This video discusses three major breakthroughs in mathematics: a new proof for the sphere packing problem in all dimensions (using random packing methods), a significant advance on the problem of arithmetic progressions in dense sets, and a proof of the geometric Langlands conjecture (a decades-long effort connecting disparate branches of mathematics). The sphere packing proof utilizes a random approach, challenging the assumption that ordered structures are always optimal. The arithmetic progression work improves bounds on the density of sets avoiding long progressions. The geometric Langlands proof, a monumental 800-page work, establishes a major connection within mathematics using analogies to Fourier theory.