Complex algebraic dynamics and geometry

Polynomials and rational functions of a single variable provide basic examples of non-invertible, algebraic dynamical systems. Even the simplest families of examples, such as the quadratic polynomials fc(z) = z2 + c with c 2 C, exhibit complicated dynamical features that we have yet to understand. The primary goal of this project is to explore connections between the algebra and the geometry of dynamical systems on algebraic varieties, with an eye towards applications in Diophantine geometry. The secondary goal of the project is a study of rational maps in dimension one, f : P1 ! P1, particularly an exploration of the \canonical shape" of the dynamical system. The projects proposed (both the questions and the proposed solution strategies) combine ingredients from complex analysis and arithmetic or algebraic geometry. Intellectual merit. In the last decade, there has been an explosion of new research in the area of arithmetic dynamics: using algebraic geometry and number theory to address dynamical questions, and more strikingly, using methods from dynamical systems to answer questions in arithmetic geometry. The PI's most recent results illustrate deep connections between the subjects; she has developed new methods of proof incorporating tools from both areas of mathematics. In this proposal, the PI outlines a research program to study: (1) the arithmetic properties of elliptic curves and abelian varieties, with dynamical methods; (2) the substantial conjectures about \unlikely intersections" in dynamical systems; (3) the stability of complex-algebraic dynamical systems; and (4) the 3-dimensional Euclidean geometry of a map f : P1 ! P1. The projects and guiding questions of this proposal should have impact on multiple areas of mathematics, from number theory to computational geoemetry, and they should pave the way for the study of further connections between these subjects. Broader impact. The PI is actively involved in the organization of conferences, workshops, and seminars, designed for discussions between researchers and for the education of students. Her work in particular promotes an exchange of mathematical ideas, primarily between researchers in dynamical systems and those in number theory or arithmetic geometry. The PI travels often in the United States and abroad to lecture, attend conferences, and work with collaborators, and she regularly delivers plenary lectures to general (mathematical or non-mathematical) audiences. She is also working with students at all levels: many of the PI's projects (experimental and theory-development) can be adapted into research projects for beginners. She is currently guiding the research of 4 PhD students and mentoring 3 postdocs, and she will continue to lead research projects with undergraduates. As a woman in mathematics, the PI is acutely aware of the small numbers of women at the top research institutions, especially at the most senior level. With this project, the PI intends to maintain a high level of visibility in the mathematical community.