Singularity Formation in Kahler Geometry and Yang-Mills Instantons

本计画将研究由非线性偏微分方程所控制的特殊几何结构的奇异性形成。这些结构是爱因斯坦引力方程和麦克斯韦电磁方程解的推广，在现代理论物理学（例如弦论和M理论）中发挥着重要作用。这些问题的解决将增进对数学各研究方向之间深层互动关系的理解。与此同时，也有许多有趣的相关问题可以作为研究项目，培养对该领域感兴趣的研究生。PI将从两个具体方向研究问题。首先，PI想研究卡-丘度量的坍缩，以及与代数几何和模紧化的联系。PI和合作者以前的工作已经在最小退化的情况下产生了令人满意的描述，PI建议研究更一般的情况。这就需要从坍缩理论中利用现有的技术，并进一步发展这些技术。其次，PI将研究杨-米尔斯瞬子的奇异性分析。在Kahler流形上的Hermitian-Yang-米尔斯联络的情况下，PI和Xuemiao Chen最近的工作给出了切锥的完整代数几何描述，PI希望通过发现更精细的结构和研究G2瞬子的情况来进一步推动这些。这需要建立代数几何框架，以及将其中一些扩展到更多的分析设置。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This project will study singularity formation of some special geometric structures governed by non-linear partial differential equations. These structures are generalizations of solutions to the Einstein’s equation for gravity and Maxwell’s equation for electromagnetism, and play significant roles in modern theoretical physics (for example string theory and M-theory). The solution to these problems will enhance the understanding of deep interaction between various research directions in mathematics. In the meantime, there are many interesting related questions that may serve as research projects for the training of graduate students interested in this area.The PI will study problems in two specific directions. First, the PI would like to study collapsing of Calabi-Yau metrics, and connections with algebraic geometry and moduli compactification. Previous work of the PI and collaborators has yielded a satisfactory description in the case of minimal degenerations and the PI proposes to study the more general situation. This requires exploiting the existing techniques from collapsing theory, and further developing those. Secondly, the PI will study singularity analysis for Yang-Mills instantons. In the case of Hermitian-Yang-Mills connections over Kahler manifolds, recent work of the PI and Xuemiao Chen gives a complete algebro-geometric description of the tangent cones, and the PI wants to push these further by discovering more refined structures, and by investigating the case of G2 instantons. This requires building algebro-geometric framework, as well as extending some of these to the more analytic setting.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.