Yang-Mills Flow and Applications

数学规范理论是现代几何的一个分支，部分植根于高能物理。它的中心对象是1954年由杨振宁和米尔斯发现的核力理论中出现的规范场，数学家们已经很熟悉了。杨-米尔斯场方程随后对几何学产生了巨大的影响，特别是通过唐纳森关于四维奇异光滑结构的工作。杨-米尔斯流是一个自然演化过程，旨在求解杨-米尔斯方程，就像汉密尔顿的利奇流对爱因斯坦方程所做的那样。这项研究的目的是通过改进和扩展PI关于Yang-Mills流动分析行为的结果来寻求数学规范理论的新应用。在第四维中，还需要建立无限时间Uhlenbeck极限的唯一性以及泡点的位置，这就需要执行Leon Simon在存在奇点时的著名收敛技巧。流动在四维规范理论中的应用主要有两个方向：第一个方向是应用于四维球面上杨-米尔斯方程的非极小解，其中Willmore猜想(Marques和Neves在2012年证明)的类似性仍然未知。其次是对4-球面或K3曲面上瞬时子模空间稳定拓扑的Atiyah-Jones猜想。更简单的3-流形上的Yang-Mills泛函的情况在很大程度上也是未被探索的。最后，PI打算将Yang-Mills Flow发展为Donaldson-Thomas计划中的工具，用于具有特殊完整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智力优势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.