This project proposes the further study of the geometric complex analysis, mostly, through pluri-potential theory. Research will be conducted in the following topics: the existence of a "general" potential theory containing all the necessary background; the complex geometry of infinite-dimensional spaces related to classical problems for plurisubharmonic functions; the complex geometry of compact sets via the pluri-potential theory on such sets; the geometry of domains and multipole Green functions (the classical analog is the Martin-Caratheodory compactification) and Bernstein-Walsh and Markov inequalities for generalized polynomials. Other geometric questions will also be considered.
In this project research will be conducted on geometric complex analysis. 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: 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.
