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细胞样 四维分解的最终目标是 提供了一个四维流形的特征， 在更高的维度中可用。 目前的重点是一个版本的 一个不相交的磁盘属性，尽管它的明显 简单性，即使在 比较“简单”的例子 该项目包括调查 分解空间的上同调局部连通性 与由绝对邻域组成的分解相关联 收缩和调查的局部同调紧有限- 维绝对邻域收缩，目标是 解决了Bing-Borsuk的猜想，即齐次有限- 推广了维绝对邻域收缩 流形

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.