Research on several geometry from the view point of sinuglarity theory

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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.