Geometric Limits in Higher Teichmueller Theory

流形是一种抽象的数学对象，在小范围内看起来像我们生活的空间，但在大范围内可以具有不同的全局属性。流形上的几何结构是一种测量量的方法，这些量在一定的对称组下是不变的，同样的距离和角度在旋转或平移下不会改变。在许多情况下，在固定流形上，可以考虑具有相同底层对称组的不同几何结构。可能的这种结构的世界就是模空间。一个例子是经典的Teichmuller空间，它在某些二维流形上参数化了常数负曲率的度量。高等Teichmuller理论研究更复杂流形上更一般的几何结构及其动力学性质。该奖项为研究高等Teichmuller理论中的基本问题提供了资金：我们如何从模空间的参数开始描述这些物体的几何形状？特别是，当参数在模空间中留下所有紧集时，我们能理解这些结构是如何退化的吗？与本研究相关的一些数值实验将作为本科生研究在PI的指导下进行。PI计划利用希格斯束、调和映射和表示理论的技术，研究具有2阶半简单李群对称的低维流形上的几何结构，如反德西特、凸投影和洛伦兹共形平面。更准确地说，PI将研究这些结构的几何极限，当参数离开相应模空间中的每个紧化集时，目的是找到这些模空间的（部分）紧化，并给出边界点的几何解释。另一项研究与Gromov关于具有固定收缩约束的表面上最小面积度量的存在性和规律性的长期问题有关，他将利用凸优化工具来研究这个问题。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。

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.