Challenges in modular representation theory

Abstract: This will be a broad survey talk on interactions between geometry and representation theory, with a focus on representations in positive characteristic (“modular representation theory”). I will outline several basic questions (e.g. for modular representations of the symmetric group) which appear very difficult, and have resisted direct algebraic approaches for over a hundred years. Over the last two decades, a new approach has emerged via geometric representation theory. It turns out that subtle questions concerning torsion in cohomology control these problems, and this allows some progress to be made. Many open questions remain, but at this point there can be no doubt that a fascinating and deep theory awaits us.