Square free monomial ideals give powerful connections between algebra and other areas of pure and applied mathematics.

Important examples are Stanley-Reisner ideals and Villarreal’s edge ideals which connect with combinatorics and graph theory respectively.

These allow one to use algebra to study simplicial complexes and graphs, e.g., leading to Stanley’s proof of the upper bound theorem for simplicial spheres. They also allow one to use simplicial complexes and graphs to study ideals, e.g., enabling us to *see* certain homological properties and invariants in the simplicial complexes and graphs. In this talk I will briefly survey these ideas and discuss two lesser known constructions, one using closed neighborhoods in graphs and another using codewords in binary linear codes.