Explore advanced geometric concepts in this mathematical lecture from the Geometry of Matroid Workshop where Kris Shaw from the University of Oslo investigates which tropicalisations of subvarieties of the torus contain information about the cohomology of the original variety. Delve into linear embeddings of complements of hyperplane arrangements as motivating examples, and discover how tropicalisation reveals cohomological information when specific conditions are met. Learn about the relationship between local tropical Poincaré duality, "wunderschön" varieties, and their mixed Hodge structures, building upon the work of Itenberg, Katzarkov, Mikhalkin, and Zharkov to understand how these properties influence families of varieties and their tropicalisations.