Lecture 1. Foundations of $p$-adic Geometry: From Tate to Huber

This lecture establishes the foundational framework for geometry over $p$-adic fields. We begin by exploring the topological crisis of non-Archimedean spaces and John Tate's pioneering algebraic solution via rigid analytic geometry and affinoid algebras. We will then trace the evolution of these ideas into Roland Huber's theory of adic spaces, providing the robust topological and geometric language required for modern $p$-adic geometry.