Moduli spaces of complex dynamical systems

多项式和单变量有理函数提供了不可逆动力系统的基本例子。 即使是最简单的例子族也表现出复杂的动力学行为;最著名的是复二次多项式族，其中的曼德尔布罗特集仍然困扰着研究人员。 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。 所提出的项目（包括问题和所提出的解决方案）结合了复分析和算术或代数几何的联合收割机成分。 在一个方向上，PI旨在研究模空间内的后临界有限有理映射的分布。 在与马修·贝克的联合工作中，PI在算术几何中制定了安德烈-奥尔特猜想的动力学模拟。这种类型的问题不仅是类比：例如，PI旨在使用动力学技术来恢复Masser和Zannier关于椭圆曲线族中扭点的结果。 主要的工具是最近的发展，在分析和动力学上的一个Berkovich analyticspace。 在一个稍微不同的方向，PI正在研究模空间内的分歧集和分歧措施。 这里的技术主要是分析性的。 新的问题源于实验工作，使用新的计算机程序动力学探索者开发的博伊德博伊德。 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;随着高级研究人员以非常不同的背景进入这个学科，问题变得更加精致，我们发现了与许多数学领域的联系。 粗略地说，代数动力学是对保持基本代数结构的动力系统的研究。 这样的系统在应用中自然出现（例如，一维逻辑族是代数的），它们在数论家研究的算术对象的分析中发挥作用（例如，定义与算术簇相关的高度函数）。 PI积极参与数论家和动力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小扰动不敏感？ 另一方面，她研究的一类特殊的动力系统具有丰富的代数结构，将动力学特征带入算术几何的悠久历史中。 最后，该项目有一个实验部分，与学生和初级研究人员合作得很好; 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.