Title:  Chromatic Levels in the Homotopy Groups of Spheres

Abstract:  Understanding the homotopy groups of spheres $\pi_nS^k$ is one of the great challenges of algebraic topology. One of the fundamental theorems in this field is the Freudenthal suspension theorem. It states that $\pi_{n+k}S^k$ is isomorphic to $\pi_{n+k+1}S^{k+1} $ when $k$ is large. Homotopy theorists call this phenomena \emph{stabilization}. The stable homotopy groups of spheres are defined to be these families of isomorphic groups. They form a ring, commonly denoted by $\pi_*S$. Despite its simple definition, this ring is extremely complex; there is no hope of computing it completely. However, it carries an amazing amount of structure. A famous theorem of Hopkins and Ravenel states that it is filtered by simpler rings called the \emph{chromatic layers}. There are many structural conjectures about the chromatic filtration. In this talk, I will give an overview of chromatic theory and talk about one of the structural conjectures, the \emph{chromatic splitting conjecture}.