Understanding the stable homotopy groups of spheres is one of the great challenges of algebraic topology. They form a ring which, despite its simple definition, carries an amazing amount of structure. It is filtered by simpler rings called the chromatic layers. Each layer corresponds to a periodic family whose element repeat at a certain frequency. The terminology is meant to remind us of the electromagnetic spectrum. This point of view organizes the homotopy groups and reveals patterns.

There are many structural conjectures about the chromatic filtration. I will talk about one of these conjectures, the chromatic splitting conjecture, which concerns the gluing data between the different layers of the chromatic filtration.