# Project Details

### Description

OVERVIEW:

The primary goal of this project is to explore connections between the dynamical theory of polynomials and rational maps in dimension 1 -- focused on stability and bifurcations in algebraic families -- and problems in Diophantine Geometry related to heights and rational points on arithmetic varieties -- focused on intersection theory and counting problems. Even the simplest families of examples, such as the quadratic polynomials $f_c(z) = z^2+c$ with $c\in \mathbb{C}$, exhibit complicated dynamical features that we have yet to understand. Similarly, there remain deep unanswered questions about the seemingly simple structure of torsion points on elliptic curves. The projects proposed here, and their proposed solutions, combine ingredients from complex analysis and arithmetic geometry.

INTELLECTUAL MERIT:

The PI has developed new methods of proof incorporating tools from both complex dynamics and arithmetic geometry. The main objective of this proposal is to exploit these combined methods to address problems in Diophantine Geometry about height functions and some new problems about the dynamics of maps on the Riemann sphere inspired by the arithmetic questions. She is working towards:

1) uniform versions of Unlikely Intersection problems about algebraic dynamical systems, with applications to uniform Manin-Mumford statements;

2) a Bogomolov-Zhang Conjecture about curves in a family of abelian varieties, to characterize which curves can intersect many points of "small" canonical height;

3) the Critical Orbit Conjecture, about the geometry of postcritically finite maps within the moduli space holomorphic maps of a given degree on $\mathbb{P}^1$;

4) the conjectured rationality of canonical heights for dynamical systems over function fields in characteristic zero, and connections to transcendence problems; and

5) equidistribution statements for families of maps on $\mathbb{P}^1$ and for families of elliptic curves.

The projects and guiding questions of this proposal should have impact on multiple areas of mathematics, including number theory, geometry, and dynamics.

BROADER IMPACTS:

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 and 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 (both experimental and theory-development) can be adapted into research projects for beginners. She is currently guiding the research of 3 PhD students and mentoring 4 postdocs. 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.

The primary goal of this project is to explore connections between the dynamical theory of polynomials and rational maps in dimension 1 -- focused on stability and bifurcations in algebraic families -- and problems in Diophantine Geometry related to heights and rational points on arithmetic varieties -- focused on intersection theory and counting problems. Even the simplest families of examples, such as the quadratic polynomials $f_c(z) = z^2+c$ with $c\in \mathbb{C}$, exhibit complicated dynamical features that we have yet to understand. Similarly, there remain deep unanswered questions about the seemingly simple structure of torsion points on elliptic curves. The projects proposed here, and their proposed solutions, combine ingredients from complex analysis and arithmetic geometry.

INTELLECTUAL MERIT:

The PI has developed new methods of proof incorporating tools from both complex dynamics and arithmetic geometry. The main objective of this proposal is to exploit these combined methods to address problems in Diophantine Geometry about height functions and some new problems about the dynamics of maps on the Riemann sphere inspired by the arithmetic questions. She is working towards:

1) uniform versions of Unlikely Intersection problems about algebraic dynamical systems, with applications to uniform Manin-Mumford statements;

2) a Bogomolov-Zhang Conjecture about curves in a family of abelian varieties, to characterize which curves can intersect many points of "small" canonical height;

3) the Critical Orbit Conjecture, about the geometry of postcritically finite maps within the moduli space holomorphic maps of a given degree on $\mathbb{P}^1$;

4) the conjectured rationality of canonical heights for dynamical systems over function fields in characteristic zero, and connections to transcendence problems; and

5) equidistribution statements for families of maps on $\mathbb{P}^1$ and for families of elliptic curves.

The projects and guiding questions of this proposal should have impact on multiple areas of mathematics, including number theory, geometry, and dynamics.

BROADER IMPACTS:

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 and 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 (both experimental and theory-development) can be adapted into research projects for beginners. She is currently guiding the research of 3 PhD students and mentoring 4 postdocs. 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.

Status | Active |
---|---|

Effective start/end date | 9/1/19 → 8/31/22 |

### Funding

- National Science Foundation (DMS-1856103)