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General study of topology

拓扑学一般研究

基本信息

批准号：
06302004
负责人：
NISHIDA Goro
金额：
$ 12.8万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Co-operative Research (A)
财政年份：
1994
资助国家：
日本
起止时间：
1994 至 1995
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-06302004/
关键词：
Knot theory Homotopy theory 4 dimensional manifold Theory of singularity Foliation Mathematical physics Transformation group Dynamical system 接触幾何学トポロジー特異点葉層構造結び目多様体

项目摘要

The first big achievement in the study of topology was the classification theory of manifolds by Thom, Milnor, Smale et al in 1960's. These researches were more or less properly topological ones in their subjects or methods, but in 70's various attempts to relate those results to other fields of mathematics were made and the Donaldson theory in 4 dim manifolds proved such attempts would be successful.In our research program, most of research groups in topology studied actively not being in the traditional fields. Yukio Matsumoto and his group studied the Seiberg-Witten theory and gave a new aspect in the theory of 4-dim manifolds. Groups of dynamical system, theory of singularities and foliation theory made not only traditional researches but also made cooperative study on some themes like topological study of algebraic variety or symplectic geometry, and in the theory of transformation groups good results on the action of algebraic groups were obtained. In the homotopy theory studies of new direction, for example, study of relations between elliptic cohomology and number theory or homotopical study of certain moduli spaces have been done. Kenji Fukaya and his group are working in a field overlapping most fields mentioned above, and have obtained remarkable results about the topological field theory, Arnold conjecture and Novikov conjecture. The group of knot theory, one of the most active group in topology, obtained numbers of results using a new method of mathematical physics as well as traditional methods. They will publish their results at the international conference on knot theory in Tokyo held in July 1996.

拓扑学研究的第一个重大成果是Thom，Milnor，Smale等人在20世纪60年代提出的流形分类理论。这些研究或多或少是正确的拓扑的主题或方法，但在70年代的各种尝试，这些结果与其他领域的数学和唐纳森理论在四维流形证明这样的尝试将是成功的。Yukio松本等人对Seiberg-Witten理论进行了深入的研究，在四维流形理论中开辟了一个新的领域。动力系统群、奇点理论和叶理理论在代数簇的拓扑研究、辛几何等方面不仅进行了传统的研究，而且还进行了合作研究，在变换群理论中对代数群的作用也取得了很好的结果。同伦理论研究的新方向，如椭圆上同调与数论关系的研究或某些模空间的同伦研究。福谷和他的研究小组所从事的研究领域与上述大多数领域重叠，并在拓扑场论、Arnold猜想和Novikov猜想等方面取得了令人瞩目的成果。纽结理论群是拓扑学中最活跃的群之一，它在传统方法的基础上，又用一种新的数学物理方法得到了许多结果。他们将于1996年7月在东京举行的国际纽结理论会议上发表他们的研究结果。