Genus of complete intersections and Ehrhart-Macdonald reciprocity for convex polytopes 2

I will describe the beautiful Ehrhart-Macdonald theorem which connects number of integral points in a convex polytope $\Delta$ and the number of integral points strictly inside $\Delta$. I will discuss the relation with Serre duality in algebraic geometry and finally show how one can use this to compute genus of complete intersections in smooth toric varieties. In a recent work (in progress, joint with A. G. Khovanskii) we generalize this to the so-called spherical varieties of any connected reductive group G. I will try to explain most of the necessary background.