Complex Algebraic Dynamics and Geometry

函数的重复应用会产生一系列的值，这些值的行为可能非常令人惊讶。这种离散动力系统作为模型出现在整个科学和工程中。多项式和单变量的有理函数通过迭代提供了不可逆、代数、离散动力系统的基本例子。即使是最简单的一类例子也表现出复杂和混沌的动力学行为；最著名的是二次多项式族，其中一个点x被映射到x的平方加上一个常数。常数可以是实数或复数。这个家族产生了曼德勃罗集，这一直困扰着研究人员。在这些系统的数学研究中，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——我们可以扰动一个系统，同时保持它的长期动态特征？本研究项目的主要目标是探索代数动力系统的稳定性与在迭代下保存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。该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。项目研究：(1)用动力学方法研究椭圆曲线和阿贝尔变的算术性质；(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；(3)一个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。本项目的主要目标是建立代数和几何在代数变量上的动力系统之间的关系，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的有理映射，特别是探索动力系统的规范形状及其稳定性。在最近的工作中，研究者和合作者开发了新的证明方法，结合了复杂和非阿基米德分析的工具，并提出了一个关于理性映射动力学的猜想，该猜想包含了关于椭圆曲线的已知结果。本项目旨在研究这一猜想的特殊情况，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。

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.