Geometry and topology of integrable systems, with computer-aided experimentation and visualization

几何学和拓扑学的研究问题使用可积系统理论的方法进行了研究。这些研究包括常平均曲率（CMC）曲面、等参流形、拉格朗日子流形、调和映射和量子上同调的分类。主要研究者获得了量子上同调的D-模结构的结果，并开始系统地研究量子上同调与可积系统之间的关系，计算机实验在该项目中发挥了重要作用。MAPLE用于微分算子和微分方程的计算，3D-XplorMath用于晶格模型模拟，CMC-Lab用于研究CMC表面。支持或部分支持了几次国内和国际研究会议，集中讨论“几何与可视化”这一主题。来自德国和美国的研究人员在这些会议上介绍了计算机实验和可视化的新技术。在日本传播和推广这种计算机技术是该项目的一个重要目标。研究助理和几名研究人员编写了日语文档，并在会议上演示了软件。作为该项目的一部分，开发了两个网站：“geom”（日本数学学会几何学分会）和“tmugs”（东京都区几何学研究活动）。在项目期快结束时举办了三次“计算机辅导班”。这些教程是基于在项目期间获得的经验，并将作为该研究领域和相关的人力资源在日本的未来发展的模式。

Research problems in geometry and topology were investigated using methods from the theory of integrable systems. These included research on the classification of constant mean curvature (CMC) surfaces, isoparametric manifolds, Lagrangian submanifolds, harmonic maps, and quantum cohomology. The principal investigator obtained results on the D-module structure of quantum cohomology, and began a systematic study of relations between quantum cohomology and integrable systems.Computer experiments played an important role in this project. MAPLE was used for computations with differential operators and differential equations, 3D-XplorMath was used for lattice model simulations, and CMC-Lab was used to study CMC surfaces. Several domestic and international research conferences were supported or partially supported, concentrating on the theme of "geometry and visualization". New techniques of computer experimentation and visualization were introduced at these conferences by researchers from Germany and the USA.Dissemination and promotion of such computer techniques in Japan was an important goal of this project. The Research Assistant and several investigators wrote Japanese documentation and demonstrated software at conferences. Two websites were developed as part of this project : "geom" (for the Geometry Section of the Mathematical Society of Japan), and "tmugs" (for geometry research activities in the Tokyo Metropolitan area). Three "computer tutorials" were held towards the end of the project period. These tutorials were based on experience gained during the project, and will serve as a model for future development of this research area and associated human resources in Japan.