The quadratically regularized optimal transport problem (QOT) has emerged in the literature as a sparse alternative to the celebrated entropically regularized transport problem (EOT). Unlike EOT, whose solutions always have full support—even for small regularization parameters—QOT solutions, or QOT plans, tend to approximate the support of the unregularized transport problem, which concentrates on the graph of a function under mild conditions. This gives way to some natural questions that I will intend to answer in this talk, such as: Why do we care about sparsity? How does this sparsity manifest, and is it monotone? Can we efficiently approximate QOT? This is joint work with my co-advisor Marcel Nutz and Dr. Alberto González-Sanz.