I will survey applications of Hodge Theory to combinatorics, and, quite suprisingly, how Hodge-Riemann relations and Lefschetz type theorems can be proven using combinatorial methods, in settings that are beyond classical projective algebraic geometry.

I then go one step further, and discuss how many triangles a PL embedded simplicial complex in R^4 can have (the answer, generalizing a result of Euler and Descartes, is at most 4 times the number of edges). I discuss how to reduce this problem to a Lefschetz property beyond projective structure, and indicate the proof of the Lefschetz property in this case, replacing positivity with anisotropy of subspaces in the Hodge-Riemann pairing.

If time permits, I will then discuss relations to the Singer conjecture for aspherical manifolds and/or the spaces of multilinear forms of small Schmidt rank.