Heights, Capacity, and Dynamics

高度、容量和动力

• 批准号: 0300784
• 负责人: Robert Rumely
• 金额: $ 18.9万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-15 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300784&HistoricalAwards=false
• 关键词: Heights; Capacity; Dynamics

DMS-0300784Rumely, Robert S.AbstractTitle: Heights, Capacity, and DynamicsIn this project, the Principal Investigators study various aspects ofanalysis on p-adic spaces, especially those arising in connection withcanonical heights, capacity theory, and algebraic dynamical systems. The project involves the development of both theory and applications, and comprises three main topics: adelic equidistribution theorems for pointsof small height; arithmetic aspects of the dynamics of iterated rationalfunctions; and analysis on Berkovich spaces with applications to dynamicalsystems, arithmetic intersection theory, and higher-dimensional capacitytheory. The motivation for this work is to understand and generalize certainrecently discovered properties of "height functions". Height functionsfirst arose in connection with elliptic curves, and are ubiquitous inmodern number theory. One of the goals of the project is to establishproperties of heights for arbitrary curves and arbitrary dynamical systemswhich are similar to those for elliptic curves. This will be approached byusing methods from potential theory. Another goal is to establish"p-adic" analogues of results which are known to hold in the theory ofmanifolds. When completed, the project will reveal new connectionsbetween number theory, dynamical systems, and potential theory. Fundingfor this project will support the infrastructure of the University ofGeorgia's number theory group, which has historically been very strong. The project will impact both the graduate and undergraduate programs atUGA: some of the questions raised by this research will lead to PhDdissertation topics for graduate students, and others will provide anopportunity to involve undergraduate students in cutting edge mathematicalresearch.

DMS-0300784Rumely，Robert S.摘要标题：高度、容量和动力学在该项目中，主要研究人员研究了 p-adic 空间分析的各个方面，特别是与规范高度、容量理论和代数动力系统相关的分析。 该项目涉及理论和应用的发展，包括三个主要主题：小高度点的等分布定理；迭代有理函数动力学的算术方面；伯科维奇空间及其在动力系统、算术交集理论和高维容量理论中的应用分析。 这项工作的动机是理解和概括最近发现的“高度函数”的某些属性。 高度函数首先与椭圆曲线相关，并且在现代数论中无处不在。 该项目的目标之一是建立任意曲线和任意动力系统的高度属性，类似于椭圆曲线的高度属性。这将通过使用势理论的方法来实现。 另一个目标是建立已知在流形理论中成立的结果的“p进”类似物。 完成后，该项目将揭示数论、动力系统和势论之间的新联系。 该项目的资金将支持乔治亚大学数论小组的基础设施，该小组历来非常强大。该项目将影响佐治亚大学的研究生和本科生课程：这项研究提出的一些问题将成为研究生的博士论文主题，而其他问题将为本科生参与前沿数学研究提供机会。