A Topological Approach to Questions in Dynamical Systems

数学模型被广泛用于了解我们周围的世界，并预测参数的变化将如何影响未来。理解模型的长期行为是一项重要的科学努力。在动力系统这一数学研究的子领域，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种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也适用于更复杂的模型。由该奖项资助的研究将有助于解决三个主要问题，这三个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，这位研究人员和他的合作者提供了齐次平面连续体的完整分类，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体的分类问题。参数空间通常是极其复杂的，人们对它们知之甚少。瑟斯顿证明了二次复多项式的参数空间可以用二次不变薄片空间来建模。这种组合模型比Mandelbrot集更简单，而且是完全描述的。每个二次复多项式对应于该组合模型中的一个点，它在组合模型中的位置决定了该多项式的拓扑动力学。在另一个方向，研究人员将专注于了解更高次数的不变分层空间的拓扑和几何，这是扩展瑟斯顿工作的第一步。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。

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.