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Character Varieties and Cluster Mutations

性状变异和簇突变

基本信息

批准号：
1711405
负责人：
Christian Zickert
金额：
$ 18.61万
依托单位：
University of Maryland, College Park
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1711405&HistoricalAwards=false
关键词：
Character Varieties Cluster Mutations

项目摘要

Every surface can be approximated by a space obtained by gluing together triangles. This fact is used, for example, in computer graphics where animated figures are represented by moving polyhedra. Similarly, curved spaces in three dimensions, which are important in Einstein's relativity theory, can be represented by spaces obtained by gluing together tetrahedra. This project will study abstract properties of a space by using the concrete combinatorics of the gluing pattern. This approach is well suited for computer experimentation, and students participating in the research can perform experiments without fully understanding the abstract theory. The investigator and his students will explore new structures, derive new formulas for invariants, perform extensive computer experiments, and extend the underlying theory to higher dimensions.The project will study certain configuration spaces of flags introduced by Fock and Goncharov. These configuration spaces have concrete coordinates and very interesting structures. The coordinates give rise to explicit coordinates on higher Teichmuller spaces for triangulated surfaces, and to coordinates on the (decorated) character variety of triangulated 3-manifolds. The cluster structure of the coordinates also gives rise to explicit formulas for invariants such as the Chern-Simons invariant. The work will investigate the structure of the configuration spaces and explore their relationship to polylogarithms. The project aims to derive new formulas for invariants and extend the theory to higher dimensions.

每个表面都可以通过将三角形粘合在一起获得的空间来近似。例如，在计算机图形学中使用了这一事实，其中动画人物由移动多面体表示。同样，在爱因斯坦相对论中很重要的三维弯曲空间可以用四面体粘合在一起获得的空间来表示。该项目将通过使用粘合图案的具体组合来研究空间的抽象属性。这种方法非常适合计算机实验，参与研究的学生可以在不完全理解抽象理论的情况下进行实验。研究者和他的学生将探索新的结构，推导出不变量的新公式，进行广泛的计算机实验，并将基础理论扩展到更高的维度。该项目将研究福克和冈察洛夫引入的某些旗帜的配置空间。这些配置空间有具体的坐标和非常有趣的结构。这些坐标产生了三角曲面的更高 Teichmuller 空间上的显式坐标，以及三角 3 流形的（装饰的）字符变体上的坐标。坐标的簇结构还产生了不变量的显式公式，例如 Chern-Simons 不变量。这项工作将研究配置空间的结构并探索它们与多对数的关系。该项目旨在推导出不变量的新公式，并将该理论扩展到更高的维度。