Yang-Mills Flow and Applications

数学规范理论是现代几何学的一个分支，部分源于高能物理学。它的中心对象是出现在C.-发现的核力理论中的规范场。N. Yang和R.米尔斯在1954年，这已经是熟悉的数学家。杨-米尔斯场方程随后对几何学产生了巨大的影响，特别是通过唐纳森对四维奇异光滑结构的研究。杨-米尔斯流是一个自然演化过程，旨在解决杨-米尔斯方程，很像汉密尔顿的里奇流爱因斯坦方程。本研究的目标是通过改进和扩展PI关于杨-米尔斯流的分析行为的结果来追求数学规范理论的新应用。在四维空间中，我们需要确定无限时间乌伦贝克极限的唯一性以及起泡点的位置，这需要在奇点存在的情况下执行Leon Simon的著名收敛技术。在四维规范理论中应用流动有两个主要方向：第一个是在四维球面上的杨-米尔斯方程的非极小解，其中Willmore猜想的类似物（由Marques和Neves在2012年证明）仍然未知。第二个是关于4-球面或K3曲面上瞬子模空间的稳定拓扑的Atiyah-Jones猜想。在3-流形上的杨-米尔斯泛函的简单情况也在很大程度上未被探索。最后，PI计划将杨-米尔斯流作为Donaldson-Thomas计划中的工具，用于具有特殊完整性的高维流形上的规范理论。该奖项反映了NSF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。

Mathematical gauge theory is a branch of modern geometry partly rooted in high-energy physics. Its central objects are the gauge fields appearing in the theory of nuclear forces discovered by C.-N. Yang and R. Mills in 1954, which were already familiar to mathematicians. The Yang-Mills field equations have subsequently had an enormous influence in geometry, notably through Donaldson's work on four-dimensional exotic smooth structures. Yang-Mills flow is a natural evolution process designed to solve the Yang-Mills equations, much as Hamilton's Ricci flow does for the Einstein equations. The goal of this research is to pursue novel applications to mathematical gauge theory by refining and extending the PI's results on the analytic behavior of Yang-Mills flow. In dimension four, it remains to establish the uniqueness of the infinite-time Uhlenbeck limit together with the position of the bubbling points, which requires carrying out the well-known convergence technique due to Leon Simon in the presence of singularities. There are two main directions for applying the flow within 4-dimensional gauge theory: the first is to non-minimal solutions of the Yang-Mills equations on the 4-sphere, where the analogue of the Willmore conjecture (proved by Marques and Neves in 2012) is still unknown. The second is to the Atiyah-Jones conjecture on the stable topology of instanton moduli spaces on the 4-sphere or a K3 surface. The simpler case of the Yang-Mills functional over 3-manifolds is also largely unexplored. Lastly, the PI intends to develop Yang-Mills flow as a tool within the Donaldson-Thomas program for gauge theory on higher-dimensional manifolds with special holonomy.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.