Modular representation theory via categorification and knot theory

We will study the category of modular representations of the special linear Lie algebra in positive characteristic; we focus on the case with a central Frobenius character given by a nilpotent whose Jordan type is a two-row partition. Building on work of Cautis and Kamnitzer, we construct a categorification of the affine tangle calculus using these categories; the main technical tool is a geometric localization-type result of Bezrukavnikov, Mirkovic and Rumynin. Using this, we give combinatorial dimension formulae for the irreducible modules, composition multiplicities of the simples in the baby Vermas, and a description of the Ext spaces. This Ext algebra is an "annular" analogues of Khovanov's arc algebra, and can be used to give an extension of Khovanov homology to links in the annulus. This is joint with Rina Anno and David Yang.