The Kardar-Parisi-Zhang (KPZ) universality class is a broad collection of probabilistic models including stochastic PDEs, random growth models and directed polymers. Models in this class show unusual, non-Gaussian fluctuations, which for some special classes of initial data are described by objects coming from random matrix theory (RMT). While the connection between KPZ and RMT is well understood in the case of curved initial data, the case of flat initial data has remained a bit of a mystery. After introducing these topics, I will present a result about a system of $N$ Brownian motions conditioned not to intersect on a finite time interval. The result shows that the distribution of the squared maximal height of the top path in this system coincides with that of the largest eigenvalue of a certain (finite) random matrix, known as a real Wishart or LOE matrix. I will describe how this result provides an explanation for the connection between KPZ and RMT in the flat case, and how it generalizes some older results concerning its $N \to \infty$ scaling limit. Based on joint work with Gia Bao Nguyen.