Heights, Capacity, and Dynamics

DMS-0300784 Rumely，Robert S.摘要标题：高度，容量和动力学在这个项目中，主要研究人员研究p-adic空间分析的各个方面，特别是那些与规范高度，容量理论和代数动力系统有关的分析。 该项目涉及理论和应用的发展，并包括三个主要议题：adelic equidistribution theorems为pointsof small height;算术方面的动态迭代rationalfunctions;和分析上的Berkovich空间与应用dynamicalsystems，算术交叉理论，和高维capacitytheory。 这项工作的动机是理解和推广最近发现的“高度函数”的某些性质。 高度函数最早出现在椭圆曲线中，并且在现代数论中无处不在。 该项目的目标之一是建立与椭圆曲线相似的任意曲线和任意动力系统的高度性质。这将通过使用方法从潜在的理论。 另一个目标是建立“p-adic”类似的结果是已知的理论ofmanifold举行。 完成后，该项目将揭示数论，动力系统和潜在理论之间的新联系。 该项目的资金将支持佐治亚大学数论小组的基础设施，该小组在历史上一直非常强大。该项目将影响研究生和本科生课程atUGA：本研究提出的一些问题将导致博士论文的研究生课题，和其他人将提供一个机会，让本科生参与尖端的prosticalresearch。

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.