Explore a lecture by Ivan Corwin from Columbia University on the mathematical analysis of the Kardar-Parisi-Zhang (KPZ) class of stochastically growing interfaces in (1+1)-dimensions. Discover how the Yang-Baxter equation serves as a fundamental tool for proving convergence of integrable models to universal scaling limits. Learn about the progression of breakthroughs in this field, from identifying free-fermionic integrable models and their single-point limiting distributions to constructing the conjectural full space-time scaling limit known as the directed landscape. The lecture details a method that has been successfully applied to the Asymmetric Simple Exclusion Process and the Stochastic Six Vertex Model, potentially applicable to all known integrable representatives of the KPZ class. This talk is part of Harvard CMSA's program on Classical, quantum, and probabilistic integrable systems.