General study of topology

拓扑学研究的第一个重大成果是Thom，Milnor，Smear等人在1960年的S提出的流形分类理论。这些研究在他们的学科或方法上或多或少都是恰当的拓扑学，但在70年代S做了各种尝试，将这些结果与其他数学领域联系起来，证明了这种尝试是成功的。在我们的研究计划中，拓扑学的大多数研究小组都是积极研究的，而不是在传统的领域。松本幸雄和他的团队研究了Seiberg-Witten理论，并给出了4维流形理论的一个新方面。动力系统群、奇点理论、分片理论等在传统研究的基础上，对代数簇的拓扑学、辛几何等主题进行了合作研究，并在变换群理论中对代数群的作用进行了较好的研究。在新方向的同伦理论研究中，如研究椭圆上同调与数论的关系，或研究某些模空间的同伦等。深谷健二和他的团队正在一个与上述大多数领域重叠的领域工作，并在拓扑场论、Arnold猜想和Novikov猜想方面取得了显著的成果。纽结理论群是拓扑学中最活跃的群之一，它除了用传统的方法外，还用一种新的数学物理方法得到了许多结果。他们将在1996年7月在东京举行的纽结理论国际会议上发表他们的研究结果。

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.