One of the central objects in the theory of optimal transport is the Brenier map: the unique monotone transformation which pushes forward an absolutely continuous probability law in R^d onto any other given law. A large body of recent work has studied the question of estimating Brenier maps on the basis of random samples. This question was initiated by Hütter & Rigollet (2021), who derived an estimator that achieves the optimal L^2 rate of convergence toward the true Brenier map. Unfortunately, their estimator is computationally intractable in general dimension. In this talk, we derive the first estimator of smooth Brenier maps which is both computationally tractable, and achieves the optimal rate of convergence. Furthermore, we show that this estimator enjoys a pointwise central limit theorem. This result provides a first step toward the question of performing uncertainty quantification for Brenier maps. Our proofs hinge upon a quantitative linearization of the Monge-Ampère equation governing the optimal transport problem, which may be of independent interest. This talk is based on joint work with Sivaraman Balakrishnan, Jonathan Niles-Weed, and Larry Wasserman.