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The $f$-vector of a simplicial complex is the vector whose entries record the number of faces in each dimension. It is typically convenient to study an integer linear transformation of the $f$-vector, called the $h$-vector, which naturally appears in the Hilbert series of an algebra associated to the simplicial complex. When we impose additional topological or combinatorial property on the complex, we observe linear relations or inequalities among the entries of the $h$-vector. For instance, when the complex triangulates a sphere, the $h$-vector is nonnegative, palindromic and unimodal. In this talk I will focus on a special family of triangulated spheres called flag and I will introduce a conjecture due to Gal which predicts precise linear inequalities which are stronger then thos ein the case of arbitrary spheres. Gal's inequalites can be conveniently encoded as statements of nonnegativity of a third integer vector, called $\gamma$-vector. I will show that a natural algebraic generalization of this conjecture does not hold (joint work with Alessio D'Alì) and comment on a reduction strategy proposed by Chudnowsky and Nevo (joint work in progress with Geva Yashfe). Finally, if time permits, I will present recent progress on a different $\gamma$-positivity problem involving triangulations of symmetric edge polytopes of a graph (joint work with Alessio D'Alì, Martina Juhnke-Kubitzke and Daniel Köhne).