Lattices: Theory and Applications

This lecture covers the theory of lattices, defining them as finitely generated Z-modules which are discrete in R and generated as Q-vector spaces. It explores the properties of lattices in different contexts, such as open-compact Zp-modules and rational lattices, emphasizing the local-global principles. The instructor discusses the bijection between lattices in open-compact Zp-modules and rational lattices, providing proofs and detailed explanations. The lecture concludes by examining the coordination properties of lattices in various bases and their applications in different mathematical scenarios.