Moduli Spaces of Higgs Bundles, Gauge Theory, and Related Topics

模是描述特定几何或代数结构变化的参数。模空间的构造使我们更深入地了解哪些几何结构在族中表现良好，模空间本身的几何分析揭示了它们参数化对象的不变性。这个概念在几何学和物理学中起着重要的作用。它也被证明对某些应用领域很有用，比如机器人。该项目旨在扩展PI先前在基本粒子规范理论中自然产生的某些模空间上的工作。该奖项支持的研究将进一步加深我们对模空间的几何、解析和代数性质之间关系的理解。PI将继续指导本科生和研究生，以及其他早期职业研究人员。他将共同组织研讨会和会议，向学生介绍该领域的最新发展，并为最近的博士学位获得者提供一个平台，宣布他们的研究成果。这个项目的具体目标在于与全纯束、规范理论和模问题相关的复杂几何的三个领域。第一篇论文继续了π在黎曼曲面上希格斯束模空间上的研究。这包括三个子课题：(1)抛物型lambda-连接的保角极限；(2)变换黎曼曲面上希格斯束的泛模空间的解析构造；(3)模空间的渐近结构及其拓扑性质。在提议的研究的第二个领域，PI将进一步证明他对复杂的陈-西蒙斯束的研究结果。他将研究模空间上与超全纯束的联系，并澄清他的构造与先前关于上同调行列式的Atiyah代数的重要工作的关系。第三个项目是研究Yang-Mills方程的高维推广及其与全纯束复几何的关系。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。

Moduli are parameters describing the variation of a particular geometric or algebraic structure. The construction of a moduli space brings with it a deeper understanding of which geometric structures behave well in families, and the geometric analysis of the moduli space itself reveals invariant properties of the objects they parametrize. This notion plays an important role in geometry and physics. It has also proved useful to certain applied fields such as robotics. This project seeks to extend the PI’s previous work on certain moduli spaces that arise naturally from the gauge theory of elementary particles. The research supported by this award will further our understanding of the relationship between the geometric, analytic, and algebraic properties of moduli spaces. The PI will continue mentoring undergraduate and graduate students, as well as other early career researchers. He will co-organize workshops and conferences to introduce students to the latest developments in the field and to give a platform to recent Ph.D. recipients to announce the results of their research.The specific goals of this project lie in three areas of complex geometry related to holomorphic bundles, gauge theory, and moduli problems. The first continues work of the PI on moduli spaces of Higgs bundles on Riemann surfaces. These include three subprojects: (1) conformal limits for parabolic lambda-connections; (2) an analytic construction of the universal moduli space of Higgs bundles over varying Riemann surfaces; and (3) the asymptotic structure of the moduli space and its topological properties. In the second area of proposed research, the PI will prove further results surrounding his work on complex Chern-Simons bundles. He will investigate the link with hyperholomorphic bundles on moduli spaces, and also clarify the relationship of his constructions to important previous work on Atiyah algebras of determinants of cohomology. The third project is the study of higher dimensional generalizations of the Yang-Mills equations and their relationship to the complex geometry of holomorphic bundles.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.