Toplogical Field theory, Moduli spaces of connections and Geometric Langlands correspondence

M.Inaba和我构造了非奇异射影曲线上的稳定抛物联络的模空间，其中形式类型固定的正则或不规则奇点为非奇异代数簇。此外，我们还证明了这些情形下的Riemann-Hilbert对应给出了一个解析同构，它蕴含着等轴线性或等Stokes非线性微分方程组的几何Painleve性质。我们和Frank Loray一起构造了射影直线上二阶稳定抛物联络的模空间及其紧化的一个很好的模型。我们发现了连通模空间的两个拉格朗日性质与射影空间及其对偶射影空间的经典对偶之间的意想不到的关系。S.Szabo和我正在发展一种关于联系和Higgs丛的表观奇点的理论，并试图理解几何上的朗兰兹对应。

M. Inaba and I constructed the moduli spaces of stable parabolic connections over a nonsingular projective curve with fixed formal type of regular or irregular singularities as non singular algebraic varieties. Moreover,we proved that the Riemann-Hilbert correspondence for these cases give an analytic isomorphism, which implies the geometric Painleve property for iso-monodromic or iso-Stokes non-linear differential equations. Together with Frank Loray, we constructed a good model of the moduli space of rank 2 stable parabolic connections over the projective line as well as its compactification. We found unexpected relations between two Lagrangian fibrations of the moduli space of connections and the classical duality of the projective space and its dual projective space. S. Szabo and I are developing a theory of apparent singularities of connections and Higgs bundles and try to understand the geometric Langlands correspondence.