Mathematical Sciences: Dynamical Behaviour of Vortices and Monopoles in Classical Gauge Theories

这个项目是关于随时间变化的Yang-Mills-Higgs方程。这些是非线性双曲型偏微分方程，在洛伦兹群和无限维规范群下是不变的。研究者证明了在某些情况下，可以根据位形空间上的有限维哈密顿系统构造涡或单极子运动方程的解。已知这些近似在长而有限的时间间隔内是有效的。本研究目前的目的是将这种理解扩展到更一般的情况，特别是单极子散射，它是由模空间上的Atiyah-Hitchin度量的测地线描述的。长期目标是将理解扩展到无限时间——可以想象，解渐近地接近最大范数中构型空间上根据哈密顿系统运动的漩涡和单极子的组合。产生这种猜想的原因是辐射的色散。另一个长期目标是扩展对广义相对论方程的理解，在广义相对论中，黑洞取代了漩涡和单极子。这项工作除了具有数学意义外，还可能应用于超导和粒子物理。

This project is on the time-dependent Yang-Mills-Higgs equations. These are nonlinear hyperbolic partial differential equations which are invariant under the Lorentz group and the infinite dimensional gauge group. The investigator has proved that under certain situations it is possible to construct solutions to the equations which correspond to motion of the vortices or monopoles according to a finite dimensional Hamiltonian system on the configuration space. These approximations are known to be valid on long but finite time intervals. The present aim of this investigation is to extend this understanding to more general situations, in particular to monopole scattering which is described by geodesics with respect to the Atiyah-Hitchin metric on the moduli space. A longer term aim is to extend the understanding to infinite time -- it is conceivable that asymptotically the solution approaches a combination of vortices and monopoles moving according to the Hamiltonian system on the configuration space in the maximum norm. The reason for this conjecture is the dispersion of radiation. Another long term aim is to extend the understanding to the equations of general relativity in which black holes take the place of vortices and monopoles. The work has possible applications to superconductivity and particle physics in addition to being of mathematical interest.