Ramification in Homotopy Theory

I will discuss a new homotopy theoretic generalization of the idea of ramification, in the sense of number theory. Understanding the ramification of an extension of number fields is useful for calculations. By comparison, the new homotopical version leads to a greatly simplified calculation of Topological Hochschild Homology (THH) of a ring of integers in a number field. On the other hand, the new definition allows us to talk about ramification of extensions of ring spectra like ku/ko. Therefore, there are hopeful applications to computing THH and K-theory in chromatic homotopy theory settings. I will survey some of these ideas, assuming some background with spectra but not with algebraic number theory.