Summer 2015

During Summer 2015 MEGL ran a 20-participant program (its inaugural program). There were three research groups, named: Orbits, Special Words, and Polytopes. Each of these group engaged in experimental research 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. Additionally, there were development teams in Virtual Reality, 3D Printing, and Community Engagement. The summer ended with a Symposium where all teams presented their work, and with a significant participation in the 2015 Geometry Labs United Conference (where all three research teams placed in the research poster competition; including first prize).

This team included undergraduates Robert Argus, Patrick Brown, and Jermain McDermott, visiting graduate student Diaaeldin Taha, and faculty adviser Dr. Sean Lawton.

The project was to explore the nature of the periods resulting from certain dynamical systems. In particular, let $\mathfrak{X}= \mathrm{Hom}(F_r, \mathrm{SL}(2,\mathbb{F}_q))/\!\!/ \mathrm{SL}(2,\mathbb{F}_q)$ be the $\mathbb{F}_q$-points of the character variety, where $F_r$ is the free group of r letters, and $\mathbb{F}_q$ is the finite field of order q. Then $\mathrm{Out}(F_r)$ acts on $\mathfrak{X}$ by $\chi\mapsto \chi\circ \alpha$ for a given $\alpha\in \mathrm{Out}(F_r)$. Upon fixing $\alpha$ and considering the resulting dynamical system for a fixed $\chi$, we were interested in the length of the orbits and how it changes as a function of $q$.

Symposium Slides

Conference Poster

General Information: REGS & Interns