I will discuss the heat equation on the unitary group $U(N)$ in the limit as $N$ tends to infinity, with an eye toward study of the Segal-Bargmann transform. On certain classes of functions on $U(N)$, the Laplacian greatly simplifies as $N$ gets large. Indeed, the large-$N$ limit of the Laplacian satisfies a first-order product rule, meaning that the cross terms in the usual product rule for the Laplacian become negligible. As a result, we are able to obtain a limiting Segal-Bargmann transform on this class of functions. These results are joint work with Bruce Driver and Todd Kemp and were motivated by earlier work of Philippe Bianne.