RUI: Applications of Non-Commutative Algebra to Character Varieties

RUI：非交换代数在字符簇中的应用

• 批准号: 1309376
• 负责人: Virgil Pierce
• 金额: $ 11.82万
• 依托单位: The University of Texas Rio Grande Valley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1309376&HistoricalAwards=false
• 关键词: RUI; Applications; Non; Commutative; Algebra

This project concerns moduli spaces of representations of finitely generated groups into Lie groups. Many moduli spaces of interest arise in this fashion: flat fiber bundles over a surface, and (parabolic) Higgs bundles are examples. These spaces are sometimes called character varieties. The study of character varieties intersects the invariant theory of products of matrix groups, and so owes much of its structure to the non-commutative ring of generic matrices (matrices over polynomial indeterminates). Since character varieties may be considered spaces of equivalence classes of representations of the fundamental group of a base space (topology) into a Lie group (geometry), they find applications throughout differential geometry and mathematical physics. The central goal of the proposed work is to use the strong relationship to non-commutative ring theory to answer questions about the homotopic, arithmetic, and dynamical structure of character varieties.Mathematicians' endeavor to classify mathematical objects is not unlike, for example, the biologists' endeavor to classify species. The purpose of this project is to classify and advance the understanding of certain mathematical objects called "character varieties." Informally, character varieties provide data which encodes equivalent ways one can associate the flexible shape of a object to a rigid shape of that same object. For example, a rubber band is a flexible "circle," but once one stipulates a radius a rigid circle is determined. The study of the geometry of character varieties in some sense can be thought of as the "geometry of geometry." In part because of this, character varieties find applications throughout differential geometry and in mathematical physics. In this project, the PI and his collaborators and students will use certain non-symmetrical structures associated with character varieties as a tool to explore and answer novel questions about their shape, and how their elements relate to each other by certain motions. Another important part of this project is the training of undergraduate research assistants in the PI's Experimental Algebra and Geometry Lab, and the conducting of community outreach activities which communicate the importance and beauty of pure mathematics.

这个项目关注的模空间的表示的n-生成群到李群。许多令人感兴趣的模空间都是以这种方式出现的：表面上的扁平纤维束和（抛物）希格斯束就是例子。这些空间有时被称为字符簇。特征标簇的研究与矩阵群的乘积的不变理论相交，因此其结构在很大程度上归功于一般矩阵（多项式不定式上的矩阵）的非交换环。由于特征标簇可以被认为是一个基本空间（拓扑）的基本群到李群（几何）的等价类的空间，它们在微分几何和数学物理中都有应用。这项工作的中心目标是利用与非交换环理论的强关系来回答有关同伦、算术和特征变量的动力学结构的问题。数学家对数学对象进行分类的奋进与生物学家对物种进行分类的奋进没有什么不同。这个项目的目的是分类和推进对某些被称为“字符变体”的数学对象的理解。“非正式地说，字符种类提供的数据编码等效的方式，可以将一个对象的灵活形状与同一对象的刚性形状相关联。例如，橡皮筋是一个灵活的“圆”，但一旦规定了半径，就确定了刚性圆。字簇几何学的研究在某种意义上可以被认为是“几何学的几何学”。部分原因是，字符变体在微分几何和数学物理中找到了应用。在这个项目中，PI和他的合作者和学生将使用与角色品种相关的某些非对称结构作为工具来探索和回答有关其形状的新问题，以及它们的元素如何通过某些运动相互关联。该项目的另一个重要部分是在PI的实验代数和几何实验室培训本科研究助理，并开展社区外展活动，传达纯数学的重要性和美感。