In chromatic homotopy theory, an object like the sphere spectrum S0  is studied by means of its "localizations", much as an abelian group can be localized at each prime p.  Remarkably, the "primes" K(n)  in the homotopy setting correspond to formal groups in characteristic p; there is one for each height n.  Our main theorem gives the structure of the homotopy groups of the K(n) -local sphere up to bounded torsion.  The methods we use come almost entirely from p-adic geometry; for instance we leverage the integral p-adic Hodge theory results of Bhatt-Morrow-Scholze.   This is joint work with Tobias Barthel, Tomer Schlank, and Nathaniel Stapleton.