Explore a 50-minute lecture from the Workshop on Geometry and Analysis of Fluid Flows, featuring Barbara Keyfitz from Ohio State University. Delve into recent research conducted by Keyfitz, John Holmes, and Feride Tiglay, which investigates the representation of scalar convex conservation laws as paths in diffeomorphism spaces. Learn how this approach, inspired by Vladimir Arnold's program, applies to equations like the inviscid Burgers' equation. Discover how weak solutions transform into "weak diffeomorphisms" - bi-Lipschitz invertible mappings that themselves become solutions to conservation law systems. Gain insights into the intersection of fluid dynamics, geometry, and analysis in this specialized mathematics talk presented at Stony Brook University in January 2023.