Explore the fascinating connection between Lipschitz constants and the topological properties of maps in this 52-minute lecture by Larry Guth from MIT, presented at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the complex relationship between a map's Lipschitz constant and its degree, focusing on mappings between Riemannian manifolds. Examine the upper bound of $L^n$ for the degree of a map with Lipschitz constant $L$, and investigate how this relationship varies for different manifolds beyond the sphere. Learn about recent developments in the field, including work by Aleksandr Berdnikov and Fedor Manin, and discover clever mapping techniques related to upcoming discussions. Cover topics such as Hopf invariants, Lipschitz extension problems, upper and lower bounds, self-avoiding random walks, and the implications of these concepts in differential geometry and topology.