Complex Dynamics: Renormalization, Geometry, and Algebra

动力系统的数学领域研究物理或数学系统的长期演化。演化过程往往具有混沌性(即对初始条件的敏感依赖性)，在研究具有混沌行为的系统时，常常会出现分形集。一个重要的分形集是Mandelbrot集，它在动力系统中扮演着重要的角色，是数学中最容易识别的分形集之一。这个项目的中心主题之一是理解曼德尔布罗特集的某些自相似性方面。Mandelbrot集编码了二次多项式如何依赖于参数。它是动力学中几个基本问题的测试对象，特别是关于混沌到底是如何产生的，即使在简单的系统中也是如此。拟议的活动还包括涉及高次有理映射(即多项式的比率)的主题。这些项目将涉及与早期职业研究人员的多次合作，以及对研究生的培训。首席研究人员的目标之一是促进不同数学分支的专家之间更广泛的互动。首席研究人员将继续发展近中性重整化理论，该理论负责Mandelbrot集在其主要心形附近的自相似特征。重整化(不同尺度之间的相互作用)出现在数学和物理的许多分支中；确切的词典尚未正式确定。在二次多项式的情况下，中心猜想本质上等价于建立相关重整化的严格理论(例如，证明重整化算子的双曲性)。主要研究人员还将研究高次有理映射参数空间的几何和拓扑性质。一个具体的主题是使用近中性重整化来理解双曲分量的边界。PI将继续设计两个球面的自分支覆盖的代数和算法理论，例如通过忘记复杂结构从后临界有限有理映射获得的映射。最后一个主题与映射班级组和自相似组的理论密切相关。这一奖项反映了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 mathematical area of dynamical systems studies the long-term evolution of physical or mathematical systems. The evolution often has chaotic nature (i.e., sensitive dependence on initial conditions), and fractal sets often come up when studying systems with chaotic behavior. An important fractal set is the Mandelbrot set, which has played a major role in dynamical systems and is one of the most recognizable fractal sets in mathematics. Among central themes of this project is understanding certain self-similarity aspects of the Mandelbrot set. The Mandelbrot set encodes how a quadratic polynomial depends on a parameter. It is a test object for several fundamental problems in dynamics, in particular on how exactly chaos arises, even in simple systems. The proposed activity also covers topics involving higher degree rational maps (i.e., ratios of polynomials). The projects will involve multiple collaborations with early-career researchers, and the training of graduate students. One of the goals of the principal investigator is to promote broader interactions between experts in different branches of mathematics.The principal investigator will continue developing the near-neutral renormalization theory responsible for the self-similarity features of the Mandelbrot set near its main cardioid. Renormalization (interplay between different scales) appears in many branches of mathematics and physics; the exact dictionary is yet to be formalized. In the case of quadratic polynomials, central conjectures are essentially equivalent to establishing rigorous theories of the associated renormalizations (for example, proving hyperbolicity of the renormalization operators). The principal investigator will also study the geometric and topological properties of parameter spaces of higher degree rational maps. A concrete theme is understanding the boundaries of hyperbolic components using near-neutral renormalization. The PI will continue designing the algebraic and algorithmic theories of self-branched coverings of a two-sphere, such as maps obtained from post-critically finite rational maps by forgetting the complex structure. The last topic is closely related to the theories of mapping class groups and self-similar groups.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.