RUI: Arboreal Galois Groups and Nonarchimedean Dynamics

RUI：树栖伽罗瓦群和非阿基米德动力学

基本信息

批准号：
2101925
负责人：
Robert Benedetto
金额：
$ 19.01万
依托单位：
Amherst College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101925&HistoricalAwards=false
关键词：
RUI Arboreal Galois Groups Nonarchimedean

项目摘要

This project concerns certain open questions in arithmetic dynamics, a field bridging number theory and dynamical systems. While the number-theoretic study of rational numbers and polynomial equations lies far from the chaos and fractals that arise in the study of dynamics, the two are tied together in this setting by p-adic dynamics. In addition, the PI will supervise undergraduate students in an REU summer research project to bolster their mathematical training. Any computational data produced in the REU will be published or posted on the web, for the benefit of the larger research community. Results from the project will also be disseminated via websites such as arXiv and via publication in mathematical journals.The specific questions to be studied arise in two areas within arithmetic dynamics: first, the action of Galois groups on dynamical orbits, and second, moduli spaces of nonarchimedean dynamical systems. On the Galois side, certain p-adic dynamical features are essential to exhibiting enough Galois automorphisms to generate the complicated Galois groups in question. On the moduli space side, nonarchimedean dynamics has evolved into an established field of research, but relatively little is currently known about one-parameter families of nonarchimedean dynamical systems. The project will focus on dynamics on the Berkovich projective line, the appropriate space on which one-variable nonarchimedean systems act. The problems to be explored are new areas that are continuations of rich theories with long histories in dynamics, Galois theory, and nonarchimedean analysis. In particular, the first topic promises to provide new dynamical tools for addressing the study of absolute Galois groups, while the second promises new approaches to moduli problems in arithmetic dynamical systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目涉及算术动力学中的某些开放问题，这是连接数论和动力系统的领域。虽然有理数和多项式方程的数论研究与动力学研究中出现的混沌和分形相距甚远，但在这种情况下，两者通过 p 进动力学联系在一起。此外，PI 将指导本科生参加 REU 夏季研究项目，以加强他们的数学训练。 REU 中产生的任何计算数据都将在网络上发布或发布，以造福更大的研究社区。该项目的结果还将通过 arXiv 等网站和数学期刊上的出版物进行传播。要研究的具体问题出现在算术动力学的两个领域：第一，伽罗瓦群在动力轨道上的作用，第二，模空间非阿基米德动力系统。在伽罗瓦方面，某些 p 进动力学特征对于展示足够的伽罗瓦自同构以生成所讨论的复杂伽罗瓦群至关重要。在模空间方面，非阿基米德动力学已发展成为一个成熟的研究领域，但目前对非阿基米德动力学系统的单参数族知之甚少。该项目将重点研究伯科维奇射影线的动力学，即一变量非阿基米德系统作用的适当空间。要探索的问题是新的领域，是动力学、伽罗瓦理论和非阿基米德分析等历史悠久的丰富理论的延续。特别是，第一个主题承诺为解决绝对伽罗瓦群的研究提供新的动力学工具，而第二个主题则承诺解决算术动力系统中模问题的新方法。该奖项反映了 NSF 的法定使命，并通过评估被认为值得支持利用基金会的智力优势和更广泛的影响审查标准。