Research of new developments in moduli spaces and integrable systems

在本研究中，稻叶、岩崎和齐藤构造了射影直线上具有至多n-正则奇点的2阶稳定抛物联络的模空间及其自然紧化，并证明了模空间的光滑性。此外，通过构造射影平面上n点补的基本群的表示的模空间，证明了从连通模空间到表示的模空间的Riemann-Hilbert对应是满射真双纯态射。这一结果表明，由连接的等单调变形得到的非线性微分方程具有Painleve性质。最近，稻叶成功地将这一结果推广到任意光滑代数曲线上具有至多任意阶正则奇点的稳定抛物型联络的情形。我们还期望在具有不规则奇点…的连接的情况下得到相同的结果在此基础上，利用Painleve VI方程的代数几何构造法，并将Riemann-Hilbert对应应用于代数曲面的遍历理论，岩崎和上原证明了Painleve VI方程的非线性单程几乎环的动力学是混沌的。Hosono和Doran利用振荡积分确定了一类Calabi-Yau超曲面的周期积分的微分方程式。此外，他们还解决了与积分有关的斯托克斯现象，从镜面对称的角度观察到了一种自然的理解。Yoshioka和Hiraku Nakajima证明了瞬子数的Nekrasov猜想。此外，与Lotha Gottsche一起，他们得到了有理曲面上Donaldson不变量的一个跨墙公式，并推广了结果。Fukaya，Ono，Ohta，哦，他们正在完成一本关于拉格朗日子流形的弗洛尔理论的大书。较少

In the present research, Inaba, Iwasaki and Saito constructed the moduli space of stable parabolic connections of rank 2 over the projective line with at most n-regular singular points and its natural compactification, and showed the smoothness of the moduli space. Moreover, constructing the moduli space of the representations of the fundamental group of the complement of the n-points in projective plane, we showed that Riemann-Hilbert correspondence from the moduli space of connections to the moduli space of the representations is a surjective proper bimeromorphic morphism. This result implies that the non-linear differential equations coming from isomonodromic deformations of connections have Painleve property. Recently Inaba succeeded in generalizing this result to the case of stable parabolic connections with at most regular singularities of any rank over any smooth algebraic curve. We expect further to obtain the same result in the case of connections with irregular singular point … More s. Noumi and Yamada gave a geometric description of elliptic difference Painleve equations with Ohta, Masuda, Kajiwara, and construct the elliptic hypergeometric solutions and its degeneration.By using the algebro-geometric construction of Painleve VI equations and applying the Riemann-Hilbert correspondence to ergotic theory of algebraic surfaces, Iwasaki and Uehara showed that the dynamics of almost loops of non-linear monodromy of Painleve VI equation are chaotic. Hosono and Doran determined the differential equation of the period integrals of certain class of Calabi-Yau hypersurfaces by oscillatory integrals. Moreover solving the Stokes phenomena associated to the integral, they observed a natural understanding from the view point of Mirror symmetry. Yoshioka and Hiraku Nakajima proved the Nekrasov conjecture for the instanton numbers. Moreover with Lotha Gottsche, they obtained a wall-crossing formula for Donaldson invariants for the rational surfaces, and extend the results. Fukaya, Ono, Ohta, Oh, are completing a big book of Floer theory of Lagrangian submanifolds. Less