Mathematical theory of dynamical systems of Painleve type

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1. We have extended the theory of symmetric form for q-Painleve IV equation with (A_2+A_1)^<(1)> affine Weyl group symmetry to formulate the q-KP hierarchy. By the similarity reduction we have constructed the hierarchy of discrete systems with (A_<m-1>+A_1)^<(1)> affine Weyl group symmetry, and further, that with (A_<m-1>+A_<n-1>)^<(1)> affine Weyl group symmetry.2. We have presented a formulation of the elliptic Painleve equation and its generalizations, the former of which is located on the top of all the Painleve systems of second order. Namely, we have formulated the time evolution and the Backlund transformations as Cremona transformations on the configuration space of generic points in the complex projective space, and given their realization as birational transformations parametrized by the theta functions on the level of the τ functions. We have also formulated the time evolution as the addition formula on the moving pencil of plane cubic curves, and clarified the geometric mea … More ning of the Hamilitonians of the Painleve differential equations.3. Applying the above formulation, we have constructed the simplest hypergeometric solutions to the elliptic Painleve equation and all the q-Painleve equations, and completed the list of coalscent diagram of hypergeometric functions, starting from the elliptic hypergeometric function _<10>E_9 to the q-Airy function.4. Combining the algebro-geometric formulation of the Painleve VI equation and the ergodic theory of birational mapping on the algebraic surface by using the Riemann-Hilbert correspondence, we have shown that the nonlinear monodromy of the Painleve VI equation is chaotic along almost all the loops.5. We have shown that the entries of Hankel determinant formula for the solutions of the Painleve differential equations arise as coefficients of asymptotic expansion of the ratio of solutions to the auxiliary linear problem, and that this phenomenon originates from the structure of the KP hierarchy.6. Applying the above results we have discussed some new extentions or new solutions to the discrete and ultradiscrete Toda equation. Less