Rational singularities, Young diagram, Painleve equation

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1. Painleve equations and Special polynomialWe disocvered that the Painleve equations generate special polynomials. If we consider the motivation of the discovery of the Painleve equtions, it is surprising that they have combinatorial aspects. We presented conjectures on the special polynomials and proved them.2. Generalization of the Painleve equations based on the symmetriesNoumi considered that the conjectures should be solved in a natural frame work. To this end, he generalized the Painleve equations from the view point of theory of Lie algebra.3. Deformation of rational double points and Backhand tranhformationsWe proved that the Backhund transformations arise from deformation of rational double points.4. Infinite dimensional differential Galois theory and Painleve equationsWe applied our theory of infinite dimensional to the definition of the Painleve equations.