Bifurcations in Complex Algebraic Dynamics

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。对于随时间演变的系统，人们的目标是确定哪些扰动将保留系统的长期行为，哪些扰动将导致截然不同的结果。本课题主要研究复杂代数动力系统的稳定性和分叉问题。这样的系统由一个或几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在一起。此外，在所有定义多项式都具有例如整数系数的例子的情况下，相关的动态稳定性问题与数论和基本方程的丢番图几何有着惊人的联系。该项目将把复杂分析例子的动力稳定性理论扩展到算术几何和复杂动力学中自然产生的新环境。这个项目还为研究生和博士后提供了研究和培训的机会。这个项目发展了射影空间上解析映射族的稳定性理论，无论是在复杂的分析环境下，还是在非阿基米德值域和p-进分析的环境下。在PI和其他研究人员的早期工作中，最近发现，关于高度函数和算术交集理论的某些问题可以用复杂的动力学来分析。在算术几何最近的一系列突破中，特别是关于代数簇族上许多有理点的一致界，稳定性理论发挥了关键--尽管有些隐藏--作用。该项目旨在阐明稳定性理论的作用，并推动该理论的进一步发展。该理论提出的许多问题和应用都与阿贝尔变种家族或更一般的极化动力系统家族中出现的“不太可能的交集”有关。这个项目的具体目标包括：(1)刻画某些分支电流和测度的正性性质；(2)给出代数动力系统不变子簇的几何上界；(3)在p元解析映射族的背景下建立分支理论。根据该奖项开展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优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.