Christopher Marks (U of California, Santa Cruz/MPI)

The main purpose of this talk is to explain the connection between vector-valued modular forms for SL(2,Z) and certain ordinary differential equations in the punctured unit disk, whose coefficient functions are holomorphic modular forms. An important tool arising in this context is a modular version of the familiar Wronskian from the classical ODE theory; I will discuss, for example, how this modular Wronskian provides a lower bound for the weight of a nonzero holomorphic form associated to a given representation of SL(2,Z). In the remaining time, I will describe my own work, which attempts to use this ODE theory -- in conjunction with some recent joint work with G. Mason, and a classification theorem of Tuba and Wenzl concerning representations of Artin's Braid group -- to give an algebraic classification of vector-valued modular forms, for irreducible representations of SL(2,Z) of dimension less than six.