Monopole moduli spaces and manifolds with corners

非阿贝尔磁单极子是生活在普通三维空间上的微分方程解的粒子型。众所周知，一些解决方案对应于广泛分离的粒子，但随着分离的减小，这些粒子失去了它们的个体身份，不再能被区分。这些单极子是物理理论(杨-米尔斯-希格斯理论)的一部分，该理论无法精确求解，但它预测单极子在运动时将相互作用。当它们以低速运动时，它们的运动很好地由某一弯曲空间上的测地线来近似，就像爱因斯坦广义相对论中的粒子在重力的影响下运动一样。这个项目将研究这个弯曲空间M背后的距离函数(度量)，特别是它在大距离下是如何看起来的。通过对度量的详细研究，我们也将能够研究定义在空间M上的自然微分方程。这在一般的微分方程研究中是有兴趣的，因为M的大尺度结构相当复杂，并且只能递归建立。微分方程组在M上的行为是重要的，因为它们对于理解最初的杨-米尔斯-希格斯理论的量子理论是必要的。这里的重要因素仍然是M及其度规的大尺度结构。这项研究不仅对于理解杨-米尔斯-希格斯理论的单极子和特征将是重要的，而且对于开发一套复杂的工具来理解这种类型的微分方程也是重要的。这些工具在类似的问题中将是有用的，可以预期它们的重要性超出了它们对单极子理论的应用。

Non-abelian magnetic monopoles are particle-like solutions of a differential equation that live on ordinary three-dimensional space. It is known that some of the solutions correspond to widely separated particles, but that as the separations decrease, the particles lose their individual identity and can no longer be distinguished. These monopoles are part of a physical theory (Yang--Mills--Higgs theory) which cannot be solved exactly but which predicts that monopoles will interact non-trivially with each other when they move. When they move at low speeds, their motion is well approximated by geodesics on a certain curved space, much as particles in Einstein's theory of general relativity move under the influence of gravity. This project will study the distance function (metric) underlying this curved space M, and will study in particular how it looks at large distances.By undertaking a detailed study of the metric, we shall be also be able to study natural differential equations which are defined on the space M. This is of interest in the study of differential equations generally, since the large-scale structure of M is rather complicated, and can only be built up recursively. The behaviour of differential equations on M is important since they are necessary for the understanding of the quantum theory of the original Yang--Mills--Higgs theory. The important ingredient here again is the large-scale structure of M and its metric.This research will be important not only for understanding monopoles and features of the Yang--Mills--Higgs theory of which they are a part, but also for the development of a suite of sophisticated tools for understanding differential equations in this type of setting. These tools will be useful in similar problems and can be expected to have an importance beyond their applications to the theory of monopoles.