Many central conjectures in modern combinatorics, such as Rota’s log-concavity conjecture for the coefficients of the characteristic polynomial, Grünbaum’s conjecture on embeddings of simplicial $k$-complexes in $\mathbf{R}^{2k}$, or McMullen’s $g$-conjecture characterizing face numbers of simplicial spheres, can be reduced to Lefschetz and Hodge type theorems for certain combinatorial structures naturally connected for the problem. \
A problematic point at this stage is that algebraic geometry only gives us the latter theorems in rather restricted settings, beyond which all classical algebraic methods fail. Not giving up hope, it is useful to go back and replace these techniques with combinatorial ones, proving Hodge and Lefschetz type theorems in completely combinatorial settings. I will survey some new developments in this subject. \
Joint work with June Huh and Eric Katz.