Professors Arhangelskii and Eisworth will focus on open problems regarding the interaction between compactness and topological homogeneity. The project is loosely organized around a famous question of van Douwen: Does there exist a homogeneous compactum containing a family of disjoint open sets of size greater than the continuum? Prior work has yielded a meager supply of techniques for building homogeneous compacta, all of which have fallen short of answering van Douwen's question. Similarly, research over the past twenty years has provided a few techniques for proving that a space is not homogeneous, but still general methods are lacking. Work of Arhangelskii has started to illuminate the connections between various counterexamples, and the project uses this work as a springboard toward attacking more fundamental questions.
General Topology is one of the great unifiers of modern mathematics, and its language and ideas are ubiquitous throughout the mathematical canon. The subject is concerned with finding general ideas and structures that are common to diverse areas of mathematics, and two of the most fundamental such general ideas are homogeneity and compactness. Loosely speaking, a structure is homogeneous if it "looks the same everywhere", while compactness is a property capturing "completeness" properties of the unit interval [0,1]. The interaction between these two topics is not well-understood --- known results have demonstrated that the relationship is complex, but the subject resisted the development of general techniques for constructing examples and counterexamples. Recent work of Arhangelskii has begun to shed light on some of these problems, and the project integrates this together with Eisworth's work on applications of set theory in topology.
