Complex Algebraic Dynamics and Geometry

复杂代数动力学和几何

• 批准号: 1600718
• 负责人: Laura DeMarco
• 金额: $ 37.5万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600718&HistoricalAwards=false
• 关键词: Complex; Algebraic; Dynamics; Geometry

The repeated application of a function produces sequences of values whose behavior can be quite surprising. Such discrete dynamical systems occur as models throughout science and engineering. Polynomials and rational functions of a single variable provide basic examples of non-invertible, algebraic, discrete dynamical systems by iteration. Even the simplest families of examples exhibit complicated and chaotic dynamical behavior; the most famous is the family of quadratic polynomials where a point x is mapped to x "squared" plus a constant. The constant can be a real or complex number. This family gives rise to the Mandelbrot set, which continues to baffle researchers. A fundamental problem in the mathematical study of these systems is to characterize their stability: under what circumstances -- and by how much -- can we perturb a system while maintaining its long-term dynamical features? The primary goal of this research project is to explore connections between the stability of algebraic dynamical systems and the algebraic or geometric structures that are preserved under iteration. The first connections between the algebra and dynamics of these models were discovered in the 19th century. This project will exploit modern tools from arithmetic geometry and dynamical systems to strengthen these connections and deepen our understanding of the systems themselves. The project studies: (1) the arithmetic properties of elliptic curves and abelian varieties, with dynamical methods; (2) a series of substantial conjectures about "unlikely intersections" in moduli spaces of dynamical systems; and (3) the 3-dimensional Euclidean geometry of a rational function, with curvature form equal to its measure of maximal entropy. The main goal of this project is to build relations between the algebra and the geometry of dynamical systems on algebraic varieties, with an eye towards applications in Diophantine geometry. The second goal of the project is a study of rational maps in dimension one, particularly an exploration of the canonical shape of the dynamical system and its stability properties. In recent work, the investigator and collaborator developed new methods of proof incorporating tools from both complex and non-Archimedean analysis and formulated a conjecture about the dynamics of rational maps that encompasses known results about elliptic curves. This project aims to investigate particular cases of this conjecture while developing the theory to connect these results to the study of the symmetries and stability and invariants of complex dynamical systems in dimensions one and two.

函数的重复应用会产生一系列的值，其行为可能非常令人惊讶。这样的离散动力系统在整个科学和工程中都以模型的形式出现。单变量多项式和有理函数通过迭代提供了不可逆的、代数的、离散的动力系统的基本例子。即使是最简单的一族例子也表现出复杂而混乱的动力学行为；最著名的是一族二次多项式，其中一个点x被映射到x的“平方”加上一个常量。常量可以是实数，也可以是复数。这个家庭产生了曼德尔布洛特布景，这仍然让研究人员感到困惑。对这些系统进行数学研究的一个基本问题是描述它们的稳定性：在什么情况下--以及在多大程度上--我们可以在保持系统长期动力学特征的同时，对系统进行扰动？这个研究项目的主要目标是探索代数动力系统的稳定性与迭代下保持的代数或几何结构之间的联系。这些模型的代数和动力学之间的第一次联系是在19世纪发现的。这个项目将利用算术几何和动力系统的现代工具来加强这些联系，并加深我们对系统本身的理解。该项目研究：(1)用动力学方法研究椭圆曲线和阿贝尔簇的算术性质；(2)关于动力系统模空间中“不可能相交”的一系列实质性猜想；以及(3)有理函数的三维欧几里德几何，其曲率形式等于其最大熵的度量。这个项目的主要目标是在代数簇上建立动力系统的代数和几何之间的关系，并着眼于在丢番图几何中的应用。该项目的第二个目标是研究一维有理映射，特别是探索动力系统的标准形及其稳定性。在最近的工作中，研究人员和合作者开发了新的证明方法，结合了复数和非阿基米德分析的工具，并提出了一个关于有理映射动力学的猜想，其中包含了关于椭圆曲线的已知结果。这个项目的目的是研究这种猜想的特殊情况，同时发展理论，将这些结果与一维和二维复杂动力系统的对称性、稳定性和不变量的研究联系起来。