Grant: Discovery Grants Program - Individual

This proposal is devoted to research at the intersection of two fields of pure mathematics, Topology and Set Theory. Topological ideas are central to areas of mathematics where notions of distance play a role, and is concerned with those properties of abstract geometric objects unaffected by changes or deformations that preserve nearness relationships. This is a beautiful area of mathematics with great power to simplify and clarify ideas from seemingly divergent areas. For example, fundamental notions from Calculus are simple consequences of topological ideas. Set Theory, on the other hand, forms the foundations for mathematics as a whole. A typical foundational question from Set Theory is, ""what are the basic assumptions (axioms) that we use and that are needed in mathematics?"" A consequence of Godel's Incompleteness Theorem is that for any acceptable axiom system for mathematics there are mathematical statements that can neither be proven nor refuted from the system. Knowing that a particular mathematical conjecture is independent in this sense is not only of obvious intrinsic interest (knowing a conjecture is independent means we can stop looking for a proof of it), but it also establishes a clear connection to the area of Set Theory. Establishing connections between apparently disparate mathematical fields has always produced interesting and important mathematics. This research proposal is focused on problems concerning the fundamental structure of topological spaces where deep set theoretic connections have either already been established or where we believe they exist. The results of this proposal should lead to the development of new and important ideas in both set theory and general topology, and we hope will find applications to other areas of mathematics. Questions formulated in this proposal are of great interest to experts in the field not only in Canada but also internationally. In addition to expanding the frontier of knowledge regarding topological spaces, this research proposal will continue to promote Canada as a centre of research in pure mathematics.