Geometric structure and integrable systems in mathematical physics

1.将四维和五维超对称规范理论的瞬子和作为随机（平面）分拆模型，应用可积族的自由费米子方法导出Seiberg-Witten曲线. 2.指出一类特殊的共形映射形变过程与一类无色散可积系统之间的关系.研究了这类可积系统的一种求解方法（速端图法）。3.从可积族出发，导出了几个新的无色散可积准经典极限。systems.as一个例子是与修改的KP（和户田）层次的q-类似物。另一个例子是从两个组件的BKP层次。此外，我们还可以将所谓亏格为零的万有Whitham族确定为KP族的多分量类似物的准经典极限（Calogero-Moser系统，Sutherland系统及其变体），例如：平衡位形，形状不变性，创生-湮没算符（如量子力学），直接积分法5.得到了Grassmann流形、非交换代数簇、低维流形的不变量、与双曲锥有关的超几何方程、6.对随机矩阵、Seiberg-Witten可积系统、等单值变形、向量丛模空间的可积系统等作了一些研究。

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.