The Pluri-Potential Theory and Its Applications

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Proposal Number: DMS-0200743PI: Evgeny PoletskyABSTRACTThis 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 containingall the necessary background; the complex geometry of infinite-dimensional spaces related to classical problemsfor plurisubharmonic functions; the complex geometry ofcompact sets via the pluri-potential theory on suchsets; the geometry of domains and multipole Green functions (the classical analog is the Martin-Caratheodorycompactification) and Bernstein-Walsh and Markov inequalities for generalized polynomials. Other geometricquestions will also be considered.In this project research will be conducted on geometriccomplex analysis. Geometry, as a part of mathematics,aims to describe qualitative links between different objects. For example, parallel lines do not meet and theheights in a triangle meet at the same point. When Euclidean objects: points, lines and planes are replacedby more complicated structures: functions, surfaces andsets, the research is of a more delicate flavor.It happens because the mechanism providing links betweenobjects is not transparent. In our proposal we will lookfor such a mechanism in the form of potentials similar tothe energy levels of electrical charges.