# "The Hole Story": how to solve problems in multiply connected domains

This talk will survey recent developments in providing novel constructive methods for the solution of

boundary value problems in multiply connected, or “holey", planar domains.

All the investigations are driven by problems arising in applications and examples will be given.

Two distinct mathematical approaches will be described: one based on special function theory, and another centred around generalized transform methods.

The setting for the special function theory approach tackles the problems on a compact Riemann surface naturally associated to any planar multiply connected domain known as the Schottky double, and makes use of the Schottky-Klein prime function associated therewith.

On the other hand, our new transform approach for multiply connected domains generalizes and combines classical transform ideas of Fourier and Mellin with more recent developments pioneered by A. S. Fokas and collaborators.