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，计算或实际角度来看，动态系统的稳定性可以说是其最重要的特征。对于随着时间发展的系统，旨在确定哪种扰动将保留系统的长期行为，哪些扰动将导致根本不同的结果。该项目涉及复杂代数动态系统的稳定性和分叉。这样的系统由一个或几个变量中的多项式公式定义。定义方程式的代数性质将动态研究与丰富的代数几何理论联系起来。此外，在示例的情况下，所有定义多项式都具有整数系数，相关的动态稳定性问题具有与数字理论以及基础方程的Diophantine几何形状的惊喜联系。该项目将将复杂分析示例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环境。该项目还为研究生和博士学位提供了研究和培训机会。该项目在复杂的分析环境和非架构的非座位值和P-ADIC分析的环境中开发了投影空间上地图的稳定理论。最近，在PI和其他研究人员的早期工作中发现，可以使用复杂的动力学分析有关身高功能和算术交集理论的某些问题。在一系列算术几何形状的一系列突破中，尤其是关于代数品种家族的合理点的统一界限，稳定理论起着至关重要的（即使有些隐藏）。该项目的目的是对稳定理论的作用进行新的启示，并进一步推动理论。该理论的许多提出的问题和应用与阿贝尔品种家庭或更一般的两极分化动态系统家族中的“不太可能的交叉点”有关。该项目的具体目标包括（1）表征某些分叉电流和度量的阳性特性； （2）为代数动态系统的不变亚变量的几何形状提供界限； （3）在地图的P-ADIC分析家族的环境中形成分叉理论。预计根据该奖项进行的研究活动有望影响数学的多个领域，包括数字理论，几何和动态。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，通过评估被认为是宝贵的支持。