Topics in Representation Theory

MATH 527 treats representations of finite groups through the lens of noncommutative ring theory and category-level functor constructions, the framework you need once ordinary character theory runs out of room and modular phenomena take over. The semester moves from semisimple and Artinian rings into block theory, defect groups, and source algebras, then reframes everything through Mackey, biset, and correspondence functors built on spans and pushouts. Work is roughly half problem sets and half reading-seminar style, with a presentation, a midterm essay, and a written final asking you to push these tools rather than just recite them. It sits at the boundary between algebra and category theory and is essentially the entry point for doing research in modular or categorical representation theory.