New developments and interaction between Algebraic Geometry and Integrable Systems

给出了具有无分支奇点曲线上稳定抛物连接的模空间的代数-几何构造，并给出了黎曼-希尔伯特对应的基本性质。这些结果证明了非线性等同构微分方程的几何Painleve性质，并建立了连接的等同构变形的几何性质，使我们能够深入研究微分方程的相空间，如经典Painleve方程的Okamoto初始条件空间。这些结果与高维几何和镜像对称几何领域的进展一起，揭示了代数几何与可积系统之间的深刻关系。

We gave an algebro-geometric construction of the moduli spaces of stable parabolic connections over curves with unramified singularities, and showed the fundamental property of the Riemann-Hilbert correspondences. These results showed the geometric Painleve property of the nonlinear isomonodromic differential equations and established the geometry of isomonodromic deformations of connections, which enables us to investigate the phase space of differential equations deeply such as Okamoto's space of initial conditions for classical Painleve equations. Together with the progress in the field of higher dimensional birational geometry and the geometry related to mirror symmetry, these results reveal deep relations between algebraic geometry and integrable systems.