Special Year Workshop on p-adic Arithmetic Geometry

In chromatic homotopy theory, an object like the sphere spectrum $S^0$ 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.