Braids, Surfaces, and Polynomials

辫子、曲面和多项式

基本信息

批准号：
2203431
负责人：
Dan Margalit
金额：
$ 39.6万
依托单位：
Georgia Tech Research Corporation
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-09-01 至 2024-03-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203431&HistoricalAwards=false
关键词：
Braids Surfaces Polynomials

项目摘要

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.

每当我们建模物理，生物学或化学系统时，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，通常会出现表面。例如，机械系统的一组配置或大尺寸数据集的基础模板。每当我们在表面上移动的点的集合，例如恒星和行星在我们的视野内移动，或多项式的根部随着参数而变化时，就会发生辫子。为了理解这些现象，必须研究表面的对称性集，这也称为表面的映射类群。这是一个美丽而丰富的理论，在过去的一个世纪中一直是激烈研究的重点。这项研究的目的是研究表面，辫子和多项式，以及这些对象之间的相互作用。一个项目是提供快速算法来确定映射类组元素的基本属性。算法计算的属性之一是熵，它决定了表面上发生的混合量。除了这些研究目标外，PI计划继续从事几个直接影响研究生，本科生和高中生的项目。第一个是非常成功的拓扑专业学生研讨会，该会议既是研究生拓扑的研究会议，又是专业发展研讨会。第二个是基本线性代数的免费，在线交互式教科书，称为交互式线性代数。 PI还计划将外展活动扩展到本地的K-12教室。PI将学习Thurston地图，编织组和地图课程组。 Thurston地图是自身上面的复杂平面（或Riemann Sphere）的分支封面，并具有有限的临界后集合。许多多项式是瑟斯顿地图。复杂动力学中的一个基本识别问题是：给定瑟斯顿地图，它等于多项式，如果是，哪一个？在先前的工作中，PI和他的合作者提供了一种新的几何算法来解决此识别问题。 PI计划调查此类识别问题宇宙的新结构描述。具体而言，该项目将从映射班级组的理论中建立最BestVina汉语算法的版本，该算法适合于瑟斯顿地图的设置。该项目还将提供Birman精确序列的版本（再次来自映射课程组的理论）。映射课程组理论中的一个项目是给出一种二次时间算法，该算法将其作为映射类组的发电机的输入，并确定该产品的Nielsen-Thurston类型。该算法还会产生有关发电机产品的更精细信息，例如减少曲线和熵。第三个项目是对编织组之间的同态分类。主要的新工具是由PI及其合作者开发的完全对称集的理论。通过对这些同态分类，我们可以深入了解不同程度的多项式之间的关系。该奖项反映了NSF的法定使命，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估标准来评估值得支持的。