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

M. Inaba和我构造了非奇异投影曲线上稳定抛物连接的模空间，该曲线具有固定形式类型的正则或不规则奇点作为非奇异代数变异体。此外，我们证明了这些情况下的Riemann-Hilbert对应给出了一个解析同构，这意味着等单项式或等stokes非线性微分方程具有几何Painleve性质。与Frank Loray一起构造了2阶稳定抛物连接在射影线上的模空间的一个很好的模型及其紧化。我们发现了连接模空间的两个拉格朗日颤振与射影空间及其对偶射影空间的经典对偶之间意想不到的关系。S. Szabo和我正在发展一个关于连接和希格斯束的表观奇点的理论，并试图理解几何朗兰兹对应。

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.