Geometry and topology of 3-manifolds II

本课题旨在观察3-流形上各种几何结构的相互作用，寻找其背后的物理规律等主要思想，促进3-流形相互作用几何与拓扑的跨学科研究。在2003年3月终止的第一个项目的基础上，目前的项目已经建立了4年，到目前为止我们已经度过了3年。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项目，因为鉴于当前工作主题的最新进展，这样做似乎是必要的。幸运的是，新项目被授予一个持续的项目，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，我们作为课题组的成员和合作者，通过进行研究交流、组织会议和发表研究报告来推动项目的发展。综上所述，我们课题组的成员取得了许多重要的进展，同时也为这一领域的进步做出了贡献。研究成果在包括国际会议在内的学术会议上发表。包括合作者的详细信息可以在附件的报告中找到。综上所述，根据项目的目的，我们越来越明确了用几何方法研究3-流形拓扑的需要。回顾近3年来，我们必须注意到历史性的进步，如明斯基-布洛克-加那利的末层猜想的解决，Agol和Calegari-Gabai的马登猜想的解决，以及汉密尔顿和佩雷尔曼的几何化方法，其中至少包括庞加莱猜想的解决。在此基础上，我们以“3流形的几何与不变量”为题，建立了后几何化。

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".