Jan-Erik Roos: Homological properties of algebras and spaces

During the last 35 years several questions and conjectures by Kostrikin, Safarevic, Govorov, Serre, Kaplansky and others have been answered and proved, thanks to the efforts of many mathematicians: Löfwall, Sjödin, Backelin, Fröberg, C. Jacobsson, Bogvad, Levin, Lech, Avramov, Golod, Tate, Serre, Anick, Gulliksen, Lemaire, Lescot, Halperin, Felix, Kathryn Hess, Anna Larsson, Snellman, Sköldberg, Berglund and many others (including myself).
But the results obtained also point to new developments. I will give an elementary talk about this, also mentioning possible results in the future.