We will discuss a geometric characterization of classes of positive volume on the boundary of the Kahler cone of a compact Kahler manifold. As an application, we will show that finite time singularities of the Kahler-Ricci flow always form along analytic subvarieties. Time permitting, we will also discuss some applications to algebraic geometry, specifically to augmented base loci of nef and big line bundles on projective varieties. This work is joint with Valentino Tosatti.