The focus of this thesis is the use of techniques from geometry, analysis, and optimization to several concrete problems in probability theory, and also to some problems in statistics and machine learning. Growing interest in the intersections of these fields is motivated by several recent developments: the recognition of the utility of optimal transport in probability and statistics, the large number of modern statistical applications where one encounters non-Euclidean data, and more.

The first part of the thesis studies couplings. Its main results include a form of infinite-dimensional linear programming duality for a rich class of coupling problems involving equivalence relations, and consequences of this abstract theory for various coupling problems encountered in stochastic calculus.

The second part studies the canonical notion of central tendency for probability measures on metric spaces, the Fréchet mean (also called the barycenter or the center of mass). The several chapters in this part establish: a limit theory for Fréchet means in a general class of infinite-dimensional metric spaces; a development of large deviations theory for Fréchet means in the Bures-Wasserstein space; a statistical optimality result for estimating Fréchet mean sets in a general metric space; and, an optimal adaptive algorithm for Fréchet mean set estimation in the space of phylogenetic trees.

The third part studies clustering, and provides asymptotic guarantees for k-means clustering and variants thereof. In particular, it establishes consistency results for adaptive variants of k-means (k-means when k is chosen according to the elbow method, k-medoids, and more) and it proves further limit theorems for the classic k-means problem.