SAMS Subject Classification Number: 9, 20

A Hermitian holomorphic line bundle over a Hermitian complex manifold carries a fundamental geometric object called the Bergman kernel. This kernel can be thought of as the ?correlation function? for pairs of points on the manifold and leads to the concept of coherent states. I will give an overview of the usefulness of the coherent states approach to well-known constructions such as the Riemann mapping theorem. I will introduce coherent loop states, obtained by integrating coherent states around Bohr-Sommerfeld loops, and give an overview of their applications to the study of inner products of modular forms and to the representation theory of SU(2) (this latter is work of my PhD student Nzaganya Nzaganya).