Stratification, exterior power structure, asymptotic behavior and connection coefficients for confluenthypergeometric functions

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In this research we planed to study confluent hypergeometric functions as deformations of weighted arrangements of hypersurfaces. In particular, we planed to analyse the behavior of confluent hypeigeometric functions via geometrical structures of the moduli space. We obtained the following results.1.Construction of cycles by confluence : We had already constructed 1-dimensional cycles. We generalized the theorem on convergence of the limit in the process of confluence. Also we tried to construct higher-dimensional cycles by confluence, and noticed that they should be constructed in the flag variety.2.Section of rigid local systems : We proved the existence of integral representations of sections of rigid local systems in a constructive way. The result is different from one by Katz et al. We also analysed Katz integral representations, and noticed that they are special cases of Selberg type integrals. We studied on the basis of the (co) homology.3.Evaluation of Stokes multipliers : By using integral representations of sections of rigid local systems, we evaluated the Stokes multipliers for the system obtaind by Laplace transform.4.Rigidity of local systems on higher dimensional spaces : We came to examples of local systems on higher dimensional spaces whose indexes of rigidity vary by coordinate changes. This implies that we need some intrinsic definition of the ridigity for the higher dimensional case.5.Application to the deformation theory of differential equations : The operations used in constructing rigid local systems can be applied to the study of deformations of differential equations. Some results are obtained.