The theory of Rees algebras of monomial ideals has been extensively studied, and as a consequence, many (sometimes partial) equivalences between algebraic properties of monomial ideals, and combinatorial properties of simplicial complexes and hypergraphs are known. In this talk, we will see how this theory can be used to find interesting examples in the theory of Lefschetz properties. We will also explore the consequences of known results from Lefschetz properties to the Rees algebras of squarefree monomial ideals. Applications include a connection between symbolic powers and f-vectors of simplicial complexes, and the positivity of mixed multiplicities of some families of squarefree monomial ideals.
