Commutative algebraists have been using free resolutions since the 1890s to study ideals (thanks, Hilbert!). One tool that was very hip among commutative algebraists in the late 1990s and early 2000s was cellular resolutions of ideals: free resolutions that are supported by cell-complexes. Many interesting results in this direction came from people who passed through or are still at Cornell, and then the subject seemed to go out of style. In recent years, however, cellular resolutions seem to be making a comeback.

This talk will begin with motivation and examples of free resolutions of ideals. Then we will turn to examples of free resolutions that are cellular and show how topological invariants of the supporting cell-complex correspond to homological invariants of the ideal. Along the way, we will survey work done by many people, including current and former professors and graduate students at Cornell. Time permitting, I will also give a brief overview of work inspired by cellular resolutions in the last five years, including some of my own.