For a framed (4k+2)-manifold M, there is associated Kervaire invariant (zero or one) under framed cobordism which measures whether M is cobordant to any manifolds homotopy equivalent to a sphere. In 2009, Hill-Hopkins-Ravenel solved the problem by showing that the dimensions must be 2, 6, 10, 14, 30, 62 and 126 for Kervaire invariant one. This problem also turns out to be related to the Adams spectral sequence and the stable homotopy groups of spheres. In this week's talk I will survey the history and context of the problem, and next week I will explain the outline of the proof. Speaker(s): Yunze Lu (University of Michigan)