Abstract: We will construct Buser's example of two Schreier graphs that are isospectral but not isomprphic. The proof of this has a wonderful connection to representation theory, and some useful pre-trace formulae will be reviewed. We will then start the construction of isospectral, non-isometric planar domains that will be concluded in part 5 of this series of talks.

Abstract: Originally aimed at understanding the relationship between classical and quantum mechanics, geometric quantization is a construction whose ideas originated in representation theory, in the work of Kirillov, Kostant and Souriau during the late 1960's. From the symplectic geometry perspective, geometric quantization produces representations of compact Lie groups starting from `geometric' (i.e. Hamiltonian) group actions of such Lie groups. Some ideas surrounding this construction will be discussed, including their adaptation to the theory of 'group-valued' moment maps, which is a finite-dimensional model of the theory of Hamiltonian loop group actions. Applications in this context (re)produce so-called Verlinde formulas of conformal field theory for simply connected compact Lie groups. For non-simply connected Lie groups, Verlinde formulas have been conjectured, and this approach verifies the conjectured formulas for $G=SO(3)$.