Geometry and Analysis of Moduli Spaces of Holomorphic Bundles

这个项目的主要目标是研究数学物理中出现的某些非线性微分方程解在各种几何环境下的行为。在这个问题的分析和代数方面之间存在着丰富的相互作用，研究人员发现这对理论和计算目的都很有用。目前的项目将进一步了解这种类型的几何流的长期演变，并将提供这些方程解的奇异结构的精确确定。这项研究的第二个重要方面是整个解空间(称为模空间)的全局性质。了解模空间的结构揭示了统一和解释个别例子的特定方面的潜在现象。杨-米尔斯方程和希格斯丛的模空间是PI感兴趣的主要例子。它们也构成了数学和理论物理的重要交汇点。这个项目的研究进展将对纯数学以外的领域产生影响。该项目概述了与全纯丛和模问题有关的复几何的三个主要领域的研究。第一个继续PI关于Kaehler流形上的Yang-Mills流，Hermitian-Einstein度量空间的紧化，以及奇异集的结构的工作。第二部分研究了Higgs丛的模空间的渐近几何和整体几何。这是目前一个非常活跃的领域，受到数学物理和低维拓扑中的不变量的构造的推动。第三个项目寻求扩展和应用PI关于多尔博特·拉普拉斯的决定因素的工作。这将为具有附加抛物结构的丛的模空间上的Quillen度量提供一个统一的框架。国际和平研究所在这一领域有工作经验，并发表了与当前提案主题直接相关的专题的研究文章。

The principal objective of this project is the study of the behavior of solutions to certain nonlinear differential equations arising in mathematical physics in various geometrical settings. There is a rich interplay between the analytic and algebraic aspects of this problem which researchers have found useful for both theoretical and computational purposes. The current project will further the understanding of the long time evolution of geometric flows of this type, and it will provide a precise determination of the singular structure of solutions to these equations. A second important aspect of this research is the global nature of the entire space of solutions (called the moduli space). Understanding the structure of moduli spaces reveals underlying phenomena that unify and explain particular aspects of individual examples. The Yang-Mills equations and moduli spaces of Higgs bundles are the major examples of interest to the PI. They also form an important point of intersection between mathematics and theoretical physics. Progress on the research in this project will have an impact outside of pure mathematics. The project outlines research in three principal areas of complex geometry related to holomorphic bundles and moduli problems. The first continues work of the PI on the Yang-Mills flow on Kaehler manifolds, compactifications of spaces of Hermitian-Einstein metrics, and the structure of singular sets. The second studies the asymptotic and global geometry of moduli spaces of Higgs bundles. This is currently an area of intense activity motivated by constructions from mathematical physics and invariants in low dimensional topology. The third project seeks to extend and apply the PI's work on determinants of Dolbeault laplacians. This will provide a unified framework for Quillen metrics over moduli spaces of bundles with additional parabolic structures. The PI has experience working in this field and has published research articles on topics directly related to the subject matter of the current proposal.