A Topological Approach to Questions in Dynamical Systems

数学模型被广泛用于理解我们周围的世界，并预测参数的变化将如何影响未来。了解模型的长期行为是一项重要的科学奋进。在动力系统中，数学的一个研究子领域，人们研究这样的数学模型，并试图理解它们的长期行为以及参数变化的后果。研究复杂数学模型的一个标准方法是将它们扰动为更简单的模型，并证明对于更简单的模型的某些结论对于更复杂的模型也是有效的。该奖项资助的研究将有助于解决三个主要问题，这些问题涉及动力系统的基本方面，并为创建新模型提供见解。许多数学模型都是基于齐次空间的。在最近的工作中，研究者和他的合作者提供了均匀平面连续统的完整分类，解决了一个可以追溯到20世纪20年代的问题。在这个项目中，这些技术将被用来解决所有一维均匀连续统的分类。参数空间通常非常复杂，人们对它们知之甚少。Thurston证明了二阶复多项式的参数空间可以用二阶不变叠层空间来建模。这种组合模型比Mandelbrot集更简单，并且得到了完整的描述。每个二阶复多项式对应于该组合模型中的单个点，并且其在组合模型中的位置决定多项式的拓扑动力学。在另一个方向，研究人员将专注于理解更高程度的不变叠层空间的拓扑结构和几何结构，这是扩展Thurston工作的第一步。该奖项反映了NSF的法定使命，并被认为值得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来支持。

Mathematical models are widely used to understand the world around us and to predict how changes in the parameters will affect the future. Understanding the long-term behavior of models is an important scientific endeavor. In dynamical systems, a research sub-area in mathematics, one studies such mathematical models and attempts to understand their long-term behavior as well as consequences of changes in parameters. A standard approach to studying complicated mathematical models is to perturb them to simpler models and to show that certain conclusions for the simpler models are also valid for the more complicated models. Research funded by this award will contribute to the solution of three main problems that address basic aspects of dynamical systems and offer insights for the creation of new models. Many mathematical models are based on homogeneous spaces. In recent work, the investigator and his collaborators provided a complete classification of homogeneous plane continua, solving a problem that dates back to the 1920s. In this project these techniques will be exploited to address classification of all one-dimensional homogeneous continua. Parameter spaces are in general extremely complicated and very little is known about them. Thurston showed that the parameter space of complex polynomials of degree two can be modeled by the space of degree two invariant laminations. This combinatorial model is simpler than the Mandelbrot set and is completely described. Every complex polynomial of degree two corresponds to a single point in this combinatorial model and its location in the combinatorial model determines the topological dynamics of the polynomial. In another direction, the investigator will focus on understanding the topology and geometry of the spaces of invariant laminations of higher degrees, which is the first step in extending Thurston's work.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.