Problems in Potential Theory and Dynamics in Several Complex Variables

这个项目涉及多势理论和多复变量的动力学问题。Pluri位势理论是位势理论在复平面上的高维版本。本文拟从复Monge-Ampere算子的叶结构出发，结合代数几何中的一些问题，研究复复Monge-Ampere算子的一类多重Green函数。该项目的第二个目标是考虑与整个超越函数的图相关的Bernstein-Walsh型多项式估计。研究者计划从这个角度研究整个函数的某些性质。这些性质在先验数论中也很有趣。最后，研究人员关注于研究高维多项式自同构类的动力学。一维和更高维复分析与许多纯数学和应用数学领域有重要联系，甚至与其他科学也有重要联系。通常情况下，人们一旦在复数的背景下描述了具体的问题，就会对它们有更深入的了解。这是因为在复分析中全纯函数的研究中发展了强大的方法和工具。研究复杂的动力系统被证明对于理解数学、物理和生物学中出现的更复杂的真实动力系统的行为是重要的。

This project deals with questions from pluri-potentialtheory and dynamics in several complex variables. Pluri-potential theory is the higher dimensional versionof 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 theirfoliation structure and also in relation to some questionsfrom algebraic geometry. A second goal of the project is to consider polynomial estimates of Bernstein-Walsh typerelated to graphs of entire transcendental functions.The investigator plans to study certain properties ofentire functions from this point of view. Such propertiesare also of interest in transcendental number theory.Finally, the investigator is concerned with studying thedynamics of classes of polynomial automorphisms inhigher dimensions.Complex analysis in one and in higher dimensions hasimportant connections with many fields of pure and appliedmathematics and, indeed, with other sciences. It is oftenthe 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 developedin the study of holomorphic functions in complex analysis.Studying complex dynamical systems proved to be importantfor understanding the behavior of more complicated real dynamical systems arising in mathematics as well as in physics and biology.