Research

Throughout mathematics, there is frequently a gulf between what we know exists and what we can explicitly define. This is particularly true when studying the infinite; many counterintuitive objects can be proved to exist (e.g., paradoxical decompositions of the sphere, well-orderings of the reals, ultrafilters, Bernstein sets, etc), while those objects which we can reasonably define are comparatively tame. This fact suggests that the language we use to define mathematical objects places constraints or, more positively, yields structure on those objects. My research concerns the tension between the wild, but hard to define, on the one hand and the more explicitly definable, and well-structured, on the other, mainly in the context of the real line, and infinite-dimensional vector and Banach spaces.

Many of the wild objects alluded to above are constructed using the tools of combinatorial set theory, like the Axiom of Choice and Well-Ordering Theorem, the Continuum Hypothesis, or the method of forcing. The study of the definable subsets of the real line and their properties is known as descriptive set theory. Both sit within the larger framework of mathematical logic, which considers language and definability across mathematics as a whole.

Work in progress

Publications & preprints