Monopole moduli spaces and manifolds with corners

非阿贝尔磁单极子是存在于普通三维空间中的微分方程的粒子状解。我们知道，有些解对应于分离很远的粒子，但随着分离的减小，粒子失去了它们的个体特性，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，但预测单极子在运动时会相互作用。当它们以低速运动时，它们的运动可以很好地近似于某个弯曲空间上的测地线，就像爱因斯坦广义相对论中的粒子在引力的影响下运动一样。本课题将研究弯曲空间M的距离函数（度量），特别是研究在大距离下的距离函数。通过对度量的详细研究，我们还可以研究定义在空间M上的自然微分方程。这在一般微分方程的研究中是有意义的，因为M的大规模结构相当复杂，只能递归地建立。微分方程的行为M是重要的，因为它们是必要的理解量子理论的原始杨-米尔斯-希格斯理论。这里的重要成分还是M的大尺度结构及其度规，这项研究不仅对于理解磁单极和杨-米尔斯-希格斯理论的特征（它们是其中的一部分）非常重要，而且对于开发一套复杂的工具来理解这种设置中的微分方程也非常重要。这些工具将是有用的类似的问题，可以预期有一个重要性超出其应用的理论的单极子。

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.