Braids, Surfaces, and Polynomials

每当我们为物理、生物或化学系统建模时，数学和科学中就会出现多项式。当我们研究一个系统的几何形状时，曲面经常出现；例如，机械系统的配置集或大维度数据集的底层模板。每当我们在一个表面上有一组点移动时，例如恒星和行星在我们的视野内移动，或者多项式的根随着参数的变化而变化，就会出现辫子。为了理解这些现象，有必要研究曲面的对称集，也就是曲面的映射类群。这是一个美丽而丰富的理论，在过去的一个世纪里一直是激烈研究的焦点。本研究的目标是研究表面、辫状结构和多项式，以及这些物体之间的相互作用。一个项目是给出确定映射类组元素基本属性的快速算法。该算法计算的属性之一是熵，它决定了表面上发生的混合量。除了这些研究目标之外，PI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生研讨会，这个会议既是面向研究生的拓扑研究会议，也是一个专业发展研讨会。第二个是免费的，在线的，交互式的基本线性代数教材，叫做交互式线性代数。PI还计划将外展活动扩大到当地的K-12教室。PI将研究瑟斯顿地图、编织组和地图类组。Thurston映射是复平面（或Riemann球）在自身上的分支覆盖，具有有限后临界集。许多多项式是瑟斯顿映射。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，是哪个？在之前的工作中，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描述。具体来说，该项目将从映射类群的理论出发，建立一个适合Thurston地图设置的Bestvina-Handel算法版本。该项目还将提供Birman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器的乘积作为输入，并确定该乘积的Nielsen-Thurston类型。该算法还可以产生关于生成器乘积的更精细的信息，例如减少曲线和熵。第三个项目是对编织组之间的同态进行分类。主要的新工具是由PI和他的合作者开发的完全对称集理论。通过对这些同态进行分类，我们可以深入了解不同阶多项式之间的关系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.