In this talk, I will discuss a technique for studying the stable homotopy groups of spheres called the Mahowald invariant. This technique takes an element in the stable homotopy groups of spheres and produces a nontrivial element in a higher stable homotopy group. In the first part of the talk, I will review some classical computations in stable homotopy theory, introduce the Mahowald invariant, and state Mahowald and Ravenel’s computation of the Mahowald invariants of 2^i for i ≥ 1.
In the second part of the talk, I will discuss a generalization of this technique to motivic stable homotopy theory, or stable homotopy theory for schemes. I will give a brief introduction to motivic homotopy theory, explain some of the differences between the classical and motivic Mahowald invariants, and conclude with some motivic Mahowald invariants. Time permitting, I will discuss a part of the computation which uses a new motivic cohomology theory.