Modular transformations, representation theory and physics

Abstract: 

One of the core consistency requirements of two-dimensional conformally invariant quantum field theory involves the modular group SL(2;Z).  This leads to many deep and beautiful relationships between the underlying algebraic structures and number theory.  The aim of this talk is to introduce some of these relationships through examples and discuss how recent research is forcing us to rethink the paradigms that these examples have suggested.