Pluripotential Theory and Applications to Geometry, Number Theory, and Dynamics

提案DMS-0900934， PI: Dan coman本项目研究多能理论中的一些问题，其中一些问题在超越数论、复几何和代数几何中有重要的应用。一个统一的主题是，其核心是多次谐波函数和正闭合电流，它们要么是主要的研究对象，要么是主要的研究工具。一个研究方向是紧复流形上的多势理论问题，其中有不同于局部环境的有趣的新现象。主要目的是研究复Monge-Ampere算子和相应的Green函数。另一个方向是复欧几里德空间中的多势理论问题。要考虑的问题包括正闭合电流的几何性质及其解析变分逼近，复数格林函数的研究及其与代数几何问题的联系。第三个研究方向是分析多项式沿超越解析变量的行为和研究整个函数的代数独立性。预计这将继续应用于超越数论。该项目还包含复杂动力学的问题，涉及复杂欧几里得空间的多项式自同构，其中多元势理论提供了重要的工具。复分析和势理论是数学的核心领域。多年来，他们提供的方法和强大的工具，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，生物等。由于复数分析的强大方法，在具体问题的研究中往往是通过首先在复数的背景下公式化来取得进展的。该项目涉及新技术的发展和进一步应用，从复杂分析和势理论到现代数学的几个重要领域的问题，如数论，复杂和代数几何，动力系统，以及数学物理的可能应用。所研究的问题属于当前研究的几个复杂变量的主流问题。在这些问题上取得进展将有助于提高这一领域的知识和理解。拟议的研究将通过对两名研究生的夏季资助来影响人力资源发展。他们将在调查员的指导下完成论文。对与本项目相关的课题进行监督。通过这种方式，该项目将研究和教育结合起来。

Abstract for Proposal DMS-0900934, PI: Dan ComanThis project addresses problems from pluripotential theory, some of which have important applications to transcendental number theory, complex geometry, and algebraic geometry. A unifying theme is that at its core lie plurisubharmonic functions and positive closed currents, either as main objects of investigation or as main tools to be employed. One direction of research deals with problems in pluripotential theory on compact complex manifolds, where there are new interesting phenomena, different from the local setting. The main goals are the study of the complex Monge-Ampere operator and of the corresponding Green functions. Another direction is concerned with problems from pluripotential theory in the complex Euclidean space. The questions to be considered involve geometric properties of positive closed currents and their approximation by analytic varieties, the study of pluricomplex Green functions and their connection to problems in algebraic geometry. A third direction of research is to analyze the behavior of polynomials along transcendental analytic varieties and to study the algebraic independence of entire functions. It is expected this will continue to have applications to transcendental number theory. The project also contains problems from complex dynamics, concerning polynomial automorphisms of complex Euclidean spaces, where pluripotential theory provides important tools.Complex analysis and potential theory are central areas of Mathematics. Over the years, they have provided methods and powerful tools that helped solve many important problems from other fields of pure and applied Mathematics, as well as from Physics, Biology, etc. 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. This project deals with the developing and further applications of new techniques from complex analysis and potential theory to problems in several important areas of modern Mathematics, such as number theory, complex and algebraic geometry, dynamical systems, as well as possible applications to Mathematical Physics. The problems to be studied belong to the main stream of current research in several complex variables. Making progress on these problems will contribute to the advancement of knowledge and understanding in the field. The proposed research will impact human resources development through summer funding of two graduate students. They will work for their dissertation under the investigator?s supervision on topics related to this project. In this way the project integrates research and education.