A subspace embedding is a random linear transformation that maps high-dimensional vectors to a lower dimension that with high probability preserves the norms of all vectors in a low-dimensional subspace up to a small relative error. Due to their simplicity and computational efficiency, subspace embeddings have been the central dimensionality reduction technique in numerical linear algebra problems such as least squares regression and low-rank approximation for the past two decades. Yet, despite their popularity, there remains a fundamental theory-practice gap in our understanding of subspace embeddings, which is characterized by a conjecture of Nelson and Nguyen (FOCS 2013) about the optimal embedding dimension that can be attained efficiently. In this talk, I will discuss recent efforts in closing this theory-practice gap by analyzing the universality properties of sparse random matrices, which has led to resolving the conjecture up to sub-polylogarithmic factors (SODA 2026). Based on joint works with Shabarish Chenakkod, Xiaoyu Dong, and Mark Rudelson.