Geometry of Groups and Moduli Spaces (2)

黎曼曲面的模空间和图的模空间及其相关的模群，如曲面的映射类群和自由群的自同构群，是代数几何、复分析、微分几何、拓扑学和数学物理等多个数学分支的重要研究课题。本文主要从拓扑学的角度对这些模空间和模群进行了研究，得到了许多有趣的结果。此外，我们还得到了与三维流形、四维流形和横辛叶理论密切相关的新结果和猜想。我们还向包括数论在内的新的研究方向提出了一个深层次的问题。

The moduli space of Riemann surfaces and the moduli space of graphs, as well as their associated modular groups such as the mapping class groups of surfaces and the automorphism groups of free groups, are among the most important research subjects of diverse branches of mathematics including algebraic geometry, complex analysis, differential geometry, topology and mathematical physics. The present project investigated these moduli spaces and modular groups, mainly form the point of view of topology, and obtained many interesting results. Furthermore, we obtained new results as well as conjectures in the closely related theories of 3 and 4 dimensional manifolds and transversely symplectic foliations. We also proposed a deep problem towards new directions of our research including number theory.