Grant: Discovery Grants Program - Individual

I am interested in various problems at the intersection of dimension theory, geometric topology and general topology. The extension theory was developed during the last fifteen years. The major results in this theory are mainly due to Dranishnikov, Dydak and Scepin. The extension theory provides a unification of the covering and cohomological dimensions, as well as a new perspective for obtaining more powerful techniques in dimension theory. It is a challenging area since it requires a combination of methods from geometric topology, general topology and homology theory. One of the major problems in extension theory is whether there exists a universal space for the class of all metric compacta of a given extension dimension. Another one is about the validity of the Dranishnikov-Dydak extensivity criterion for some classes of infinite-dimensional spaces. Developing methods to address these two problems is one of the objectives of this proposal. Another objective is the study of common properties of finite-dimensional metric homogeneous continua and Euclidean manifolds. This is motivated by the well known Bing-Borsuk conjecture stating that any absolute neighborhood homogeneous retract of a finite dimension is an Euclidean manifold. Our recent result with Karasev and Krupski that finite-dimensional homogeneous continua have the strong Cantor manifold property (a property which is possessed by Euclidean manifolds) provides another motivation for studying homogeneous spaces. Continuous selections of multi-valued maps and their applications in dimension theory is also a proposed area of research. A number of my results in dimension and extension theory are based on selection theorems. This is a theoretical research and the anticipated results would improve our knowledge in mathematics, specially in topology.