Semigroup and Poincaré series for divisorial valuations and unimodality for integrally closed polytopes

This thesis consists of two parts. The first part is about the semigroup of values SV and the Poincaré series PV(t) associated to a finite set of divisorial valuations coming from a modification of Kd, where K is any field. When K is infinite, we can prove that SV is finitely generated whenever there exists some finite generating sequence L for V . The existence of such a finite L also implies that PV(t) is a rational function whose denominator can be expressed in terms of the valuation vectors of the elements of L. Here K can even be a finite field. However, a finite generating sequence does not always exist. This is the case for the modification of C3 where we blow up in nine very general points on the first exceptional divisor. In that specific example, the semigroup of values is not finitely generated.The second part is about lattice polytopes. It is a famous open question whether every integrally closed reflexive polytope has a unimodal Ehrhart delta-vector. We generalize this question to arbitrary integrally closed lattice polytopes and we prove unimodality for the delta-vector of integrally closed polytopes of small dimension and for lattice parallelepipeds. This is the first non-trivial class of integrally closed polytopes. Moreover, we suggest a new approach to the problem for reflexive polytopes via triangulations.