## Project Details

### Description

Polynomials and rational functions of a single variable provide basic examples of non-invertible dynamical systems. Amazingly, even the simplest families of examples exhibit complicated dynamical behavior. In the setting of complex polynomials, the most famous family is that of the quadratic polynomials ffc(z) = z2 + c : c 2 Cg, 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.
Intellectual merit. The main object of study is the moduli space Md of complex rational functions, defining dynamical systems on P1, under the equivalence relation of conformal conjugacy. The postcritically-finite maps in Md form a Zariski dense subset; apart from the flexible Lattes maps, they form a countable set, defined over Q (by Thurston's Rigidity Theorem). They play a role in Md analogous to that of elliptic curves with complex multiplication in the modular curve. The PI aims to investigate their distribution within Md. In joint work with Matthew Baker, the PI has formulated a conjectural analogue to the well-known Andre-Oort conjecture in arithmetic geometry. Questions of this type about Md are not simply 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.
In a slightly different direction, the PI is studying bifurcation sets and bifurcation measures in distinguished subvarieties within Md. Here the techniques are predominantly analytic. The questions themselves stemmed from experimental work, using the new program DE Tool (developed by Boyd & Boyd, made available in 2012), to illustrate Julia sets and bifurcation loci. For example, the PI is interested in classical problems about the existence and classifcation of symmetries of rational functions. Recently, methods from algebra and arithmetic geometry have been brought to bear on this question.
Research in the area of \arithmetic dynamics" has exploded in the last five or ten years; but only now is it becoming less of a list of analogies and more of a genuine collaboration between number theorists and dynamicists. The PI is actively involved in this exchange of mathematical ideas.
Broader impact. Many of the PI's projects (experimental or otherwise) can be adapted into research projects for students. The PI is organizing several conferences each year and runs regular seminars in her home institution. She travels often in the United States and abroad to lecture, attend conferences, and work with collaborators. As a woman in mathematics, the PI is acutely aware that there are few women at the top research institutions. With this project, the PI intends to maintain a high level of visibility in the mathematical community. She is working with students at all levels and actively trying to improve her own department. Her first three PhD students are expected to complete
their degrees in Spring 2013.

Status | Finished |
---|---|

Effective start/end date | 9/15/14 → 10/31/17 |

### Funding

- National Science Foundation (DMS-1517080)

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