I will introduce chromatic homotopy theory which uses Bousfield localization with respect to Morava K-theories K(n) to filter the category of spectra. This filtration by height allows us to simplify calculations of stable homotopy groups of spheres by working one prime and one chromatic height at a time. I will introduce the main tools from number theory that help with these computations.

Then I will talk specifically about current work at height 2 and describe short self-dual resolutions of the  sphere at height 2 and how they can be used to find invertible objects and study duality in this K(2)-local category.