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Higgs Bundles, Surface Groups, and Conformal Limits

希格斯丛、表面群和共形极限

基本信息

批准号：
2103685
负责人：
Brian Collier
金额：
$ 14.75万
依托单位：
University of California-Riverside
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2103685&HistoricalAwards=false
关键词：
Higgs Bundles Surface Groups Conformal

项目摘要

This project is based on recent advancements in geometry which generalize hyperbolic geometry, and lie at the interface of mathematics and physics. There are many distinguished subvarieties of moduli spaces of Higgs bundles and flat connections which this project ties together and analyzes how these spaces change as defining data is varied. The PI uses tools coming from Higgs bundle theory, deformation theory and geometric representation theory to tackle these questions. Since both the objects and the tools used involve the interaction of many different mathematical fields, this works has cross-disciplinary implications. The award also supports out-reach programs aimed at youths in demographics traditionally underrepresented in STEM disciplines.This project centers on the interaction between the geometry of surface group representations, representation theory of Lie groups, and algebraic and analytic techniques in Higgs bundles theory. In these projects, a central role is played by the PI's work on Global Slodowy slices, which globalizes certain Lie theoretic spaces to moduli spaces of Higgs bundles and holomorphic connections. The main goals of the project center on elucidating which aspects of the nonabelian Hodge correspondence change as the Riemann surface is deformed and to identify structures and subvarieties which are constant under such deformations. As a result, this will deepen the understanding of geometries associated to surface groups and their relation to Higgs bundles and representation theory. A special focus is placed on generalizing the Fuchsian locus in the space of Quasi-Fuchsian representations to higher rank and explaining the role of the conformal limit.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.

这个项目是基于几何学的最新进展，它推广了双曲几何，并位于数学和物理的交界处。Higgs丛和平坦联系的模空间有许多不同的子种类，本项目将它们联系在一起，并分析这些空间如何随着定义数据的不同而变化。PI使用了希格斯丛理论、形变理论和几何表示理论的工具来解决这些问题。由于所使用的对象和工具都涉及许多不同的数学领域的相互作用，这项工作具有跨学科的影响。该奖项还支持针对传统上在STEM学科中代表性不足的人口统计学中的年轻人的外展项目。该项目集中于表面群表示的几何、李群的表示理论以及希格斯丛理论中的代数和分析技术之间的相互作用。在这些项目中，PI在Global Slodowy切片上的工作发挥了核心作用，它将某些Lie理论空间全球化为Higgs丛和全纯连接的模空间。该项目的主要目标是阐明非阿贝尔霍奇对应的哪些方面随着黎曼面的变形而变化，并确定在这种变形下保持不变的结构和亚种。因此，这将加深对与表面群相关的几何以及它们与希格斯丛和表示理论的关系的理解。特别关注的是将准福克斯表示空间中的福克斯轨迹推广到更高的级别，并解释保角极限的作用。这一裁决反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。