Explore recent mathematical advances in the study of groups of area-preserving homeomorphisms through this 58-minute conference talk delivered at the Centre International de Rencontres Mathématiques in Marseille, France. Delve into two key areas of progress: the (non)-simplicity of these previously poorly understood groups and the existence of quasimorphisms. Learn about proof techniques utilizing symplectic topology tools, presented through collaborative research with Dan Cristofaro-Gardiner, Cheuk-Yu Mak, Sobhan Seyfaddini, and Ivan Smith. Access this recording from the "Foliations and Diffeomorphism Groups" thematic meeting through CIRM's Audiovisual Mathematics Library, featuring chapter markers, keywords, abstracts, bibliographies, and Mathematics Subject Classification for enhanced learning and reference.