Systems of Holonomic Differential Equations

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1.Hypergeometric differential equations : (1) We obtained the system of differential equations satisfied by integrals associated with a non-degenerated quadratic hypersurface and n hyperplanes in the (k-1)-dim. complex projective space and we studied certain sym-metries of the system.(2) We made clear geometrically the processes of confluence of general confluent hypergeometric functions on Grassmann manifolds.2.Painleve systems : (1) We found processes of conflunce among the spaces of initial con-ditions of Painleve systems. The processes are compatible with the well known ones.(2) We proved the irreducibility of the second and the fourth Painleve functions exept the known classical functions, by determining ivariant devisors of Hamiltonian vector fields associated with the Painleve systems.3.Quantum groups and q-functions : (1) We realized a family of quantum complex projective spaces as one of quantum homogeneous spaces associated with a family of coideals, and we expressed the zonal spherical functions in terms of Askey-Wilson polynomials. (2) We solved afflrmatively the integrality conjecture of Macdonald for the (q, t)-Kostka coefficients, by constructing raising operators for Macdonald's symmetric polynomials.4.Percolation problem : We obtained the order of the spectral gap in the case where + and - spins are randomly distributed on the boundary. The order is the same as that in the case where there are on spins no the boundary.5.We showed that the affine geometry of surfaces in the 3-dim. projective space works aiso in the case where some invariants are degenerated, and we classified projective homogeoenus surfaces.