Pluripotential Theory and Applications to Complex Dynamics and Number Theory

提案DMS-0500563摘要， PI：Dan Coman这个项目解决了几个复杂变量的潜在理论和动力学问题。复杂动力学中的问题使用多能理论的工具，有些可能需要开发新的工具。研究的第一个方向是研究复Monge-Ampere算子的基本解，称为复绿色函数。这将有助于理解电流的奇异性和代数几何中的某些问题。第二个方向的研究涉及的动力学多项式自同构的复杂欧氏空间，在正常的情况下，并为特殊类别的不规则映射。主要的兴趣在于获得详细的了解的动力学，并在遍历性的动态绿色电流和措施的研究。本计画的第三个研究方向是分析多项式沿着超越解析簇的行为。这将用于研究整函数的算术性质，以及超越函数的值集的代数独立性。复分析和势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，帮助解决了许多重要的问题，从其他领域的纯数学和应用数学，以及从物理学，生物学，经济学等这个项目涉及的发展和进一步应用新技术，从分析和潜在的理论在几个复杂的变量问题，在动力系统和数论。由于强大的复分析方法，在研究具体问题时，往往首先在复数的背景下将其公式化，从而取得进展。

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.