Bifurcations in Complex Algebraic Dynamics

复杂代数动力学中的分岔

• 批准号: 2246630
• 负责人: Laura DeMarco
• 金额: $ 44.69万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246630&HistoricalAwards=false
• 关键词: Bifurcations; Complex; Algebraic; Dynamics

The stability of a dynamical system is arguably its most important feature, from a theoretical, computational, or practical point of view. For systems that evolve with time, one aims to determine which perturbations will preserve the system’s long-term behavior and which perturbations will lead to radically different outcomes. This project concerns stability and bifurcations in the setting of complex algebraic dynamical systems. Such systems are defined by polynomial formulas in one or several variables. The algebraic nature of the defining equations connect the dynamical study with the rich theory of algebraic geometry. Moreover, in the case of examples where all of the defining polynomials have, for example, integer coefficients, the relevant dynamical stability questions have surprising connections to number theory and to the Diophantine geometry of the underlying equations. The project will extend the theory of dynamical stability for complex analytic examples to new settings that arise naturally in arithmetic geometry and complex dynamics. The project also provides research and training opportunities for graduate students and postdocs.This project develops the theory of stability for analytic families of maps on projective spaces, in both a complex analytic setting and in the setting of non-archimedean-valued fields and p-adic analysis. It was recently discovered, in earlier work of the PI and of other researchers, that certain questions about height functions and arithmetic intersection theory can be analyzed using complex dynamics. In a series of recent breakthroughs in arithmetic geometry, especially concerning uniform bounds for numbers of rational points on families of algebraic varieties, stability theory played a crucial--if somewhat hidden--role. This project aims to shed new light on the role of stability theory and to push the theory further. Many of the proposed problems and applications of the theory are related to the occurrence of `unlikely intersections’ in families of abelian varieties or in more general families of polarized dynamical systems. Specific goals of this project include (1) to characterize positivity properties of certain bifurcation currents and measures; (2) to provide bounds on the geometry of invariant subvarieties for algebraic dynamical systems; and (3) to formulate a theory of bifurcations in the setting of p-adic analytic families of maps. The research activity conducted under this award is expected to impact 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.

从理论、计算或实践的角度来看，动力系统的稳定性可以说是其最重要的特征。对于随时间演化的系统，我们的目标是确定哪些扰动将保持系统的长期行为，以及哪些扰动将导致完全不同的结果。该项目涉及复杂代数动力系统设置中的稳定性和分岔。此类系统由一个或多个变量的多项式公式定义。定义方程的代数性质将动力学研究与丰富的代数几何理论联系起来。此外，在所有定义多项式都具有整数系数的示例的情况下，相关的动态稳定性问题与数论和基础方程的丢番图几何具有令人惊讶的联系。该项目将把复杂分析例子的动态稳定性理论扩展到算术几何和复杂动力学中自然出现的新设置。该项目还为研究生和博士后提供研究和培训机会。该项目开发了射影空间上的解析族映射的稳定性理论，无论是在复杂的分析环境中还是在非阿基米德值域和 p-adic 分析的环境中。最近，在 PI 和其他研究人员的早期工作中发现，有关高度函数和算术交集理论的某些问题可以使用复杂的动力学进行分析。在算术几何最近的一系列突破中，特别是在代数簇上有理点数量的统一界限方面，稳定性理论发挥了至关重要的（尽管有些隐藏的）作用。该项目旨在阐明稳定性理论的作用并进一步推动该理论的发展。该理论提出的许多问题和应用都与阿贝尔簇族或更一般的极化动力系统族中“不太可能的交叉”的发生有关。该项目的具体目标包括（1）表征某些分岔电流和措施的正特性； (2) 为代数动力系统的不变子族的几何提供界限； (3) 在映射的 p-adic 解析族的设置下制定分岔理论。该奖项下进行的研究活动预计将影响数学的多个领域，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。