Topics in complex approximation theoryChenoweth, Brett Simon (Avtor) Forstnerič, Franc (Mentor) Stein manifoldOka manifoldholomorphic mapCarleman approximationbounded exhaustion hullsminimal surfacedirected holomorphic curveIn this thesis, we investigate problems in complex approximation theory motivated by recent developments in Oka theory, minimal surface theory, and contact geometry. Primarily, our focus lies in proving approximation results in the spirit of Carleman’s theorem, that is, better than uniform approximation on noncompact sets. The original research of this dissertation begins in Chapter 3, where we prove a generalisation of Carleman’s theorem for maps from Stein manifolds to Oka manifolds. Then, in Chapter 4, we prove a version of Carleman’s theorem for directed holomorphic immersions and minimal surfaces. Under suitable hypotheses, we may even ensure that the approximating maps have desirable global properties, including completeness and properness. As an application of these results, we give an approximate solution to a Plateau problem for divergent Jordan curves in Euclidean spaces. Finally, Chapter 5 is concerned with approximation by solutions of systems of differential equations. We adapt the tools and techniques that have successfully been applied in the single equation, contact case. Period dominating sprays play an instrumental role.20202020-10-13 11:31:24Doktorsko delo/naloga121510UDK: 517.5VisID: 113391COBISS_ID: 32648963sl