Explore the intricacies of the local Langlands program and its connection to characteristic p in this 58-minute conference talk by Christophe Breuil. Delve into the challenges of understanding and constructing representations of GLn(K) in characteristic p, where K is a finite extension of the field of p-adic numbers Qp. Trace the 20-year history of this complex problem, focusing on its significance in number theory and its application to n-dimensional representations of Galois groups and infinite-dimensional representations of GLn. Examine the specific case of GL2(K), which remains largely unsolved, and gain insights into recent developments in this field of mathematical research.