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。我的程序的第二部分涉及无限维拓扑。无限维空间在数学的各个领域及其在物理学中的应用中扮演着重要的角色。有不同类型的无限维空间。我的目标是找到其中一些类型的特征性质，并提高我们对这些类型之间的差异的理解。我的建议的第三部分致力于渐近维度。拓扑学中最经典的方法是研究空间、它们的性质和“小尺度”上的不变量，而渐近拓扑学则分析空间的“大尺度”结构。有几种类似的类维不变量被设计成在大规模类别中工作。其中一些不变量与性质A有关，该性质由余国良在2000年引入。我希望更好地理解其中的一些联系，并在渐近维度的情况下从通常的维度理论中推广某些定理和构造。拓扑学的研究，特别是维度理论的研究，有助于推进基本的科学知识。它在物理学、数据分析、经济学、可视化等领域也有应用。

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.The second part of my program is concerned with infinite-dimensional topology. Infinite-dimensional spaces play an important role in various areas of mathematics and its applications in physics. There are different types of infinite-dimensional spaces. My goal is to find characterizing properties of some of these types and improve our understanding of differences between these types.The third part of my proposal is devoted to asymptotic dimension. While the classical approach in topology is to study spaces, their properties, and invariants on the "small scale", asymptotic topology analyzes "large scale" structure of spaces. There are several analogs of dimension-like invariants that are designed to work in the large scale category. Some of these invariants are related to Property A, instroduced by Guoliang Yu in 2000. I wish to better understand some of these connections as well as to generalize certain theorems and constructions from usual dimension theory on the case of asymptotic dimension.Research in topology, and in particular in dimension theory, helps to advance fundamental scientific knowledge. It also has applications in physics, data analysis, economics, visualization, and other areas.