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.
