Dimensions, universal spaces, and continua

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The concept of a universal object is very important in mathematics. It appears in algebra, analysis, geometry, and topology. What does it mean for an object X to be universal for a certain family of objects? One standard requirement is that X is an element of the family itself. Other requirements may vary. Nevertheless, the general idea is that X must contain in itself a copy of each object from the family, or each object from the family can be obtained from X by means of a certain standard procedure. I study universal objects for various versions of dimensions. One of my main goals is to find out whether a universal compactum exists in the case of cohomological dimension. To solve this problem, I intend to apply methods of extension dimension, and, in particular, further develop the theory of quasi-finite complexes that I have introduced. Another part of my program is concerned with infinite-dimensional topology. Infinite-dimensional spaces are important in many applications of mathematics. For example, Hilbert space is an essential ingredient of quantum mechanics. I study various classes of infinite-dimensional spaces. I wish to find "nice" characterizations for these classes to better understand the structure of spaces. The third part of my proposal is devoted to spans of continua. The simplest example of a continuum is a metric graph, which can be visualized as a network of roads. The span of such graph can be described as the maximal distance two travellers can keep between each other while simultaneously traversing the graph. I am going to investigate properties of span, such as relations between types of spans. I also want to find out whether continua with span zero are similar to a line segment. The concluding part of my program is devoted to the study of various analogs between commutative and non-commutative topology. By studying properties of algebras of functions and their connection to the properties of underlying spaces I hope to discover new properties of C*-algebras.