Mathematical Sciences: The Topology of Lattice Gauge Fields

格点规范场是物理学家发明的离散场 主纤维丛中的连接模型。 安东尼 菲利普斯以前的工作（与大卫斯通）的计算 拓扑电荷导致了一个局部的一般方案 计算表示真实的特征类的上循环。 计划中的研究的一部分涉及将该计划扩展到 更广泛的结构群。 另一部分涉及 修改该方案以检测SU（n）-中的U（1）-磁单极 理论在某种程度上是可靠的和有效的格 计算。 菲利普斯还计划调查拓扑结构 和G-丛中H-单极子的几何 H. 最后，他打算解决一个组合问题， 列举了一个特定类别的迷宫，其中出现在他的前 研究格点规范场 总的来说，这个项目采用了一些 微分几何中最现代的数学工具 和拓扑学来分析一种结构， 量子力学的公式

Lattice gauge fields were invented by physicists as discrete models of connections in principal fibre bundles. Anthony Phillips' previous work (with David Stone) on calculations of topological charge led to a general scheme for a local calculation of cocycles representing real characteristic classes. Part of the planned research involves extending this scheme to a wider class of structure groups. Another part involves modification of this scheme to detect U(1)-monopoles in SU(n)- theories in a way that is both reliable and efficient for lattice computations. Phillips also plans to investigate the topology and geometry of H-monopoles in G-bundles for general G including H. Finally, he intends to attack a combinatorial problem, the enumeration of a certain class of mazes which arose in his prior work on lattice gauge fields. Overall, this project employs some of the most modern mathematical tools of differential geometry and topology to analyze a type of structure which arises in one formulation of quantum mechanics.