Tom Braden (UMass): Intersection cohomology without spaces

We survey three settings in which dimensions of intersection cohomology groups of algebraic varieties provide deep combinatorial and representation-theoretic information, and where computations of the groups themselves have been made using combinatorial sheaves on finite posets. These settings are (1) intersection cohomology of Schubert varieties, the associated Kazhdan–Lusztig polynomials and their realizations via moment graph sheaves and Soergel bimodules; (2) intersection cohomology of toric varieties, the associated g-polynomials of convex polytopes, and their realization via the theory of intersection cohomology of fans; and (3) intersection cohomology of arrangement Schubert varieties, the associated Kazhdan–Lusztig polynomials of matroids, and their realization via intersection cohomology of matroids. In all three settings these constructions work in more general situations where the variety does not exist, leading to “intersection cohomology without spaces."