  Our main goal is to give a sense of recent developments in the (stable)
rationality problem from the point of view of unramified cohomology and
0-cycles as well as derived categories and semiorthogonal decompositions, and
how these perspectives intertwine and reflect each other. In particular, in the
case of algebraic surfaces, we explain the relationship between Bloch's
conjecture, Chow-theoretic decompositions of the diagonal, categorical
representability, and the existence of phantom subcategories of the derived
category.

Algebraic Geometry