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Various Aspects of Topology

拓扑的各个方面

基本信息

批准号：
12304003
负责人：
TSUBOI Takashi
金额：
$ 31.17万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12304003/
关键词：
Algebraic Topology Topology Foliations Mapping Class Groups General Topology Three Dimensional Manifolds Sinqularity Theory Gauge Theory

项目摘要

Topology is the mathematics to study the position and the shape. The development of topology in the last decade was promoted by the interaction between the branches of topology as well as that between differential geometry, algebra, analysis, mathematical physics and topology. In this research, we wish to promote the development much more. We did the researches in the following fields : classifying theory of singularities of mappings and algebraic varieties, various group actions on manifolds and simplicial complexes, the action of the mapping class groups of the surfaces on their Teichmuller spaces, the dynamical study of complex analytic maps, the dynamical study of vector fields on manifolds and foliations, the classifying theory of hyperbolic 3 dimensional manifolds and hyperbolic spaces with singularities, the differentiable structures and symplectic structures on 4 dimensional manifolds, the conformal field theory and invariants of dimensional manifolds, the topology of the moduli spaces of connections of various principal bundles, the Poisson manifolds and contact manifolds, equivariant generalized (co)homology theories and homotopy theories, invariants of knots and links and their classification, general topology theory for wild spaces. These researches were done successfully with their own results. In order to promote the interaction between these researches we held "Topology Symposium" each year as well as many conferences on the above research fields. These are done as Topology Projects in the collaboration with researchers in topology in Japan. In particular, the series of the meetings "Encounter with Mathematics" were held in order to promote the interaction with researchers in other fields as well as graduate students. By these research activities, the direction of new development for the next project became clear.

拓扑学是研究物体的位置和形状的数学。拓扑学各分支之间的相互作用，以及微分几何、代数、分析、数学物理与拓扑学之间的相互作用，促进了拓扑学在近十年的发展。希望通过本课题的研究，进一步推动我国生物医学的发展。我们在以下领域进行了研究：映射和代数簇的奇点分类理论，流形和单纯复形上的各种群作用，曲面的映射类群在其Teichmuller空间上的作用，复解析映射的动力学研究，流形和叶理上向量场的动力学研究，三维双曲流形和双曲空间的奇异性分类理论，四维流形上的可微结构和辛结构，共形场论和维流形的不变量，各种主丛联络的模空间的拓扑，Poisson流形与切触流形，等变广义（上）同调理论与同伦理论，纽结与链环的不变量及其分类，Wild空间的一般拓扑理论。这些研究都取得了成功，并取得了自己的成果。为了促进这些研究之间的互动，我们每年举办“拓扑研讨会”以及上述研究领域的许多会议。这些都是与日本拓扑学研究人员合作完成的拓扑学项目。特别是，举行了一系列会议“与数学相遇”，以促进与其他领域的研究人员以及研究生的互动。通过这些研究活动，下一个项目的新发展方向变得清晰。