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，一个空间被称为齐次的，如果对于它的每两个点存在一个同胚，将其中一个点发送到另一个点。直观地说，这意味着空间在其每个点周围看起来都是一样的。许多拓扑空间，模型的现实生活中的现象，是齐次的，例如欧几里得流形，无限维流形，Menger流形，等等，许多这些空间也表现出分形的行为。然而，并非所有的分形都是均匀的。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。我的计划的目标之一是研究齐次空间。特别是，我感兴趣的结构“好”齐性空间，即那些是绝对邻域收缩（ANR）。请注意，所有流形，更一般地说，多面体都是ANR。然而，有许多奇异的ANR例子与流形有很大的不同。其中一个中心问题，我想工作的是宾博苏克猜想：是每一个有限维齐次ANR-紧流形？雅各布谢在1980年表明，宾-博苏克猜想暗示著名的庞加莱猜想，最近证明了格里戈里佩雷尔曼。 另一个目标是发展一个统一的方法来构造齐次空间。 我的程序的第二部分是关于无限维拓扑的。无限维空间在数学的各个领域及其在物理学中的应用中起着重要的作用。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属性，并提高我们对这些类型之间差异的理解。 我的建议的第三部分致力于渐近维数。拓扑学的经典方法是研究空间、它们的性质和“小尺度”上的不变量，而渐近拓扑学分析空间的“大尺度”结构。有几个类似于维度不变量的设计用于大尺度类别。这些不变量中的一些与性质A有关，性质A由Guoliang Yu在2000年引入。我希望更好地理解这些联系，以及推广某些定理和建设，从通常的维数理论的情况下，渐近维数。 拓扑学的研究，特别是维数理论，有助于推进基础科学知识。它还在物理学，数据分析，经济学，可视化和其他领域中应用。

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.