Geometric Limits in Higher Teichmueller Theory

流形是一个抽象的数学对象，在小的时候看起来像我们生活的空间，但在大的时候可以有不同的全局属性。流形上的几何结构是一种测量在特定对称群下不变的量的方法，就像距离和角度在旋转或平移下不会改变一样。在许多情况下，在一个固定的流形上，人们可以考虑具有相同基本对称群的不同几何结构。这种可能结构的世界就是模空间。一个例子是经典的Teichmuller空间，它参数化了某些二维流形上的常负曲率度量。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。该奖项提供资金的研究，重点是在更高的Teichmuller理论的基本问题：我们如何描述这些对象的几何形状开始在模空间中的参数？特别是，我们能理解当参数离开模空间中的所有紧集时，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由PI监督的本科研究进行。PI计划使用Higgs bundle，调和映射和表示论的技术研究具有秩2半单对称李群的低维流形上的几何结构，例如anti-de Sitter，凸投影和Lorentzian共形平坦。更确切地说，PI将研究这些结构的几何极限，当参数离开对应模空间中的每个紧集时，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。 Gromov的另一项研究涉及到一个长期存在的问题，即在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，他将利用凸优化工具对此进行研究。该奖项反映了NSF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.