Another fun one (from the many examples above!) is a new paper I am working on at the moment as well, but in the intersection of physics and math: it turns out you can phrase a bunch of physics problems as optimization problems, then use some very simple tools in optimization to derive fundamental limits of how well any device could ever perform.

The preprint is here: https://arxiv.org/abs/1811.12936

An associated blog post is here: https://guille.site/physics-impossibility-results.html