Homogeneous spaces and dimension theory

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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.