RUI: Arboreal Galois Groups and Nonarchimedean Dynamics

这个项目涉及算术动力学、场桥数论和动力系统中的一些公开问题。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学相去甚远，但在这种背景下，这两者通过p-进动力学联系在一起。此外，PI将指导REU暑期研究项目的本科生，以加强他们的数学训练。REU产生的任何计算数据都将发布或发布在网络上，以造福于更大的研究社区。该项目的结果还将通过arxiv等网站和在数学期刊上发表。需要研究的具体问题出现在算术动力学的两个领域：第一，伽罗瓦群在动力学轨道上的作用，第二，非阿基米德动力系统的模空间。在伽罗瓦方面，某些p-进动力特征对于展示足够的伽罗华自同构以产生所讨论的复杂伽罗瓦群是必不可少的。在模空间方面，非阿基米德动力学已经发展成为一个成熟的研究领域，但目前对单参数非阿基米德动力系统的了解相对较少。该项目将专注于伯克维奇投影线上的动力学，该投影线是单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，它们是具有悠久历史的动力学、伽罗瓦理论和非阿基米德分析的丰富理论的延续。特别是，第一个主题承诺提供新的动力学工具来解决绝对伽罗瓦群的研究，而第二个主题承诺提供新的方法来解决算术动力系统中的模数问题。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.