Fall 2016

During Fall 2016 MEGL ran a program with 18 participants. There were four research groups (Orbits, Special Words, Polytopes, Embedding Graphs), three visualization groups (Geometric Surfaces, Virtual Reality, Double Pendulum), and one public outreach group (Outreach). The research and visualization groups engaged in experimental exploration involving faculty, graduate students, and undergraduates. Teams met weekly to conduct experiments generating data, to make conjectures from data, and to 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 public libraries. We concluded with final reports from all teams and an end of term symposium.

We want 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 $SL_2(\mathbb{C})$ character variety of $F_r$ by looking at the length of the largest orbit. In particular, we are interested in elements of $Out(F_2)$ that exhibits a $p\text{log }p$ orbit growth as we increase the prime $p$ defining the finite field. We conjecture that the elements that shows this growth are pseudo-Anosov/hyperbolic elements and we try to get experimental evidence for small primes using Mathematica.
Another problem of interest is to define the notion of arithmetic ergodicity motivated from the definition in terms of measurable sets and use it to conjecture that for a compact surface $\Sigma$, $Out(\Sigma)$ acts arithmetically ergodically on the character variety $\mathfrak{X}_{\Sigma}(G)$, for all connected compact Lie groups $G$. Recent results by Bourgain, Gamburd and Sarnak have shown some light towards a special case which inspired the formulation of the conjecture.

View our symposium presentation here Presentation