Interaction between Geometry and various mathematics

1995年：Fukaya和Y.oh研究了余切丛上的亏格零开弦，并发现它等价于有理同伦理论。Fukaya也研究了高等亏格的情况，发现它与Chern-Simons微扰理论有关。作为基于这一资助的主要活动，我们举办了一个名为“无限群与几何”的研讨会，并发表了一份250多页的报告。这对几何学家了解几何群论的最新发展有很大的帮助。1996：Fukaya和K.Ono证明了Arnold关于周期哈密顿系统的周期轨道数的猜想的同调版本。Fukaya还写了一篇关于无穷范畴的同调代数及其在3个有边界流形的Floer同调中的应用的论文。作为这一授权的主要活动，Fukaya，Furuta等人举办了一系列的理论物理研讨会(特别是超对称规范理论和超弦理论)。在这些研讨会的基础上，我们在3月底举办了关于Seiberg Witten理论等的研讨会，以提高数学家对规范理论、弦理论等物理学的理解，希望将其应用于未来的数学研究。我们还发表了250页关于规范理论、弦理论等的报告。这是数学家就这一主题撰写的为数不多的文章之一。

1995 : Fukaya with Y.Oh studied genus zero open string on the cotangent bundle and ind that it is equivalent to the rational homotopy theory. (This work will be published from Asian J.Math.)Fukaya also studied higher genus case and find that it is related to Chern-Simons perturbation theory.As a main activity based on this grant, we had a workshop "Infinite group and geometry" and publish a report over 250 pages. It helps a lot for geometers to know the recent development of geometric group theory.1996 : Fukaya with K.Ono proved a homology version of Arnold's conjecture on the number of periodic orbit of the periodic hamiltonian system. They also gave a construction of Gromov-Witten in variant for general symplectic manifolds.Fukaya also wrote a paper on the homological algebra of A infinity category and its application to the Floer homology of 3 manifolds with boundary.As a main activity based on this grant, Fukaya, Furuta etc have a series of seminars on theoretical physics, (especially supper symmetric gauge theory and supper string theory). Based on these seminars we had a workshop at the end of March on Seiberg Witten theory etc to improve mathematician's understanding of the physics of Gauge theory, String theory etc. hoping to apply it to the future research in Mathematics.We also published 250 pages of reports on Gauge theory String theory etc. This is one of a few article writtne by Mathematician on this subjects.