Pluripotential Theory and Applications to Complex Dynamics and Number Theory

多能理论及其在复杂动力学和数论中的应用

• 批准号: 0500563
• 负责人: Dan Coman
• 金额: --
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500563&HistoricalAwards=false
• 关键词: Pluripotential; Theory; Applications; Complex; Dynamics

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.

DMS-0500563 提案摘要，PI：Dan Coman 该项目解决了多个复杂变量中的势理论和动力学问题。复杂动力学中的问题需要使用多能理论的工具，有些问题可能需要开发新的此类工具。第一个研究方向涉及复数 Monge-Ampere 算子的基本解的研究，称为复复格林函数。这将有助于理解电流的奇点和代数几何中的一些问题。研究的第二个方向涉及复杂欧几里德空间的多项式自同构的动力学，在常规情况下，以及特殊类别的不规则映射。主要兴趣在于获得对动力学的详细了解，以及研究动态格林电流和测量的遍历特性。该项目的第三个研究方向是分析多项式沿超越分析簇的行为。这将用于研究整个函数的算术性质，以及超越函数值集的代数独立性。复分析和势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，帮助解决了纯数学和应用数学其他领域以及物理学、生物学、经济学等领域的许多重要问题。该项目涉及新数学的开发和进一步应用从多个复杂变量的分析和势论到动力系统和数论问题的技术。由于复分析的强大方法，通常情况下，通过首先在复数的背景下制定具体问题，才能在具体问题的研究中取得进展。