Volumes and lattice counts in polyhedra

In the second session, we use the formalism exposed in the first talk to various valuations. The exponential valuation allows to compute the volume of polytopes through Brion's theorem, and allows to give a meaning to the notion of volume for polyhedra without lines. Next, lattice points count of polyhedra is considered, where a discrete analog of the exponential valuation is defined via generating functions. We recover then Brion's theorem for lattice point count.