The PI investigates independence results in the study of countable convergence questions in the class of compact spaces. Some of the questions remain unresolved even in the presence of the Proper Forcing Axiom. The sequential order of a space is a natural ordinal invariant that measures the complexity of a sequential space; the question of whether there can be a finite or countable bound on this invariant for compact spaces has only been resolved under CH. We will also study continuous maps on the remainder of the Stone-Cech compactification of the integers which are finite-to-one. We will try to determine if such maps have to be "somewhere trivial" and whether the range space will be homeomorphic to the domain. This question has been fully resolved under PFA but not, for example, under MA. Two other questions about the remainder of N under PFA: can it be covered by nowhere dense P-sets and can there be non-trivial copies of the remainder?
When one formalizes the limiting process of an infinite sequence of steps one is naturally lead to the notion of topological space. Points in physical space can be analyzed using the notion of a converging sequence, but just as mathematics has a need for a much broader understanding of points (e.g. functions or processes themselves lying in a space of like processes), so too has the need for a broader understanding of limit and topological space been developed to handle convergence in the more general settings. The investigation into the nature of generalized convergence, connections between conditions that may imply the existence of (the much easier to understand and apply) classical convergent sequences, and the behavior of the convergence structure under deformations is on-going and productive. These notions are very sensitive to the foundational axioms of mathematics and the questions serve as both a testbed for foundational study and a generator of new development. One of the most famously indeterminate questions of mathematics, the continuum hypothesis (the "size" of the real number line) is intimately connected to these questions of convergence.
