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Research on several geometry from the view point of sinuglarity theory

奇点理论视角下的几种几何学研究

基本信息

批准号：
15204002
负责人：
IZUMIYA Shyuichi
金额：
$ 32.2万
依托单位：
HOKKAIDO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

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.

该项目的目的是改进拓扑和微分几何中的几个领域，这些领域是奇点理论的根源，为奇点理论的应用建立新的领域，并将奇点理论的最新成果应用于拓扑和微分几何。为此，我们对以下几个问题进行了研究，并取得了许多成果：高阶接触几何、辛拓扑、动力系统和叶理理论、纽结理论、离散群上的拓扑、拓扑动力学和一般拓扑、同伦代数的几何应用、群的拓扑同态、拓扑场论和黎曼几何，为了交流以上研究成果，我们每年组织全国拓扑学会议。我们还就上述主题组织了几次会议。在奇点理论的研究中，我们研究了辛奇点理论、光滑映射的整体奇点理论、代数曲线的基本群以及奇点理论在微分几何中的应用，并由此确立了本课题的目的。特别是2003年9月与北海道大学COE计划“通过奇点的非线性数学”共同组织了第12届MSJ-IRI国际研究所“奇点理论及其应用”。作为会议的结果，我们决定了研究项目的目的。在最后一年，我们与上述COE计划共同组织了“奇点理论”特别月（2005年9月至2005年12月）。在具体的月份中，我们确认了项目的结果和未来的问题。