Research on several geometry from the view point of sinuglarity theory

该研究项目的目的是改进拓扑和微分几何中作为奇点理论根源的几个领域，为奇点理论的应用建立新的领域，并发展奇点理论在拓扑和微分几何中的最新成果。为此，我们在高阶接触几何、辛拓扑、动力系统与叶理理论、结理论、离散群拓扑、拓扑动力学与一般拓扑、同局部代数的几何应用、微分同构群、拓扑场论与黎曼几何、四维拓扑等方面进行了研究，取得了许多成果。为了交流上述研究成果，我们每年组织召开全国拓扑学会议。我们还就上述主题组织了几次会议。在奇点理论的研究方面，我们研究了辛奇点理论、光滑映射全局奇点理论、代数曲线基本群以及奇点理论在微分几何中的应用。因此，我们确定了该项目的目的。特别是在2003年9月，我们与北海道大学COE项目“基于奇点的非线性数学”联合举办了第12届MSJ-IRI国际研究所“奇点理论及其应用”。作为会议的结果，我们决定了这个研究项目的目的。在最后一年，我们与上述COE项目联合组织了“奇点理论”专月（2005年9月- 2005年12月）。我们在特殊月份确认了项目的结果和未来的问题。

The purposes of the reasearch project were improvements of several areas in Topology and Differential Geometry which are roots of Singularity theory, constructions of new field for the applications of Singularity theory and developments of the applications of recent results of Singularity theory to Topology and Differential Geometry. For the purpose, we made a study on the following subjects and got many resutls : The higher order contact geomety, Symplectic Topology, Dynamical system and the theory of foliations, Knot theory, Topology on discrete groups, Topological dynamics and General Topology, geometric applications of Homotopical Algebra, groups of diffeomorphisms, Topological field theory and Riemannian geometry, Four dimensional topology.I order to exchange the above researches, we organized the national conference on Topology on each year. We also organized several conferences on the above subjects. As researches on Singularity theory, we have studied Symplectic singularity theory, Global theory on singularities of smooth mappings, Fundamental group of algebraic curves and applications of singularity theory to Differential Geometry.As a concequence, we established the puropose of the project. Especially, we organized the 12th MSJ-IRI International research institute "singularity theory and its applications" on September,2003 jointly with COE program of Hokkaido University "Nonlinear mathematics via singularities". We decided the puropose of the research project as the result of the conference. In the final year, we organized the special months "Singularity theory" (September,2005-12,2005) jointly with the above COE program. We confirmed the results of the project and the future problems in the speical months.