Topology of Algebraic Varieties

Inspired by the elementary notion of graph chordality, we introduce the notion of toric chordality, which naturally gives a tool to to study the geometry and combinatorics of cohomology classes of toric varieties and the weight algebras of polytopes. Moreover, I will sketch some interesting applications:
-- The notion of toric chordality naturally provides new proofs and generalizations of a variety of crystallization theorems arising in commutative algebra.
-- Moreover, I will discuss a quite unprecedented relation between primitive Betti numbers of toric varieties and the shape of a convex polytope, and finally
-- I will discuss some applications to the study of deformation spaces of polytopes.