Mathematical Adventures
Welcome to Mathematical Adventures! This is perhaps a cheesy name, but it wasn’t chosen thoughtlessly: doing math can feel like a freely creative, exciting, and even playful adventure (notwithstanding the dry, formal prose of some mathematical writing, including some of my own). This blog hopes to document the kind of exploration that takes a routine excercise, asks a seemingly endless stream of follow-up questions, and ends up landing in more or less current areas of research; the current series, Pathological Periodicity, attempts exactly that. It will also be a place for reflections on the politics of the discipline, on relevant philosophy, and on my personal relationship with math (see Returning not Repeating).
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Start here: Pathological Periodicity
Nonconstant continuous periodic functions cannot have dense period sets, and if two continuous functions have no (nonzero) periods in common, their sum won’t be periodic. Neither of these facts holds in general. This series explores the surprisingly rich family of pathological counterexamples that emerges once continuity, and then measurability, are abandoned, and ties these to the Periodic Decomposition Problem and to various “fragments” of the Axiom of Choice.
