The object of this project is to solve a number of related problems concerning convergence of sequences, which have arisen during the past twenty years. The methods to be employed include techniques from topology, set theory and mathematical logic. The main problem asks: Is every product of sequentially compact spaces countably compact? It is one of the few remaining, long-standing, unsolved problems concerning ultrafilters on the natural numbers. This simple-sounding problem remains unsolved after more than 20 years of work. A related problem, just as old, asks: Is the product of two countably compact topological groups always countably compact? Other problems concerning sequences will also be considered. For example, does there exist a first countable, countably compact separable, non-normal Hausdorff space? The main problem and others to be considered have formulations in both set theory and topology, and solutions to these problems would advance areas of mathematics where convergence of sequences is relevant. While it is unlikely that applied mathematics will benefit immediately, clarification of basic issues is always welcome, and a clearer view of the real world and its various mathematical models may result.
