Differential equations for periods and super string and susy gauge theories

在这个项目中，三位研究者都是在密切合作下独立进行研究的。如果我们互相鼓励，我们就能取得比最初目标更大的成功。Yamada研究了二维量子引力的可积结构与奇点理论之间的关系。他发现了KP方程、Drinfeld-Sokolov方程和Painleve方程之间的深刻联系，并在Painleve方程及其对称性的理论上取得了成功的结果。这一结果有望应用于数学物理中的其他问题. S.K.Yang用明确的例子系统地研究了N = 2 susy规范理论中的Seiberg-Witten曲线（或更一般的几何）。阐明了规范群与奇异性理论的深刻关系，确定了特殊规范群的几何结构.细野慎太郎继续了镜像对称性的研究。他作出了重要贡献的评价Gromov-Witten不变量的椭圆或K3纤维卡-丘流形。从理论和计算的角度来看，高亏格曲线的计数也取得了一些进展。

In this project, each of the three investigators carried out his research independently with in close co-operation. Inspiring with each other, we can obtain successful results more than our initial aim.1. Y.Yamada studied the relation between the integrable structure in the two dimensional quantum gravity and singularity theories. He found deep connections between the KP equations, Drinfeld-Sokolov equations and Painleve equations, and obtained successful results on the theory of Painleve equations and their symmetries. This result is expected to be applied to other problems in mathematical physics.2. S.K.Yang investigated the Seiberg-Witten curves (or more general geometries) in N = 2 susy gauge theories in systematic way with explicit examples. He clarify the profound relation with the singularity theories and determine the geometrical structures for exceptional gauge groups.3. S.Hosono has continued the research on the mirror symmetry. He made important contribution for the evaluation of the Gromov-Witten invariants for elliptic- or K3-fibered Calabi-Yau manifolds. There is also some progress in counting the higher genus curves from theoretical and computing points of view.