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某些非线性微分方程在各种几何环境下的解的行为。这个问题的分析和代数方面之间有着丰富的相互作用，研究人员发现这对理论和计算都很有用。目前的项目将进一步了解这种类型的几何流动的长期演变，它将提供一个精确的确定， 解的奇异结构。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。理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包的模空间是PI感兴趣的主要例子。它们也形成了数学和理论物理之间的重要交叉点。这个项目的研究进展将在纯数学之外产生影响。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。 第一个继续工作的PI杨米尔斯流Kaehler流形，紧空间的厄米-爱因斯坦度量，和结构的奇异集。 第二部分研究了Higgs丛模空间的渐近几何和整体几何。这是目前的一个领域的激烈活动的动机建设从数学物理和不变量的低维拓扑结构。 第三个项目旨在扩大和应用PI的工作决定Dolbeault拉普拉斯算子。 这将提供一个统一的框架上的Quillen度量模空间的额外的抛物结构的丛。主要研究者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.