Mathematical Sciences: Geometric Questions in Algebraic Topology

曼教授将研究代数拓扑学中的两个独立项目。这些项目研究了瞬子的模空间和分段线性流形和边界模上的分段线性群作用。第一个项目是与C.P.Boyer教授的合作，延续了Boyer教授和Mann教授正在进行的计划，通过使用迭代环空间及其相关的同调运算来研究在Yang-Mills规范理论的微分几何公式中出现的瞬子的模空间的几何。第二个项目代表与E.Y.Miller教授的合作，主要基于Mann和Miller教授在等变函数空间方面的工作，即为等变分段线性结构和相关动作映射的空间分类。这项工作将使曼教授能够回答几何问题，尽管这些问题对于光滑流形是众所周知的，但对于分段线性流形现在是未知的。令人欣慰的是，在第一个项目中，理论物理和拓扑学是相互促进的。

Professor Mann will investigate two independent projects in algebraic topology. These projects study moduli spaces of instantons and piecewise linear group actions on piecewise linear manifolds and bordism modules. The first project, representing joint work with Professor C.P. Boyer, continues the ongoing program of Professors Boyer and Mann to study the geometry of moduli spaces of instantons, which arise in the differential geometric formulation of Yang-Mills gauge theory, by using the theory of iterated loop spaces and their associated homology operations. The second project, representing joint work with Professor E.Y. Miller, is based in large part on the work of Professors Mann and Miller on equivariant function spaces, classifying spaces for equivariant piecewise linear structures and associated action maps. This work will allow Professor Mann to answer geometric questions which, while well-known for smooth manifolds, are now unknown for piecewise linear manifolds. It is gratifying to find theoretical physics and topology stimulating each other as in the first project.