Sheaf theoretic methods in modular representation theory

This thesis concerns the use of perverse sheaves with coefficients in commutative rings and in particular, fields of positive characteristic, in the study of modular representation theory. We begin by giving a new geometric interpretation of classical connections between the representation theory of the general linear groups and symmetric groups. We then survey work, joint with D. Juteau and G. Williamson, in which we construct a class of objects, called parity sheaves. These objects share many properties with the intersection cohomology complexes in characteristic zero, including a decomposition theorem and a close relation to representation theory. The final part of this document consists of two computations of IC stalks in the nilpotent cones of sl₃and sl₄. These computations build upon our calculations in sections 3.5 and 3.6 of (31), but utilize slightly more sophisticated techniques and allow us to compute the stalks in the remaining characteristics.