Mathematical Sciences: Decompositions in Dimension Four, Local Connectivity at Decomposition Spaces, Homogeneity and Generalized Manifolds

沃尔什教授将继续研究第四维的胞状分解，最终目标是提供一个可与更高维的流形相媲美的四维流形的特征。目前的焦点是一种不相交的磁盘特性，尽管它表面上很简单，但即使在相对“简单”的例子中也可能极难检测到。该项目包括研究由绝对邻域收缩构成的分解空间的上同调局部连通性和紧致有限维绝对邻域收缩的局部同调，目的是解决Bing-Borsuk关于齐次有限维绝对邻域收缩是广义流形的猜想。

Professor Walsh will continue to investigate cell-like decompositions in dimension four with the ultimate goal of providing a characterization of four-manifolds comparable to that available in higher dimensions. Current focus is on a version of a disjoint-disks-property that, in spite of its apparent simplicity, can be extremely difficult to detect even in relatively "simple" examples. The project includes investigating the cohomological local connectedness of decomposition spaces associated to decompositions consisting of absolute neighborhood retracts and investigating the local homology of compact finite- dimensional absolute neighborhood retracts with the goal of resolving the conjecture of Bing-Borsuk that homogeneous finite- dimensional absolute neighborhood retracts are generalized manifolds.