Number theoretic and algebraic research of divergent formal solution

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In this research we introduced a new method based on Toeplitz operator theory, WKB analysis and generalized implicit function theorem in dealing with divergent formal power series solutions. Our method can be applied to global and local hypoelipticity of operators, index formula of a system of (partial and ordinary) differential operators on various function spaces and (local)normal forms of commuting systems of singular vector fields and diffeomorphisms. These results revealed transcendental phenomena in the theory of differential operators and the role or rapidly convergent iteration method with infinite loss of derivatives. This research will continue as a collaborations with Italian and English researchers. We will give short items of our results.1. Index formula of systems of irregular singular ordinary differential operators in formal Gevrey spaces via Toeplitz symbols.2. Necessary and sufficient conditions for irregular singular type partial differential operators with two independent variables.3. To determine critial Gevrey indices for solvability and regularity of degenerate parabolic operators via Newton polygones.4. Necessary and sufficient conditions for the reduction to their normal forms of commuting systems of resonant singular vector fields.5. WKB analysis to global solvability and regularity of mixed type operators on the torus.