CAREER: Higgs bundles and Anosov representations

职业：希格斯丛集和阿诺索夫表示

• 批准号: 2337451
• 负责人: Brian Collier
• 金额: $ 47万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2029-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2337451&HistoricalAwards=false
• 关键词: CAREER; Higgs; bundles; Anosov; representations

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.

该项目通过将代数对象连接到它们，从而概括其应用程序范围，重点介绍曲面的数学研究。所使用的主要概念之一是表面组表示，该概念将表面与诸如欧几里得和双曲线几何形状等经典几何形状的概括相关联。对表面的研究在许多数学和物理学领域都有令人惊讶的应用。因此，该项目在于多个学科的交集。除了尖端数学研究外，该项目还将通过针对研究生和社区外展活动的不同研讨会来促进科学和数学的进步。教育组成部分还将着重于为学生和早期职业研究人员创造一个引人入胜且包容的位置。在过去的几十年中，希格斯捆绑包和Anosov动力学的理论都在我们对表面群体几何形状的理解方面取得了重大进步。连接这些方法的最新突破是间接的，并且主要涉及较高的双曲几何形状概括，称为高级Teichmuller空间。该项目的广泛目的是通过使用Higgs捆绑识别表面组表示的亚变化，超越高级Teichmuller空间，这些束概括了flouthsian locus in quasi-fuchsian空间中。该方法的基石是希格斯捆绑包的slo缝片的作用。具体而言，PI旨在建立与Higgs束模量空间中与Slodowy Slices相关的表面组表示的ANOSOV特性。这种方法将显着将希格斯捆绑包的应用扩展到Anosov表示和（G，X）几何形状。它将完成表面群体表示模量的组件计数。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响审查标准来评估值得支持的。