Complex Dynamics and Diophantine Geometry

该项目的主要目标是探索多项式和一元有理函数的动力学理论与研究多项式方程解的算术特征的丢番图几何理论之间的联系。具体而言，主要研究人员研究了这些动力系统在复数领域定义的代数家族内的分叉和稳定性，并使用结果来解决有关算术品种上的高度函数和合理点的问题，重点是交叉理论和计数问题。 即使是最简单的例子族，如研究得很好的二次多项式族，也表现出我们尚未理解的复杂动力学特征。 类似地，关于椭圆曲线上扭点的看似简单的结构，仍然存在一些深层次的未解问题。这项研究结合了复分析和算术几何的方法。首席研究员和她的合作者开发了新的证明方法，结合了复动力学和非阿基米德分析的工具。 这个项目的主要目标是利用这些组合的方法来解决高度函数的问题和一些新的问题的黎曼球面上的映射的动力学，启发算术问题。主要研究者致力于：（1）关于代数动力系统的不可能相交问题的统一版本;（2）阿贝尔簇族中的扭点研究，以表征哪些曲线可以与许多“小”标准高度的点相交;（3）临界轨道猜想，关于有理映射的模空间内的后临界有限映射的几何;（4）特征为零的函数域上动力系统的正则高度的严格合理性，以及与超越问题的联系;（5）映射族和椭圆曲线族的等分布陈述。 这项研究应该对数学的多个领域产生影响，包括数论、几何和动力学。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.