Abstract:

Complex analysis has its roots in the eighteenth century, whereas functional analysis is a creature of the twentieth century. I'll try to capture the flavor of functional analysis by touching on some high points of its development over the past century. I'll also try to paint a picture of (single-variable) complex analysis by submitting a list of major advances in complex analysis over the same century. After sketching developments over the past century, I'll focus on one narrow area where functional analysis and complex analysis come together, and on a specific set of problems. The area is the theory of uniform algebras, and the problems will involve homomorphisms of a uniform algebra to itself. One of the key ideas involves the introduction of a pseudohyperbolic metric on the so-called spectrum of the uniform algebra and an application of the contraction mapping principle.