Geometric Limits in Higher Teichmueller Theory

高等Teichmueller理论中的几何极限

• 批准号: 2005501
• 负责人: Andrea Tamburelli
• 金额: $ 13.65万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-15 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005501&HistoricalAwards=false
• 关键词: Geometric; Limits; Higher; Teichmueller; Theory

A manifold is an abstract mathematical object that in the small looks like the space we live in, but that can have different global properties in the large. A geometric structure on a manifold is a way of measuring quantities that are invariant under a certain group of symmetries, the same way distances and angles do not change under rotations or translations. In many cases, on a fixed manifold, one can consider different geometric structures with the same underlying group of symmetries. The world of possible such structures is the moduli space. One example is the classic Teichmuller space that parametrizes metrics of constant negative curvature on certain two dimensional manifolds. Higher Teichmuller theory studies more general geometric structures on more complicated manifolds and their dynamical properties. This award provides funding for the research that focuses on fundamental questions in higher Teichmuller theory: how can we describe the geometry of these objects starting from the parameters in the moduli space? In particular, can we understand how these structures degenerate when the parameters leave all compact sets in the moduli space? Some numerical experiments related to this research will be carried out as undergraduate research supervised by the PI.The PI plans to study geometric structures on low-dimensional manifolds with a rank 2 semi-simple Lie group of symmetries, such as anti-de Sitter, convex projective and Lorentzian conformally flat, using techniques from Higgs bundles, harmonic maps and representation theory. More precisely, the PI will study geometric limits of these structures when the parameters leave every compact set in the corresponding moduli spaces, with the aim of finding a (partial) compactification of these moduli spaces and give a geometric interpretation of the boundary points. A different line of research is related to a long-standing question by Gromov about existence and regularity of minimal area metrics on surfaces with a fixed systolic constraint, which he will investigate exploiting tools from convex optimization.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.

歧管是一个抽象的数学对象，在小外观中，就像我们所生活的空间一样，但在大的整体属性中可能具有不同的全局属性。歧管上的几何结构是测量在某些对称组下不变的数量的一种方式，在旋转或翻译下，距离和角度相同的距离和角度不会改变。在许多情况下，在固定的歧管上，可以考虑具有相同基础对称组的不同几何结构。可能的结构是模量空间。一个例子是经典的Teichmuller空间，该空间参数在某些二维歧管上参数恒定负曲率的指标。较高的Teichmuller理论研究了更复杂的流形及其动力学特性的更通用的几何结构。该奖项为研究重点的研究提供了资金，该研究重点介绍了更高的Teichmuller理论中的基本问题：我们如何描述这些对象的几何形状，从模量空间中的参数开始？特别是，当参数留在模量空间中所有紧凑的集合时，我们能否了解这些结构如何退化？与PI有关的一些与这项研究相关的数值实验将作为本科研究进行。PI计划在低维歧管上研究几何学结构，并具有等级为2个半简单的对称性组，例如抗De Sitter，例如Anti-De Sitter，例如Anti-De Sitter，例如Anti-De Sitter，使用希格斯束，谐波图和代表理论的技术，凸射射击和洛伦兹的扁平态。更确切地说，当参数留下相应模量空间中的每个紧凑型时，PI将研究这些结构的几何限制，目的是找到这些模量空间的（部分）紧凑型并给出边界点的几何解释。 另一项研究与格罗莫夫（Gromov）关于具有固定收缩期约束的表面上的存在和最小面积指标的规律性有关的一个长期问题有关，他将调查从凸优化中利用工具。认为值得通过基金会的智力优点和更广泛影响的评论标准来评估值得支持。