As shown by N. Brown in 2011, for a separable $II_1-factor N$, the invariant $Hom(N,R^U)$ given by unitary equivalence classes of embeddings of $N$ in to $R^U$--an ultrapower of the separable hyperfinite $II_1-factor$--takes on a convex structure. This provides a link between convex geometric notions and operator algebraic concepts; e.g. extreme points are precisely the embeddings with factorial relative commutant. The geometric nature of this invariant provides a familiar context in which natural curiosities become interesting new questions about the underlying operator algebras. For example, such a question is the following. ``Can four extreme points have a planar convex hull?'' The goal of this talk is to present a recent result generalizing the characterization of extreme points in this convex structure. After introducing and discussing this convex structure, we will see that the dimension of the minimal face containing an equivalence class $[\pi]$ is one less than the dimension of the center of the relative commutant of $\pi$. This result also establishes the ``convex independence'' of extreme points, providing a negative answer to the above question. Along the way we make use of a version of Schur's Lemma for this context. No prior knowledge of this convex structure will be assumed.