New developments and interaction between Algebraic Geometry and Integrable Systems

给出了具有未分枝奇点的曲线上稳定抛物联络的模空间的代数几何构造，并证明了Riemann-Hilbert对应的基本性质。这些结果揭示了非线性等单项微分方程解的几何Painleve性质，建立了联络的等单项变形几何，使我们能够深入研究微分方程组的相空间，如经典Painleve方程初始条件的Okamoto空间。结合高维双曲几何和镜像对称几何领域的进展，这些结果揭示了代数几何与可积系统之间的深刻联系。

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.