Explore the intricacies of metric representations in this 56-minute lecture delivered by Anna Gilbert from Yale University at the virtual 2022 Joint Mathematics Meetings. Delve into the algorithms and geometry behind metric representations, starting with classical multi-dimensional scaling and progressing through metric repair definitions and problem formulations. Examine optimization techniques, including the Project and Forget method, and learn about metric violations through separation oracles and shortest paths. Understand the Bregman projection and theoretical results summarized in the Summary Theorem. Investigate extensions, applications, and traditional techniques for metric repair, including three repair scenarios. Conclude with insights into graph metric repair and dimensionality reduction algorithms, gaining a comprehensive understanding of this complex mathematical topic.