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 algebraic geometry, at least as long as some notion of positivity is available.

I then go one step further, and ask 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.

The main part of the talk is devoted to provide an indication how the proof works, explain the notion of transversal primes as well as Hall matching theorems for spaces of linear operators, and their connection in the Hall-Laman relations which replace our knowledge of the signature of the Hodge-Riemann relations in the Kähler case with nondegeneracy at monomial ideals.

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