Pluripotential Theory and Applications to Complex Dynamics and Number Theory

建议摘要DMS-0500563，圆周率：丹·科曼这个项目解决了几个复变量中的位势理论和动力学问题。复杂动力学中的问题使用多势理论中的工具，其中一些可能需要开发新的此类工具。第一个研究方向是研究复Monge-Ampere算子的基本解，我们称之为复复格林函数。这将在理解电流奇点和代数几何中的一些问题上有应用。第二个方向的研究涉及复欧氏空间的多项式自同构的动力学，在正则情况下，以及对于特殊的非规则映射。主要的兴趣在于获得对动力学的详细了解，以及研究动态绿流和措施的遍历特性。本项目的第三个研究方向是沿超越解析簇分析多项式的行为。它将被用来研究整函数的算术性质，以及超越函数值集合的代数无关性。复数分析和势能理论是数学的核心领域。多年来，他们提供了方法和强大的工具，帮助解决了许多来自理论数学和应用数学的其他领域的重要问题，以及来自物理、生物学、经济学等领域的许多重要问题。这个项目涉及从多个复变量的分析和位势理论到动力系统和数论问题的新技术的发展和进一步应用。由于复数分析的强大方法，在具体问题的研究中经常取得进展，首先在复数的背景下对它们进行表述。

Abstract for Proposal DMS-0500563, PI: Dan ComanThis project addresses problems from potential theory and dynamics in several complex variables. The problems in complex dynamics use tools from pluripotential theory and some are likely to require developing new such tools. The first direction of research deals with the study of the fundamental solutions of the complex Monge-Ampere operator, which are called pluricomplex Green functions. This will have applications in understanding the singularities of currents and to some questions in algebraic geometry. The second direction of research deals with the dynamics of polynomial automorphisms of complex Euclidean spaces, in the regular case, and for special classes of irregular mappings. The main interest lies in obtaining a detailed understanding of the dynamics, and in the study of the ergodic properties of the dynamical Green currents and measures. The third direction of research of this project is to analyze the behavior of polynomials along transcendental analytic varieties. This will be used to study arithmetic properties of entire functions, and the algebraic independence of sets of values of transcendental functions. Complex analysis and potential theory are central areas of Mathematics. Over the years, they have provided methods and powerful tools which helped to solve many important problems from other fields of pure and applied Mathematics, as well as from Physics, Biology, Economics, etc. This project deals with the developing and further application of new techniques from analysis and potential theory in several complex variables to problems in dynamical systems and number theory. Thanks to the powerful methods of complex analysis, it has been often the case that progress is made in the study of concrete problems by formulating them first in the context of complex numbers.