Originating in the 18th century, optimal transport is a mathematical field concerned with moving one probability distribution to another in the most efficient manner. In the past decade, interest in optimal transport has increased as many machine learning and data science tasks require quantifying some notion of disagreement between probability distributions. In particular, the entropic regularized variant of the optimal transport problem has garnered much attention as it is efficiently computable. The solution to this variant, known as the Schrödinger bridge, is an object with its own physical motivation and storied history. This talk (practice for an upcoming general exam) will present recent and current projects that aim to better quantify the behavior of the Schrödinger bridge in the small temperature regime by developing tight approximations with diffusion processes. We will then demonstrate that this tight approximation can be leveraged to create discrete approximations of Wasserstein gradient flows (curves of steepest descent) that advance understanding about the behavior of the Transformer neural architecture.