Geometric structure and integrable systems in mathematical physics

点击翻译按钮获取中文摘要

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.