Higgs Bundles, Surface Groups, and Conformal Limits

这个项目是基于几何的最新进展，它推广了双曲几何，并处于数学和物理的界面。希格斯束和平面连接的模空间有许多不同的子变体，该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。由于所使用的对象和工具都涉及许多不同数学领域的相互作用，因此这项工作具有跨学科的含义。该奖项还支持面向传统上在STEM学科中代表性不足的年轻人的外展项目。本项目主要研究表面群表示的几何、李群表示理论以及希格斯束理论中的代数和解析技术之间的相互作用。在这些项目中，PI在全局慢速切片上的工作发挥了核心作用，它将某些李论空间全局化为希格斯束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曲面的变形而变化，并识别在这种变形下不变的结构和子变种。因此，这将加深对与表面群相关的几何形状及其与希格斯束和表示理论的关系的理解。特别着重于将拟Fuchsian表示空间中的Fuchsian轨迹推广到更高的秩，并解释保角极限的作用。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 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.