Higgs Bundles, Surface Groups, and Conformal Limits

这个项目是基于几何学的最新进展，它概括了双曲几何，并位于数学和物理的界面。希格斯束和平坦连接的模空间有许多著名的子变种，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变化而变化。PI使用来自希格斯束理论，变形理论和几何表示理论的工具来解决这些问题。由于使用的对象和工具涉及许多不同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青年的外展计划。该项目的中心是表面群表示的几何学，李群表示理论以及希格斯束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在全局Slodowy切片上的工作发挥了核心作用，该切片将某些Lie理论空间全局化为Higgs丛和全纯连接的模空间。该项目的主要目标集中在阐明哪些方面的nonabelian霍奇对应变化的黎曼曲面变形，并确定结构和subvariations下是恒定的这种变形。因此，这将加深对与表面群相关的几何及其与希格斯束和表象理论的关系的理解。 一个特别的重点放在推广Fuchsian轨迹在空间中的准Fuchsian表示到更高的rank和解释的作用，共形limit.This奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.