Theory of vector-valued modular forms

Modular forms and their generalizations are one of the most central
concepts in number theory. It took almost 300 years to cultivate the
mathematics lying behind the classical (i.e. scalar) modular forms. All
of the famous modular forms (e.g. Dedekind eta function) involve a
multiplier, this multiplier is a 1-dimensional representation of the
underlying group. This suggests that a natural generalization will be
matrix valued multipliers, and their corresponding modular forms are
called vector valued modular forms. These are much richer mathematically
and more general than the (scalar) modular forms. In this talk, a story
of vector valued modular forms for any genus zero Fuchsian group of the
first kind will be told. The connection between vector-valued modular
forms and Fuchsian differential equations will be explained.