It has been an open problem since the 1960s to construct minimal free resolutions of arbitrary monomial ideals. In this talk, I will explain how to approach that in characteristic 0, without making arbitrary choices. A differential is constructed using Moore-Penrose pseudoinverses of differentials of simplicial complexes related to the monomial ideal. This differential appears to average geometrically or combinatorially defined resolutions. For example, in three variables, the pseudoinverse is the average of all differential quasi-inverses that come from spanning trees of the upper Koszul complex, and in this way, the trace minimal resolution is an average of the minimal free resolutions of the ideal by planar graphs. I will also explain how to determine the resulting minimal free resolution in three variables by looking at the staircase surface.
This is joint work with John A. Eagon and Ezra Miller.