Complex Dynamics and Diophantine Geometry

这个项目的主要目标是探索动力多项式理论和一元有理函数与丢番图几何理论之间的联系，丢番图几何理论研究多项式方程解的算术特征。具体地说，主要研究者研究了定义在复数域上的这些动力系统的代数族内的分支和稳定性，并利用这些结果来解决关于算术簇上的高度函数和有理点的问题，主要集中在交集理论和计数问题上。即使是最简单的一族例子，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的动力学特征。同样，关于椭圆曲线上扭点的看似简单的结构，仍有深层次的悬而未决的问题。这项研究结合了复杂分析和算术几何的方法。主要研究人员和她的合作者开发了新的证明方法，结合了复杂动力学和非阿基米德分析的工具。这个项目的主要目的是利用这些组合方法来解决有关高度函数的问题，以及受算术问题的启发，在黎曼球面上解决关于地图动力学的一些新问题。主要研究人员致力于：(1)关于代数动力系统的不太可能相交问题的统一形式；(2)关于阿贝尔族内的扭点的研究，以刻画哪些曲线可以与许多具有‘’小‘正则高度的点相交；(3)临界轨道猜想，关于在有理映射模空间内的后临界有限映射的几何；(4)关于特征为零的函数场上动力系统正则高度的猜想的合理性，以及与超越问题的联系；以及(5)映射族和椭圆曲线族的等价分布陈述。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。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.