Heights, Capacity, and Dynamics

DMS-0300784 Rumely，Robert S.Abstracy标题：高度、容量和动力在这个项目中，首席研究人员研究了p-进空间的各个方面的分析，特别是那些与正则高度、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，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；以及Berkovich空间的分析及其在动力系统、算术交集理论和高维容量理论中的应用。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。高度函数最早与椭圆曲线有关，在现代数论中是普遍存在的。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性质，它们类似于椭圆曲线的高度性质。我们将使用位势理论中的方法来解决这一问题。另一个目标是建立流形理论中已知的结果的“p-进”类比。该项目完成后，将揭示数论、动力系统和位势理论之间的新联系。该项目的资金将支持佐治亚大学数论小组的基础设施，该小组历来非常强大。该项目将影响UGA的研究生和本科生项目：这项研究提出的一些问题将导致研究生的博士论文主题，而其他问题将为本科生提供一个让本科生参与尖端数学研究的机会。

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.