In this project the principal investigator will study various problems in pluri-potential theory that have applications to complex analysis. The ultimate goal of this work is to construct an invariant boundary theory for plurisubharmonic functions and holomorphic functions. In particular, the principal investigator will study boundary properties of pluri- subharmonic functions with respect to the Green-Martin topology, the decomposition of plurisubharmonic functions into potentials and maximal functions, the analytic structure of polynomial hulls and Wermer compacta. The theory of subharmonic functions occupies a special place in complex function theory because of its inherent beauty and because it is important to various other areas such as the theory of partial differential equations. In this project the principal investigator will continue working on an analogous theory for what are called "plurisubharmonic" functions. Plurisubharmonic functions arise in many different contexts, and a knowledge of their properties will help us understand a number of problems in differential geometry and complex function theory.
