Homogeneous spaces and dimension theory

同质性是一个非常自然的概念，存在于科学的各个领域。在拓扑学中，如果一个空间的每两个点都存在一个将其中一个点发送到另一个点的同胚，则称为齐次空间。直观地说，这意味着每个点周围的空间看起来都是一样的。许多模拟现实现象的拓扑空间是齐次的，如欧几里得流形、无限维流形、螺线管、Menger紧致空间等，其中许多空间也表现出类分形的行为。然而，并不是所有的分形图都是均匀的。尽管这个概念很重要，但拓扑同质性仍然没有得到很好的理解。我的课程的目标之一是研究齐次空间。我特别感兴趣的是“漂亮的”同质空间的结构，也就是那些绝对邻里缩进(ANR)的空间。请注意，所有流形以及更一般的多面体都是ANR。然而，有许多奇异的ANR例子与多种形式有很大的不同。我想要研究的中心问题之一是Bing-Borsuk猜想：每个有限维的齐次ANR-紧致都是流形吗？Jakobsche在1980年证明了Bing-Borsuk猜想蕴含着著名的Poincare猜想，最近由Grigori Perelman证明了这一猜想。另一个目标是发展一种构造齐次空间的统一方法。

Homogeneity is a very natural concept, existing in various areas of science. In topology, a space is called homogeneous if for every two of its points there exists a homeomorphism that sends one of these points to the other. Intuitively, it means that the space looks the same around each of its points. Many topological spaces, that model real-life phenomena, are homogeneous, e.g. Euclidean manifolds, infinite-dimensional manifolds, solenoids, Menger compacta, etc. Many of these spaces also exhibit a fractal-like behaviour. However, not all fractals are homogeneous. Despite the importance of the concept, topological homogeneity is still not well-understood. One of the goals of my program is the study of homogeneous spaces. In particular, I am interested in the structure of “nice” homogeneous spaces, i.e. those which are absolute neighbourhood retracts (ANRs). Note that all manifolds and, more generally, polyhedra are ANRs. However, there are many exotic examples of ANRs that differ substantially from manifolds. One of the central problems I would like to work on is the Bing-Borsuk conjecture: is every finite-dimensional homogeneous ANR-compactum a manifold? Jakobsche showed in 1980 that the Bing-Borsuk conjecture implies the famous Poincare conjecture, recently proved by Grigori Perelman.Another objective is to develop a unified approach to construction of homogeneous spaces.