Singular Soergel bimodules and their diagrammatics

In this talk, I will report on work in progress, joint with B. Elias, H. Ko, and N. Libedinsky, which aims at giving a diagrammatic presentation of the category of singular Soergel bimodules. Soergel bimodules categorify the Hecke algebra of a Coxeter group and play a central role in modern representation theory. Their singular counterparts categorify the Hecke algebroid, which is governed by the rich combinatorics of double cosets in Coxeter groups. I will describe this category and explain how its diagrammatic presentation will allow us to make explicit computations and to work in great generality, for example over Z.