The basic thrust here is to characterise the analyticity of a collection of functions in algebraic and/or topological means. More generally, when is an abstract ring or algebra isomorphic to a ring or algebra of analytic functions over some manifold.
Requirements: Complex variables (possibly several), ring theory, functional analysis, possibly di?erential geometry. References: I. Richards, Axioms for analytic functions, Adv. Math. 5 (1970), 311-338.