Algebraic approximations to stable homotopy theory

The stable homotopy groups of spheres can be thought of as a derived analogue of the integers, and
they encode deep information about algebraic and differential topology as well as arithmetic geometry.
Chromatic homotopy theory reveals this hidden information and provides a systematic approach to
the computation of these groups. 

After a quick survey of the chromatic perspective, we explain recent joint work with Schlank and
Stapleton which implies that the chromatic approach is asymptotically algebraic. Our work is based
on a categorical generalization of ideas from mathematical logic which should be of interest beyond the
applications to stable homotopy theory.