting the stress more on invariants because it seems to be necessary to do so in the light of recent progress on the current working subject. Fortunately, the new project was awarded as a continuing one and thus we here report our activity of the last 3 years.In the last 3 years, we, the member of research group and collaborators, have promoted the project by doing research communication, organizing meetings and publishing research reports. As conclusion, we, the member of research group, have obtained a plenty of significant progress and simultaneously contributed the progress of this field. The research results were presented in the academic meetings including international ones. The details including one's by collaborators can be found in the attached report. In summary, according to the purpose of the project, we have darified what is needed for geometric methods to study topology of 3-manioflds more and more.Looking at last 3 years, we must notice historical progress such as the solution of the ending lamination conjecture by Minsky-Brock-Canary the solution of Marden's conjecture by Agol and Calegari-Gabai and the approach to the geometrization by Hamilton and Perelman, which at least includes the solution of Poincare conjecture. Based on these, we set up the post geometrization by renewing our project with entitling "The geometry and invariants of 3-manifolds".