Geometric structure and integrable systems in mathematical physics

1.我们考虑了四维和五维超对称规范理论的瞬时和作为随机（平面）划分的模型，并应用可积层次的自由费米子方法推导了Seiberg-Witten曲线2。指出了一类特殊的保角映射变形过程与一类无色散可积系统之间的关系。本文还研究了这类可积系统的一种求解技术（hodograph法）。我们导出了几个新的无色散可积系统。作为可积层次的准经典极限。一个例子与改进的KP（和Toda）层次结构的q模拟有关。另一个例子是从双组分BKP层次结构中得到的。此外，我们还可以将所谓的零属普遍Whitham层次识别为KP层次的多分量模拟的拟经典极限。阐明了可解多体系统（Calogero-Moser系统、Sutherland系统及其变体）的一些新特征，如：平衡构型、形状不变性、创造-湮灭算符（如量子力学）、直接积分法（如经典力学）等。在Grassmann流形、非交换代数变量、低维流形的不变量、与双曲锥相关的超几何方程等方面得到了一些几何结果。我们还对随机矩阵、Seiberg-Witten可积系统、等同构变形、与向量束模空间相关的可积系统等进行了研究。

1.We considered the instanton sum of four and five dimensional supersymmetric gauge theories as a model of random (plane) partitions, and applied the method of free fermions for integrable hierarchies to derive the Seiberg-Witten curve.2.We pointed out a relation between a special class of deformation process of conformal mapping and a kind of dispersionless integrable systems. A solution technique (hodograph method) of such integrable systems was also studied.3.We derived several new dispersionless integrable systems.as quasi-classical limit from integrable hierarchies. An example is related to a q-analogue of the modified KP (and Toda) hierarchy. Another example is obtained from the two-component BKP hierarchy. Moreover, we could identify the so called genus-zero universal Whitham hierarchy to be quasi-classical limit of a multi-component analogue of the KP hierarchy.4.We elucidated some new features of solvable many body systems (the Calogero-Moser system, the Sutherland systems, and their variants) such as : equilibrium configuration, shape invariance, creation-annihilation operator (as quantum mechanics), direct integration method (as classical mechanics), etc.5.We obtained several geometric results on Grassmann manifolds, noncommutative algebraic varieties, invariants of low dimensional manifolds, hypergeometric equations related to hyperbolic cones, etc.6.We did some other researches on random matrices, Seiberg-Witten integrable systems, isomonodromic deformations, integrable systems related to a moduli space of vector bundles, etc.