Study of Fuchsian system of differential equations from the view point of algebraic analysis and microlocal analysis

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(1) If we impose an irregularity condition due to N. Honda for a system of analytic linear differential equations (D-Module), we can define non-characteristic initial and boundary values for the corresponding Gevrey function or ultradistribution solutions. Moreover, under a (weak) hyperbolicity condition, we can prove unique solvability theorems for Cauchy and boundary value problems.(2) For any regular-specializable system, we can define general boundary values for extensible distribution or ultradistribution solutions under an irregularity condition due to H. Tahara.(3) By a joint work with T. Aoki and N. Honda, we can establish new cohomological representation and symbol theory for pseudodifferetial operators of infinite order in analytic category.