Abstract:
Many classes of graphs, matroids, and other combinatorial objects can be described in two ways. The first description tells us how this class can be constructed from a finite number of basic elements by gluing them together. The second description tells us that every object that doesn't contain certain substructures is in the class.

This structural graph theory was developed in a long series of papers by Robertson and Seymour, and celebrated conjectures have been proved using it.

In commutative algebra several classes of ideals are defined from graphs, with many examples from algebraic geometry and algebraic statistics. For these classes, the structural theory for graphs can sometimes be lifted to the algebraic setting. Recently, these theorems lifted to the algebraic setting have started to live their on life, and can be applied to ideals without a graph hiding behind.

I will discuss work with Thomas Kahle (at Max Planck, Leipzig, going to Mittag-Leffler) and Seth Sullivant (NCSU) towards a theory of structural commutative algebra.