Moduli spaces of complex dynamical systems

单变量多项式和有理函数提供了不可逆动力系统的基本例子。即使是最简单的样本族也表现出复杂的动力学行为；最著名的是复杂的二次多项式族，其中的Mandelbrot集继续困扰着研究人员。这个项目的主要目标是探索多项式和有理函数的动态模空间。建议的项目(问题和建议的解决策略)结合了复杂分析和算术或代数几何的成分。在一个方向上，PI的目的是研究模空间中后临界有限有理映射的分布。在与马修·贝克的合作中，PI制定了一个动态模拟算术几何中的安德烈-奥尔特猜想。这类问题不仅仅是类比：例如，PI旨在使用动力学技术来恢复Masser和Zannier关于椭圆曲线族中的扭点的结果。主要工具是Berkovich分析空间上的分析和动力学方面的最新发展。在一个略有不同的方向上，PI正在研究模空间内不同亚种的分支集和分叉度量。这里的技术主要是分析性的。使用博伊德和博伊德开发的新的计算机程序Dynamic Explorer，实验工作产生了新的问题。PI还对关于有理函数对称性的存在性和分类的经典问题感兴趣。在过去的五年或十年里，“代数动力学”已经成为一个非常活跃的研究领域；随着资深研究人员以非常不同的背景进入该学科，以及我们发现与数学的许多领域的联系，这些问题变得更加精致。粗略地说，代数动力学是研究保持基本代数结构的动力系统。这样的系统在应用中自然出现(例如，一维逻辑斯族是代数的)，它们在数论家研究的算术对象的分析中发挥作用(例如，定义与算术变种相关的高度函数)。国际数学家协会积极参与数字理论家和动态学家之间的数学思想交流。一方面，她的问题是关于动力系统的基本稳定性：在什么条件下系统对微小的扰动不敏感？另一方面，她研究的这类特殊的动力系统显示出丰富的代数结构，将动力特征带入了算术几何的悠久历史中。最后，该项目有一个与学生和初级研究人员合作非常好的实验部分；PI积极参与本科生、研究生和博士后研究人员的研究项目和培训计划。

Polynomials and rational functions of a single variable provide basic examples of non-invertible dynamical systems. Even the simplest families of examples exhibit complicated dynamical behavior; the most famous is the family of complex quadratic polynomials, where the Mandelbrot set continues to baffle researchers. The primary goal of this project is to explore the dynamical moduli spaces of polynomials and rational functions. The projects proposed (both the questions and the proposed solution strategies) combine ingredients from complex analysis and arithmetic or algebraic geometry. In one direction, the PI aims to study the distribution of postcritically-finite rational maps within the moduli space. In joint work with Matthew Baker, the PI has formulated a dynamical analogue to the Andre-Oort conjecture in arithmetic geometry. Questions of this type are not only analogies: for example, the PI aims to use dynamical techniques to recover a result by Masser and Zannier about torsion points in families of elliptic curves. The main tools are recent developments in analysis and dynamics on a Berkovich analyticspace. In a slightly different direction, the PI is studying bifurcation sets and bifurcation measures in distinguished subvarieties within the moduli space. Here the techniques are predominantly analytic. New questions have stemmed from experimental work, using the new computer program Dynamics Explorer developed by Boyd & Boyd. The PI is also interested in classical problems about the existence and classification of symmetries of rational functions. In the last five or ten years, "algebraic dynamics" has become an extremely active area of research; the questions have become more refined as senior researchers enter the subject with very different backgrounds and we uncover connections to many areas of mathematics. Roughly speaking, algebraic dynamics is the study of dynamical systems that preserve an underlying algebraicstructure. Such systems arise naturally in applications (for example, the one-dimensional logistic family is algebraic), and they play a role in the analysis of arithmetic objects studied by number theorists (for example, in defining height functions associated to arithmetic varieties). The PI is actively involved in the exchange of mathematical ideas between number theorists and dynamicists. On one hand, her questions are about the fundamental stability of dynamical systems: under what conditions is a system insensitive to small perturbations? On the other hand, the special class of dynamical systems she studies exhibits a rich algebraic structure, bringing dynamical features into a long history of arithmetic geometry. Finally, the project has an experimental component that works very well with students and junior researchers; the PI is actively involved in research projects and training programs for undergraduate, graduate, and postdoctoral researchers.