Analysis Seminar: Hyperbolic Analytic and Algebraic Curves

Abstract: Hyperbolic analytic curves are analytic generalizations of hyperbolic algebraic curves; ‘cut outs’ of algebraic sets intersected with bounded open sets in complex n space. To facilitate the analysis of these objects one can treat them as image sets of Riemann surfaces under holomaps; formally, holomorphic proper functions which are non-singular and injective away from an at most finite set. The algebra of functions resulting from the composition of a holomap and the holomorphic functions of the hyperbolic analytic curve is a cofinite subalgebra of the space of holomorphic functions on the Riemann surface. The theory is due to Jim Agler and John McCarthy who proved its first key result, namely that an isomorphism exists between two hyperbolic analytic curves if and only if an isomorphism of their Riemann surfaces preserves the corresponding cofinite algebras. As a result, it is of interest to classify cofinite subalgebras of Riemann surfaces, the case of petals, hyperbolic analytic curves holized by the disk, is done for low dimensions and the resulting work is then used to understand some of the function theory of hyperbolic analytic curves. Recently I have been working on extending this work to annular domains.

Note: This talk is given in partial fulfillment of the Third Year Candidacy Requirement. 

Host/Advisor: John McCarthy