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.

拓扑学是研究位置和形状的数学。拓扑学各分支之间以及微分几何、代数、分析、数学物理和拓扑之间的相互作用促进了拓扑学近十年的发展。通过这项研究，我们希望能够进一步推动这一领域的发展。我们的研究领域包括：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、流形和叶状向量场的动力学研究、双曲三维流形和奇点双曲空间的分类理论、可微结构和辛4维流形上的结构，共形场论和维流形不变量，各种主束连接的模空间拓扑，泊松流形和接触流形，等变广义（共）同调理论和同伦理论，结不变量和链接及其分类，野生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，我们每年都会举办“拓扑研讨会”以及针对上述研究领域的许多会议。这些是与日本拓扑研究人员合作作为拓扑项目完成的。特别是，为了促进与其他领域研究人员以及研究生的互动，举办了“邂逅数学”系列会议。通过这些研究活动，下一个项目的新发展方向变得清晰起来。