**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.