Quadratic vanishing cycles, reduction curves and reduction of the monodromy group of plane curve singularities

The geometric monodromy of a plane curve singularity is a quasi-finite diffeomorphism. In this paper we locate the reduction curves of the geometric monodromy and the quadratic vanishing cycles of the singularity. An application to the geometric monodromy group is given.Comment: 20 pages, 16 figure

Sub-normal Solutions to Painleve's Second Differential Equation

In a recent paper, Aimo Hinkkanen and Ilpo Laine proved that the transcendental solutions to Painleve's second differential equation w"=a+zw+w^3 have either order of growth 3 or else 3/2. We complete this result by proving that there exist no sub-normal solutions (order of growth 3/2) other than the so-called Airy solutions