Explore the geometric and topological aspects of structured matrix spaces in this conference talk by Tom Needham. Delve into concepts such as momentum maps, spaces of frames, and the frame homotopy conjecture. Examine simplec manifolds, compress sensing, and normal matrices. Learn about gradient descent techniques and their applications in this context. Investigate the topology of balancing directed graphs and consider open questions in the field. Gain insights into the intersection of geometry, topology, and matrix theory through this 38-minute presentation recorded at the Centre International de Rencontres Mathématiques in Marseille, France, during the thematic meeting "Geometric Sciences in Action: from geometric statistics to shape analysis."