Georgia Institute of Technology College of Sciences

If a finite group $G$ acts on a Cohen-Macaulay ring $A$, and the order of $G$ is a unit in $A$, then the invariant ring $A^G$ is Cohen-Macaulay as well, by the Hochster-Eagon theorem. On the other hand, if the order of $G$ is not a unit in $A$ then the Cohen-Macaulayness of $A^G$ is a delicate question that has attracted research attention over the last several decades, with answers in several special cases but little general theory. In this talk we show that the statement that $A^G$ is Cohen-Macaulay is equivalent to a statement quantified over the inertia groups for the action of G$ on $A$ acting on strict henselizations of appropriate localizations of $A$. In a case of long-standing interest—a permutation group acting on a polynomial ring—we show how this can be applied to find an obstruction to Cohen-Macaulayness that allows us to completely characterize the permutation groups whose invariant ring is Cohen-Macaulay regardless of the ground field. This is joint work with Sophie Marques.

Hadwiger (Hajos and Gerards and Seymour, respectively) conjectured that the vertices of every graph with no K_{t+1} minor (topological minor and odd minor, respectively) can be colored with t colors such that any pair of adjacent vertices receive different colors. These conjectures are stronger than the Four Color Theorem and are either wide open or false in general. A weakening of these conjectures is to consider clustered coloring which only requires every monochromatic component to have bounded size instead of size 1. It is known that t colors are still necessary for the clustered coloring version of those three conjectures. Joint with David Wood, we prove a series of tight results about clustered coloring on graphs with no subgraph isomorphic to a fixed complete bipartite graph. These results have a number of applications. In particular, they imply that the clustered coloring version of Hajos' conjecture is true for bounded treewidth graphs in a stronger sense: K_{t+1} topological minor free graphs of bounded treewidth are clustered t-list-colorable. They also lead to the first linear upper bound for the clustered coloring version of Hajos' conjecture and the currently best upper bound for the clustered coloring version of the Gerards-Seymour conjecture.