Topological theory of local systems having irregular singularities

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The aim of this research was to study the topological structure of integral representations of confluent hypergeometric functions, and to clarify the relation between the topological structure and the analytic behavior of the functions.When the local system corresponding to the integral is defined over a 1-dimensional space, we have constructed good bases of twisted (co) homology group. Since we have constructed them by confluence, the twisted Riemann's period relation for regular singular hypergeometric functions can be extended to ones for confluent hypergeometric functions.On the relation between twisted cycles and asymptotic behaviors of functions given by the integral over the cycles, we noticed that, if a cycle which give a pure asymptotic behavior suivives in the process of confluence, the cycle after confluence also will give a pure behavior. Using this understanding, we can evaluate several connection coefficients and global behaviors for confluent hypergeometric functions.For local systems defined over multi-dimensional spaces, we have constructed bases of twisted (co) homology groups by using the exterior power structure. We noticed that the asymptotic behavior of a function given by a vanishing cycles can be discribed by functions corresponding to degenerate arrangements of hyperplanes.The study of confluent hypergeometric functions corresponding to degenerate arrangements of hypersurfaces will be substantial in future, and they will be closely related to rigid local systems and GKZ hypergeometric functions.