In these lectures, I will describe the formalism of the "decomposition of the diagonal", Chow-theoretic or cohomological, and the way it controls some more classical stable birational invariants. I will then explain the specialization method and some applications.

In these lectures, I will describe the formalism of the "decomposition of the diagonal", Chow-theoretic or cohomological, and the way it controls some more classical stable birational invariants. I will then explain the specialization method and some applications.