Consider two random subspaces of a finite-dimensional vector space -- i.e. two random projection matrices P and Q. What is the dimension of their intersection? This (random) integer is almost surely equal to its minimal possible value, which corresponds to the subspaces being in general position. Many more delicate questions about the geometry of the configuration are encoded by the principle angles between the subspaces, which are determined by the eigenvalues of the operator-valued angle matrix PQP. The situation is much more complicated in infinite-dimensions. Even the question of whether two random projections are likely to be in general position is difficult to make sense of, let alone answer. Nevertheless, understanding the operator-valued angle in an infinite-dimensional setting is of critical importance to the biggest open problem in free probability theory -- the so-called ``Unification Conjecture'' -- with ramifications for operator algebras, information theory, and random matrices. In this talk, I will discuss recent and ongoing joint work with Benoit Collins, addressing the configuration of random subspaces in an infinite-dimensional context. Using a mixture of techniques from stochastic processes, PDEs, and complex analysis, we prove the general position claim and give a complete understanding of the associated geometry. This work proves an important special case of the Unification Conjecture, and has interesting implications for the original finite-dimensional setting as well.