RUI: Arboreal Galois Groups and Nonarchimedean Dynamics

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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.