Yang-Mills Flow and Applications

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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.