Finite dimensional integrable structure in systems with infinite degree of freedom

本课题旨在寻找无限自由度系统中的各种有限维可积系统，并阐明其数学结构。研究了几何和不可积系统的相关问题。首席研究员在Whitham变形方程、等单调变形、超对称/拓扑规范理论和Calogero-Moser系统中产生的系统上获得了有趣的结果，所有这些都是相互关联的。首先，他找到了惠瑟姆变形方程的显式形式，用于黎曼球上等同系统的渐近描述。其次，利用可解自旋系统中的方法和共形场论，将这类同模系统推广到环面。第三，阐明了tau函数和Whitham变形在四维超对称和拓扑规范理论中的作用。最后，他考虑了与四维超对称规范理论密切相关的椭圆型卡罗伽罗-莫泽系统，并发现了这些系统的非自治类似物给出了环面上的等同构皮肤的一个例子。项目组其他成员所取得的成果包括：低维拓扑、次椭圆算子的随机分析、复杂动力系统、与非线性动力系统相关的超越数论、元胞自动机、智能体、无限退化的次椭圆和双曲方程、

This research project aimed to search for various finite dimensional integrable systems in systems with infinite degrees of freedom, and to elucidate their mathematical structures. Related issues on geometry and non-integrable systems were also investigated.The head investigator obtained interesting results on Whitham deformation equations, isomonodromic deformations, systems arising in supersymmetric/topological gauge theories and Calogero-Moser systems, all of which are mutually related. Firsly, he could find an explicit form of the Whitham deformation equations for asymptoci description of isomonodromic systems on the Rie-mann sphere. Secondly, an extension, to a torus, of such isomonodromic systems was achieved on the basis of methods in solvable spin sysmtes and conformal field theories. Thirdly, he clarified the roles that the tau functions and the Whitham deformations play in four dimensional supersymmetric and topolotical gauge theories. Finally, he considered the elliptic Calogero-Moser systems, which are also closely related to four dimensional supersymmetric gauge theories, and discoveref that a non-autonomous analogues of those systems give an example of isomonodromic dermations on a torus.The achievement due to other members of the project group respectively ranges over low-dimensional topology, stochastic analysis of hypoelliptic operators, complex dynamical systems, transcendental number theory related to nonlinear dynamical systems, celular automata, intelligent agents, hypoelliptic and hyperbolic equations with infnite degeneracy,