  Kazhdan-Lusztig-Stanley polynomials are a combinatorial generalization of
Kazhdan-Lusztig polynomials of for Coxeter groups that include g-polynomials of
polytopes and Kazhdan-Lusztig polynomials of matroids. In the cases of Weyl
groups, rational polytopes, and realizable matroids, one can count points over
finite fields on flag varieties, toric varieties, or reciprocal planes to
obtain cohomological interpretations of these polynomials. We survey these
results and unite them under a single geometric framework.
