Braids, Surfaces, and Polynomials

每当我们对物理、生物或化学系统建模时，多项式就会出现在数学和科学中。当我们研究系统的几何形状时，通常会出现曲面；例如，机械系统的配置集或大维数据集的底层模板。每当我们有一组点在一个表面上移动时，就会出现辫子，例如，在我们的视野内移动的恒星和行星，或者多项式的根相对于一个参数变化。为了理解这些现象，有必要研究曲面的对称性集，它也称为曲面的映射类群。这是一个美丽而丰富的理论，在过去的一个世纪里一直是密集研究的焦点。这项研究的目标是研究曲面、辫子和多项式，以及这些对象之间的相互作用。一个项目是给出确定映射类群元素的基本性质的快速算法。该算法计算的属性之一是熵，它决定了在表面上发生的混合量。除了这些研究目标，PI还计划继续开展几个对研究生、本科生和高中生有直接影响的项目。第一个是非常成功的拓扑学学生研讨会，这个会议既是研究生的拓扑学研究会议，也是专业发展研讨会。第二个是免费的，在线的，交互式的基础线性代数教科书，叫做交互式线性代数。公社还计划将外展活动扩展到当地的K-12班级。公社将学习瑟斯顿地图、辫子群和地图班级小组。瑟斯顿映射是复平面(或黎曼球)在其自身上的分支覆盖，具有有限的后临界集。许多多项式都是瑟斯顿映射。复杂动力学中的一个基本识别问题是：给定一个瑟斯顿映射，它是否等同于一个多项式，如果是，是哪一个？在以前的工作中，PI和他的合作者给出了一个新的几何算法来解决这个识别问题。PI计划对这类识别问题的宇宙进行新的、结构性的描述。具体地说，该项目将根据映射类别组的理论建立一个版本的Bestvina-Handel算法，该算法适用于瑟斯顿地图的设置。该项目还将提供Bman精确序列的一个版本(同样来自映射类群的理论)。映射类群理论中的一个项目是给出一个二次时间算法，该算法将映射类群的生成元的乘积作为输入，并确定该乘积的Nielsen-瑟斯顿类型。该算法还产生关于生成器乘积的更精细的信息，例如减少曲线和熵。第三个项目是对辫子群之间的同态进行分类。主要的新工具是完全对称集理论，由PI和他的合作者开发。通过对这些同态的分类，我们深入了解了不同次数的多项式之间的关系。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。

Polynomials arise in mathematics and science whenever we model a physical, biological, or chemical system. Surfaces often appear when we study the geometry of a system; for instance, the set of configurations of a mechanical system or the underlying template for a large dimensional data set. Braids occur whenever we have a collection of points moving in a surface, for instance stars and planets moving within our field of vision, or the roots of polynomials changing with respect to a parameter. In order to understand these phenomena, it is essential to study the set of symmetries of a surface, which is also known as the mapping class group of the surface. This is a beautiful and rich theory that has been the focus of intense study over the past century. The goal of this research is to study surfaces, braids, and polynomials, and the interactions of these objects with each other. One project is to give fast algorithms for deciding basic properties of elements of the mapping class group. One of the properties that the algorithm computes is the entropy, which determines the amount of mixing happening on the surface. In addition to these research goals, the PI plans to continue work on several projects that have direct impact on graduate, undergraduate, and high school students. The first is the highly successful Topology Students Workshop, a conference that serves both as a research conference in topology for graduate students as well as a professional development workshop. The second is a free, online, interactive textbook for basic linear algebra, called Interactive Linear Algebra. The PI also plans to expand outreach activities to local K-12 classrooms.The PI will study Thurston maps, braid groups, and mapping class groups. A Thurston map is a branched cover of the complex plane (or Riemann sphere) over itself, with finite post-critical set. Many polynomials are Thurston maps. A basic recognition problem in complex dynamics is: given a Thurston map, is it equivalent to a polynomial, and if so, which one? In prior work, the PI and his collaborators gave a new, geometric algorithm to solve this recognition problem. The PI plans to investigate new, structural descriptions of the universe of such recognition problems. Specifically, the project will establish a version of the Bestvina-Handel algorithm from the theory of mapping class groups that is adapted to the setting of Thurston maps. The project will also provide a version of the Birman exact sequence (again from the theory of mapping class groups). One project in the theory of mapping class groups is to give a quadratic time algorithm that takes as input a product of generators of the mapping class group and determines the Nielsen-Thurston type of that product. This algorithm also produces finer information about the product of generators, such as reducing curves and entropy. A third project is to classify homomorphisms between braid groups. The main new tool is the theory of totally symmetric sets, developed by the PI and his collaborators. By classifying these homomorphisms we gain insight into the relationships between polynomials of different degrees.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.