Geometric methods in modular representation theory

I will start by explaining how the geometry of certain complex
algebraic varieties (nilpotent cones, Schubert varieties etc.) has
implications for modular representation theory, and vice versa, via the
theory of perverse sheaves. I will give some examples and try to give a
feeling for the fascinating interplay between algebra, geometry and
topology that arises. In the second half of the talk I will discuss joint
work with Carl Mautner and Daniel Juteau in which we introduce a new class
of sheaves, which behave in some respects like tilting modules.