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

多能理论及其在几何、数论和动力学中的应用

• 批准号: 0900934
• 负责人: Dan Coman
• 金额: $ 15.81万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900934&HistoricalAwards=false
• 关键词: Pluripotential; Theory; Applications; Geometry; Number

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.

DMS-0900934 提案摘要，PI：Dan Coman 该项目解决了多能理论的问题，其中一些问题在超越数论、复几何和代数几何中具有重要应用。一个统一的主题是，其核心在于多次谐波函数和正闭电流，它们要么作为主要研究对象，要么作为主要使用工具。研究方向之一涉及紧复流形的多势理论问题，其中存在与局部环境不同的新的有趣现象。主要目标是研究复杂的 Monge-Ampere 算子和相应的格林函数。另一个方向涉及复杂欧几里得空间中的多能理论问题。要考虑的问题涉及正闭流的几何性质及其解析簇的近似、复复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分析簇的行为并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问题，涉及复杂欧几里得空间的多项式自同构，其中多能理论提供了重要工具。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，帮助解决了纯数学和应用数学其他领域以及物理学、生物学等领域的许多重要问题。由于复分析的强大方法，它常常被通过首先在复数的背景下表述具体问题，在研究具体问题方面取得了进展。该项目涉及复数分析和势论等新技术的开发和进一步应用，以解决现代数学几个重要领域的问题，例如数论、复数和代数几何、动力系统，以及在数学物理中的可能应用。所要研究的问题属于当前多复杂变量研究的主流。在这些问题上取得进展将有助于增进该领域的知识和理解。拟议的研究将通过为两名研究生提供夏季资助来影响人力资源开发。他们将在研究者的监督下就与该项目相关的主题撰写论文。通过这种方式，该项目将研究和教育融为一体。