How algebraic is space?

A phenomena in topology is said to be stable if it occurs in all sufficiently high dimensions. As discovered by Quillen over five decades ago, such phenomena are closely related to number theory, and can often be described in terms of arithmetic objects known as formal groups. Unfortunately, in general this dictionary is not quite a one-to-one and many periodicities one sees on the arithmetic side become broken and more complex in the world of topology. In this talk, I will describe a solution to an old conjecture of Franke that the arithmetic - topology correspondence can be refined to an equivalence of categories when the ambient prime is sufficiently large.