Complex Dynamics and Diophantine Geometry

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The primary goal of this project is to explore connections between the dynamical theory of polynomials and rational functions of one variable and the theory of Diophantine geometry which studies arithmetic features of solutions to polynomial equations. Specifically, the principal investigator studies bifurcations and stability within algebraic families of these dynamical systems defined over the field of complex numbers and uses the results to address questions about height functions and rational points on arithmetic varieties, focused on intersection theory and counting problems. Even the simplest families of examples, such as the well-studied family of quadratic polynomials, 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. This research combines methods from both complex analysis and arithmetic geometry.The principal investigator with her collaborators has developed new methods of proof incorporating tools from complex dynamics and non-archimedean analysis. The main objective of this project is to exploit these combined methods to address problems about height functions and some new problems about the dynamics of maps on the Riemann sphere, inspired by the arithmetic questions. The principal investigator is working towards: (1) uniform versions of Unlikely Intersection problems about algebraic dynamical systems; (2) a study of torsion points within 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 of rational maps; (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 and for families of elliptic curves. This research should have impact on multiple areas of mathematics, including number theory, geometry, and dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.