CAREER: Higgs bundles and Anosov representations

该项目致力于通过将代数对象连接到曲面来对曲面进行数学研究，从而推广了曲面的应用范围。使用的主要概念之一是表面群表示，这是一个将曲面与欧几里得和双曲几何等经典几何的推广相联系的概念。曲面的研究在数学和物理的许多领域都有令人惊讶的应用。因此，该项目位于多个学科的交叉点。除了前沿的数学研究外，该项目还将通过针对研究生的不同讲习班以及社区外联活动来促进科学和数学的进步。教育部分还将专注于为学生和早期职业研究人员创造一个引人入胜和包容的数学互动场所。在过去的几十年里，希格斯丛和阿诺索夫动力学理论都导致了我们对曲面群几何的理解的重大进步。最近将这些方法联系在一起的突破是间接的，主要涉及双曲几何的高阶推广，称为高阶Teichmuller空间。这个项目的主要目的是通过使用Higgs丛来识别表面群表示的子簇，从而超越高阶TeichMuller空间，这些子簇将Fuchsian轨迹推广到拟Fuchsian空间中。这种方法的基石是希格斯束的偷懒切片所起的作用。具体地说，PI的目的是建立与Higgs丛模空间中的Slodowy切片相关的曲面群表示的Anosov性质。这种方法将显著地将Higgs丛的应用扩展到Anosov表示和(G，X)几何。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.