an ordinary gushel-mukai fourfold $x$ is a smooth quadric section of a linear section of the grassmannian $g(2,5)$. kuznetsov and perry proved that the bounded derived category of $x$ admits a semiorthogonal decomposition whose non-trivial component is a subcategory of k3 type. in this talk i will report on a joint work in progress with alex perry and laura pertusi, in which we construct bridgeland stability conditions on the k3 subcategory of $x$. then i will explain some applications concerning the existence of a homological associated k3 surface, and related algebraic constructions in hyperkaehler geometry.