Edge ideals of graphs in exterior algebras (Spring 2022)

A graph is a finite collection of points, called vertices, connected by some collection of line segments, called edges. One way of studying a graph G algebraically is by considering a polynomial ring whose variables correspond to the vertices of G and defining its edge ideal to be the ideal generated by all degree 2 monomials corresponding to edges of G. Much research has already been devoted to exploring how the combinatorics of a graph influences algebraic properties of its edge ideal (in particular, the minimal free resolution of the edge ideal) and vice versa.

In this project, we will explore the much less studied edge ideal of a graph in an exterior algebra where, unlike in a polynomial ring, the product of two variables x and y skew-commutes so that xy = -yx. This has the curious effect that the square of every variable is zero, and whereas every free resolution over a polynomial ring is guaranteed to stop after finitely many steps, free resolutions over exterior algebras are typically infinitely long. We will similarly seek to understand how nice combinatorial properties of a graph translate to good algebraic properties of its exterior edge ideal in spite of these extra complications.

For more information contact Matthew Mastroeni (mmastro@iastate.edu)

People:

• Matthew Mastroeni (Postdoc)

• Jason McCullough (Faculty)

• Josh Rice (Grad Student)

Pre-requisites:

• Linear algebra (MATH 317)

• Experience with rings (MATH 301 and/or MATH 495) is desirable, but not necessary

• Some programming experience (Macaulay2/Python/Matlab) is desirable, but not necessary