Matthew Stamps: Betti diagrams of graphs

The emergence of Boij-Söderberg theory has given rise to new connections between combinatorics and commutative algebra. In a recent paper, Herzog, Sharifan, and Varbaro show that every Betti diagram of an ideal with a $k$-linear minimal resolution in a polynomial ring with $n$ variables arises from the Stanley-Reisner ideal of a simplicial complex on $n$ vertices.  In this talk, we will investigate further the case of $k=2$ and, in particular, give a bijective correspondence between the non-complete threshold graphs on $n$ vertices and the Betti diagrams of ideals with $2$ linear minimal resolutions over a polynomial ring with $n$ variables.  The key observation is that the Betti diagrams of these ideals are the lattice points of a reflexive polytope which can be constructed recursively from non-complete threshold graphs.  This is joint work with Alexander Engström.