Basis problems in topology

At the start of modern mathematics a program was launched to provide a rigorous foundation for all of mathematics. Because the notion of a set is among the most primitive in mathematics, it was used as the basic fabric with which to build the more complicated objects of mathematics. Through subsequent work of Godel and Cohen, it has been realized that many aspects of infinite sets are themselves quite subtle and defy a complete description using axioms. Moreover, these complexities sometimes manifest themselves in more complex mathematical structures, such as those studied in algebra, analysis, geometry, and topology--all of which are core areas of mathematics. The aim of this grant is to further develop both understanding of set-theoretic methods and their role in topology, particularly in relation to the basic notations of compactness and convergence. It also more broadly supports graduate student research into applications of set theory to other fields of mathematics and logic. The first part of the research project aims to classify the simplest nonmetrizable compact topological spaces in the presence of PFA, an additional axiomatic assumption. It will leverage a new approach to building nonmetrizable compacta as limits of spaces which are finite in nature (for instance finite sets of points or their convex hulls). The second part of the research project aims to classify the cofinal types of uniform ultrafilters on the first uncountable cardinal. The third component of the research project is to support thesis research of graduate students working under the PI's supervision. The topics include the analysis of generic extensions of Solovay's model via ultrafilters on topological Ramsey spaces, set-theoretic aspects of homological algebra, and the interplay between proof theory and the determinacy of infinite games. The grant also provides support for research visitors and travel to facilitate collaboration and dissemination of results obtained under the grant. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.