I will introduce very carefully (without assuming much background) 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 chromatic height 2 and describe how the sphere at height 2 can be decomposed in terms of spectra related to the spectrum of topological modular forms TMF. I will talk about computing the Spanier-Whitehead dual of TMF and describe how this is useful for understanding the K(2)-local sphere.