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，现场桥接数字理论和动态系统中的某些开放问题。虽然对理性数字和多项式方程的数量理论研究远与动力学研究中出现的混乱和分形相去甚远，但两者在这种情况下由P-ADIC动力学捆绑在一起。此外，PI将在REU夏季研究项目中监督本科生，以增强其数学培训。 REU中产生的任何计算数据都将在网络上发布或发布，以使大型研究社区受益。该项目的结果也将通过网站（例如Arxiv和数学期刊上的出版物）进行传播。要研究的具体问题是在算术动力学的两个领域中出现的：首先，Galois组对动态轨道的作用，第二，第二个非近距离动力学系统的Moduli空间。在Galois侧，某些P-ADIC动力学特征对于表现出足够的Galois自动形态以产生相关的复杂的Galois组至关重要。在模量太空方面，非一切本动力学已经演变为已建立的研究领域，但是目前，关于非一切本动力学系统的单参数家族的众所周知，相对较少。该项目将重点介绍伯科维奇投射线的动态，这是一个可变性的非Archimedean Systems法案的适当空间。要探索的问题是新领域，这些领域是富裕理论的延续，在动力学，GALOIS理论和非构造分析方面具有悠久的历史。特别是，第一个主题有望提供新的动力学工具来解决绝对Galois群体的研究，而第二个主题则有望在算术动力学系统中解决模量问题的新方法。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估来通过评估来支持的，这是值得的。