Aspects of complex analysis

Complex analysis is an area of mathematics with links to many areas of pure and applied mathematics and has a rich history, yet has many modern facets which are highly active areas of research today. These include such areas as Teichmueller theory and the theory of quasiconformal mappings, which has numerous links to toplogy, geometry and geometric group theory. Complex dynamics is currently a very popular area of research, popularized by famous computer generated diagrams. Value distribution theory is a more concrete branch of complex analysis with links to the theory of ordinary differential equations and complex dynamics.The aim of this proposal is to study important research problems with connections to these aspects of complex analysis. Quasiconformal mappings in the plane have the decomposition property, that is, they can be decomposed into compositions of quasiconformal mappings of arbitrarily small complex dilatation. A central open problem is whether bi-Lipschitz mappings in the plane can be decomposed into compositions of bi-Lipschitz mappings with arbitrarily small bi-Lipschitz constant. Certain special mappings are known to have the decomposition property, however the general case remains open and the study of this problem is one of the aims of this proposal.Teichmueller spaces and asymptotic Teichmueller spaces are defined as very complicated objects, and yet it can be shown that they can be viewed via bi-Lipschitz mappings as subsets of much more intuitively simpler spaces. Another aim of this proposal is to study this connection more deeply.Finally, iteration theory for quasiregular mappings has mainly been studied for uniformly quasiregular mappings. In this case, all the iterates have uniformly bounded dilatation which implies the iterates form a normal family. This notion of normal families of iterates is central to complex dynamics in the plane. General quasiregular mappings in arbitrary dimension do not have this property, which makes carrying over some notions of complex dynamics in the plane difficult. The escaping set does, however, make sense and another aim of this proposal is to study its structure and see if it shares certain properties with the escaping set of analytic functions in the plane.