The Langlands program is a far-reaching framework in modern mathematics connecting two seemingly unrelated types of objects. These are Galois representations, which encode the symmetries of polynomials, and certain analytic functions known as automorphic forms. The relationship between these objects has traditionally been studied using techniques from representation theory. Recently, however, a groundbreaking geometric perspective introduced by Fargues and Scholze has opened up new avenues of investigation, allowing mathematicians to apply powerful tools from algebraic geometry. Despite its elegance, this geometric approach remains inexplicit and only partially understood in relation to classical representation-theoretic results. In this project, the PI will develop a new and explicit theory, of cuspidal vector bundles, that will provide a means to connect these two approaches and enhance understanding of each. Beyond advancing mathematical knowledge, the project will provide training opportunities for graduate and undergraduate students and contribute to the mathematical community through workshops and conferences. The formulation of the categorical local Langlands conjecture of Fargues and Scholze was a major breakthrough in the Langlands program for p-adic groups. However, this theory is quite inexplicit and its connections to classical representation-theoretic results are not well understood. The PI will investigate the representation-theoretic consequences of the ge