Problems in Potential Theory and Dynamics in Several Complex Variables

Syracuse University, Syracuse NY

Investigators

Abstract

This project deals with questions from pluri-potential theory and dynamics in several complex variables. Pluri-potential theory is the higher dimensional version of potential theory in the complex plane. The investigator plans to study certain classes of pluricomplex Green functions for the complex Monge-Ampere operator, from the point of view of their foliation structure and also in relation to some questions from algebraic geometry. A second goal of the project is to consider polynomial estimates of Bernstein-Walsh type related to graphs of entire transcendental functions. The investigator plans to study certain properties of entire functions from this point of view. Such properties are also of interest in transcendental number theory. Finally, the investigator is concerned with studying the dynamics of classes of polynomial automorphisms in higher dimensions. Complex analysis in one and in higher dimensions has important connections with many fields of pure and applied mathematics and, indeed, with other sciences. It is often the case that one gains insight about concrete problems once they are formulated in the context of complex numbers. This is because of the powerful methods and tools developed in the study of holomorphic functions in complex analysis. Studying complex dynamical systems proved to be important for understanding the behavior of more complicated real dynamical systems arising in mathematics as well as in physics and biology.