Research

Thompson s Group:

With Melanie Stein and Jennifer Taback of Bowdoin College, I have also become interested in finding efficient ways to calculate the word metric in Thompson`s group, F, with respect to nonstandard generating sets of the form {x_0, ... x_n}. Our methods are similar to those of Blake Fordham for the standard generating set {x_0, x_1} and has lead the way to answering questions such as whether or not F has deep pockets or is almost convex with respect to nonstandard generating sets.

REU Research:

During the summer of 2011, I was a the research mentor for the geometric group theory group at the BYU REU.  The students in my group were Alexis Johnson, Justin Halverson and  Amelia Stonesifer.  They worked on two projects.  One was to study some "new" generating sets for Thompson's group F.  In this project, they were able to determine a method for calculating length with respect to a previously un-studied generating set.  They used their new method to determine that F has dead ends with respect to this generating set, just as it does with respect to all other sets for which we know how to determine length.  The other group improved the known lower bound for the dead end depth of F with respect to the standard consecutive generating sets.  Previously the lower bound for the dead end depth of X_n was n/2.  This group improved the lower bound to 2n-7.

Graph Theory:

For the 2008 - 2009 academic year, I received a mini-grant from the Center for Undergraduate Research in Mathematics (CURM) to support work with four students, Eric LaRose, Jessica Moore, Mic Rooney and Hannah Rosenthal. We studied a property of metric spaces called "roundness" in the context of graphs. We simply investigated the question of what possible values can occur as the roundness of a finite graph. Two of the students, Eric and Hannah wrote a Java program that takes the adjacency matrix for a graph with 10 or fewer vertices and estimates its roundness (with "3" meaning "infinity"). This version of the program is good for general experimentation, but they also modified it to read in adjacency matrices from a file and then managed to compute the roundness of all of the connected graphs with 6, 7, 8, or 9 vertices. Jessica and Mic took a more theoretical approach, investigating the roundness of various infinite families of graphs. Among other things, Mic looked at cyclic graphs and Jessica looked at triangulated cyclic graphs. Both managed to find (with proof) the roundness of the graphs in their classes.

Mapping class groups and Out(Fn):

This is the field of my dissertation research at Cornell University under Karen Vogtmann. The idea is to try to understand the homology of Out(Fn) in terms of mapping class groups of surfaces, because their homology is understood much better. This involves studying actions of these groups on complexes of graphs such as Outer space and ribbon graph complexes.

Ordered groups and order trees:

With Melanie Stein, I've become involved in an effort to study partially ordered groups by way of their actions on oriented order trees. We've found a characterization of the left-invariant partial orders on a group that can arise from a action of that group on an order tree. This is a first step in studying partially ordered groups by way of their actions on various geometric objects.

Knot theory:

During the summer of 2005, I worked with Alexis Morley-Lyons on a research project in knot theory. We focused on the so-called "rational tangles." Our goal was to find an relatively easy way to determine whether or not a given tangle diagram represents a rational tangle. Alexis presented the results she found at the Spuyten Duyvil Undergraduate Mathematics Conference at Manhattan College.