Applications of algebra to a problem in topology (YouTube) at Atiyah80 was a talk by Mike Hopkins. In it he announced the solution to the Kervaire invariant one problem in all but one dimension (arXiv, Annals).
EDIT: The video seems to have been removed from Youtube but is still available as a download in the .mov format at empg.maths.ed.ac.uk/Videos/Atiyah80/Hopkins.movempg.maths.ed.ac.uk/Videos/Atiyah80/Hopkins.mov