The recent detection of gravitational waves is another spectacular confirmation that the universe is fundamentally geometric, at least on cosmological scales. This raises a natural question: Could dark matter, which makes up most of the mass of galaxies, be fundamentally geometric as well? We'll present a geometric model of dark matter called wave dark matter, also known as scalar field dark matter and Bose-Einstein condensate dark matter, which fits observations about dark matter very well. Using geometric PDE, we'll show how wave dark matter predicts spiral patterns in galaxies, and compare computer simulations with photos of actual galaxies.