Spring 2016

During Spring 2016 MEGL ran a program with 23 participants. There were five research groups (named Orbits, Special Words, Polytopes, Allen-Cahn Equation, and Embedded Graphs), two visualization groups (named 3D Printing and Virtual Reality), and one public outreach group, simply named Outreach. The research and visualization groups engaged in experimental exploration involving faculty, graduate students, and undergraduates. Teams met weekly to conduct experiments generating data, make conjectures from data, and work on theory resulting from conjectures. The outreach group involved faculty, graduate students, and undergraduates to develop and implement activities for elementary and high school students that were presented at local schools and libraries. The Spring semester ended with a Symposium where all teams presented their work.

We are trying to understand the dynamics of the action of the outer automorphism group of a free group of rank $r$, denoted $Out(F_r)$, on the finite field points of the $\mathrm{SL}_2(\mathbb{C})$- character variety of a free group. One of the questions involve ﬁnding an upper bound for the length of the largest orbit under this action for a ﬁxed element of $Out(F_r)$.

From previous computations, we know the upper bounds when the rank of the free group is two. The next question is to ﬁgure out if there are elements of $Out(F_r)$ such that the length of the orbit equals the given bound. Eventually, we would like to ﬁnd out the growth of the length of the largest orbit as the rank of the free group and the order of the ﬁnite ﬁeld varies. This work will continue into Summer 2016.

View the slides here: Orbits Symposium Slides