Systems of Holonomic Differential Equations

1.超几何微分方程组：(1)得到了(k-1)维非退化二次超曲面和n个超平面的积分所满足的微分方程组。(2)几何地阐明了一般合流超几何函数在Grassmann流形上的合流过程。2.Painleve系统：(1)我们发现了Painleve系统初始条件空间之间的合流过程。(2)通过确定与Painleve系统相关的哈密顿向量场的不同因子，证明了除已知经典函数外，第二类和第四类Painleve函数的不可约性。3.量子群和Q-函数：(1)我们将一族量子复射影空间实现为与一族上理想族相关的量子齐次空间，并用Askey-Wilson多项式来表示带状球函数。(2)通过构造Macdonald对称多项式的提升算子，解决了Macdonald关于(q，t)-Kostka系数的积分猜想。4.渗流问题：当+自旋和-自旋在边界上随机分布时，我们得到了谱间隙的阶数。5.我们证明了三维曲面的仿射几何。射影空间在某些不变量退化的情况下也起作用，我们对射影齐次曲面进行了分类。

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.