CAREER: Higgs bundles and Anosov representations

本项目着重于曲面的数学研究，通过将代数对象与曲面联系起来，从而推广其应用范围。使用的主要概念之一是曲面群表示，这个概念将曲面与经典几何（如欧几里得几何和双曲几何）的推广联系起来。表面的研究在数学和物理的许多领域都有惊人的应用。因此，该项目处于多个学科的交叉点。除了尖端的数学研究外，该项目还将通过针对研究生的不同研讨会以及社区外展活动来促进科学和数学的进步。教育部分还将侧重于为学生和早期职业研究人员创造一个有吸引力和包容性的数学互动场所。在过去的几十年里，希格斯束理论和阿诺索夫动力学都使我们对表面群几何的理解取得了重大进展。最近将这些方法联系起来的突破是间接的，主要涉及双曲几何的高阶推广，即高阶Teichmuller空间。本课题的主要目标是超越高阶的Teichmuller空间，利用希格斯束来识别曲面群表示的子变体，这些子变体推广了拟富克斯空间中的富克斯轨迹。该方法的基石是希格斯束的慢速切片的作用。具体来说，PI旨在建立与希格斯束模空间中slow切片相关的表面群表示的Anosov性质。这种方法将极大地扩展希格斯束的应用到阿诺索夫表示和（G，X）几何。它将完成曲面群表示模的分量计数。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。

This project focuses on the mathematical study of curved surfaces by connecting algebraic objects to them and thereby generalizing the scope of their application. One of the main notions used is that of a surface group representation, a concept which connects surfaces to generalizations of classical geometries such as Euclidean and hyperbolic geometry. The study of surfaces has surprising applications throughout many fields of mathematics and physics. Consequently, the project lies at the intersection of multiple disciplines. In addition to cutting edge mathematical research, the project will promote the progress of science and mathematics through different workshops aimed at graduate students as well as community outreach events. The educational component will also focus on creating an engaging and inclusive place for mathematical interactions for students and early career researchers.In the past decades, both the theories of Higgs bundles and Anosov dynamics have led to significant advancements in our understanding of the geometry of surface groups. Recent breakthroughs linking these approaches are indirect and mostly involve higher rank generalizations of hyperbolic geometry known as higher rank Teichmuller spaces. The broad aim of this project is to go beyond higher rank Teichmuller spaces by using Higgs bundles to identify subvarieties of surface group representations which generalize the Fuchsian locus in quasi-Fuchsian space. The cornerstone for the approach is the role of Slodowy slices for Higgs bundles. Specifically, the PI aims to establish Anosov properties of surface group representations associated to Slodowy slices in the Higgs bundle moduli space. This approach will significantly extend applications of Higgs bundles to both Anosov representations and (G,X) geometries. It will complete the component count for moduli of surface group representations.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.