Homotopy groups of spheres and algebraic K-theory

The homotopy groups of spheres classify continuous functions from an n-dimensional sphere to an m-dimensional sphere up to continuous deformation. They are of great interest as they control many classification problems in geometric topology, but are rich and complicated. I will explain some of what we know about them, and describe the telescopic approach to understanding them systematically. I will then explain how algebraic K-theory has refined our understanding of the telescopic approach, and can be used to obtain asymptotic lower bounds on the complexity of these homotopy groups that were previously out of reach.