Problems in Potential Theory and Dynamics in Several Complex Variables

这个项目涉及的问题，从pluripotential理论和动力学在几个复变量。复数势理论是复数平面上势理论的高维版本。研究者计划从复Monge-Ampere算子的复复绿色函数的叶状结构的角度以及与代数几何中某些问题的关系来研究复Monge-Ampere算子的复Green函数。该项目的第二个目标是考虑多项式估计的Bernstein-Walsh型相关的图形的整个超越functions.The调查计划研究某些性质的整个功能从这个角度来看。这些性质在超越数论中也很有趣。最后，研究者关心的是研究高维多项式自同构类的动力学。一维和高维的复分析与许多纯数学和应用数学领域，甚至与其他科学有着重要的联系。通常情况下，一旦在复数的背景下制定了具体的问题，人们就会对这些问题有更深入的了解。这是因为在复分析中研究全纯函数的强大方法和工具。研究复杂的动力系统被证明对于理解数学以及物理和生物学中出现的更复杂的真实的动力系统的行为非常重要。

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.