权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Complex Dynamics and Diophantine Geometry

复杂动力学和丢番图几何

基本信息

批准号：
2050037
负责人：
Laura DeMarco
金额：
$ 25.36万
依托单位：
Harvard University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2050037&HistoricalAwards=false
关键词：
Complex Dynamics Diophantine Geometry

项目摘要

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.

该项目的主要目标是探索多项式动力学理论和一个变量的有理函数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。具体来说，主要研究人员研究了在复数领域定义的这些动力系统的代数族内的分岔和稳定性，并利用结果来解决有关算术簇上的高度函数和有理点的问题，重点是交集理论和计数问题。即使是最简单的例子系列，例如经过充分研究的二次多项式系列，也表现出我们尚未理解的复杂动力学特征。同样，关于椭圆曲线上扭转点看似简单的结构，仍然存在一些尚未解答的深刻问题。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结合了复动力学和非阿基米德分析的工具，开发了新的证明方法。该项目的主要目标是受算术问题的启发，利用这些组合方法来解决有关高度函数的问题以及有关黎曼球上地图动力学的一些新问题。首席研究员正在致力于：（1）关于代数动力系统的不太可能相交问题的统一版本； (2) 对阿贝尔簇家族内的扭转点的研究，以表征哪些曲线可以与许多“小”规范高度的点相交； (3) 临界轨道猜想，关于有理映射模空间内的后临界有限映射的几何； (4) 特征零函数域上动力系统正则高度的猜想合理性，以及与超越问题的联系； (5) 地图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。