The algebra of Knight models
In 1977, Julia Knight produced a countable structure $K$ such that its Scott sentence has a model of cardinality $\aleph_1$ but no models of larger cardinality. In other words it "characterizes" $\aleph_1$. This structure consists of a linear order along with a countable family of unary functions that ensure that every element can define all of its predecessors. Greg Hjorth observed that "Knight's model" has some important properties in the field of invariant descriptive set theory. Knight's model is not canonical so we're really talking about a family of models that we call "Knight models". We'll make sense of these models by exposing their hidden algebraic information, and give an easier construction of them. We'll also talk about how, although Hjorth built on Knight's work by characterizing each $\aleph_\alpha$ for $\alpha < \omega_1$, the picture is still far from complete from the perspective of invariant descriptive set theory, and why a more direct generalization of Knight's construction is needed.

