The PI investigates independence results in the study of countable convergence questions in the class of compact spaces. The fundamental questions concern understanding the dichotomy between the assumption that infinite sequences have converging subsequences or that the space contains a copy of the Stone-Cech compactification of the integers. One of main projects is to answer Efimov's question of whether it is consistent that one or the other must be true in each compact space. Another is to complete our project of determining if there is a small upper bound to the natural notion called the sequential order of a compact (sequential) space. The structure of the remainder of the Stone-Cech compactification of the integers, known as N*, and continuing an important line of investigation following Shelah's breakthrough result on self-homeomorphisms, the behavior of continuous maps with that as the domain, will be investigated. Given the current state of knowledge, it is quite interesting that most of these questions remain unresolved in models of the celebrated Proper Forcing Axiom (PFA) and forcing extensions of models in which this axiom holds. This project will be adding to the toolbox of techniques and results in this central area of study. Unquestionably, there is a huge literature on the structure of N* and we will continue our project of investigating the analogous questions that naturally arise about an equally fundamental space, R* for the real number (half-) line R. There is a fundamental difference that arises from the fact that this space is a continuum and possible applications to topological dynamics provide additional motivation.
