This project proposes the further study of various problems in complex analysis. There are two main parts to the proposal: the pluripotential theory and the geometric function theory. The pluripotential theory is more important for complex analysis in several variables than the classical potential theory for the one-dimensional analysis because it becomes more difficult to use analytic tools as the number of dimensions grows. That is why this theory found applications in all areas of complex analysis, including complex dynamics, and far beyond it in algebraic and transcendental number theories. The geometric function theory, whose goal is to obtain qualitative characteristics of investigated objects when quantitative approach fails, is represented in the project by suggested studies of the pluripotential compactifications of domains, disk envelopes and groups of automorphisms. The project also intends to study algebraic properties of the ring of entire functions with applications to transcendental number theory.
In this project research will be conducted on geometric complex analysis and number theory. Geometry, as a part of mathematics, aims to describe qualitative links between different objects. For example, parallel lines do not meet and the heights in a triangle meet at the same point. When Euclidean objects: points, lines and planes are replaced by more complicated structures like functions, surfaces and sets, the research is of a more delicate flavor. It happens because the mechanism providing links between objects is not transparent. In our proposal we will look for such a mechanism in the form of potentials similar to the energy levels of electrical charges. We also intend to understand better the properties of the number p.
