Monday October 7, 2002

Speaker: Charles Doran (Columbia)

Title: On a theme of variations: Algebraic geometry, differential equations, and the mirror map

Abstract: We begin by recalling the classical theory of elliptic integrals for the periods of an elliptic curve. Within this classical setting we describe how algebraic variation of elliptic curves in a family defines the period mapping, the corresponding Picard-Fuchs differential equation, and the "mirror map". By applying the theory of monodromy-preserving (isomonodromic) variation of Fuchsian ordinary differential equations to these Picard-Fuchs equations, we obtain from certain families of elliptic surfaces algebraic solutions to Painleve VI equations, Garnier systems, and their generalizations, which in turn describe the algebraic variation of the mirror map.