(joint work with Assaf Naor) A key problem in metric geometry asks: given metric spaces X and Y, how well does X embed in Y? In this talk, we will consider this problem for the case of the Heisenberg group and explain its connections to geometric measure theory and computer science. The Heisenberg group H is a metric space that is hard to embed in ℝn. Cheeger and Kleiner used a metric derivative based on surfaces in H to show that it also fails to embed into L1. We will use uniform rectifiability to analyze surfaces in ℍ, provide sharp bounds on the distortion of embeddings of H and estimate the accuracy of an approximate algorithm for the Sparsest Cut Problem.