Title: The Lie Ring of Holomorphic Functions

Abstract: For an open and connected subset U of the complex plane, the collection of functions which are holomorphic on U with the usual Lie bracket is a natural Polish Lie ring. We will discover that the Lie ring structure, in most cases of interest, determines what kind of Polish ring topologies are available. In particular, for the open unit disk, we will see that there is only one Polish ring topology on its space of holomorphic functions.