Spring 2017

During Spring 2017 MEGL ran a program with 16 participants. There were three research groups (Orbits, Special Words, Polytopes), two visualization groups (Geometric Surfaces, Virtual Reality), 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 middle 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.

This project continues the study of the dynamics of the outer automorphism group of a free group of rank r, denoted $Out(F_{r})$, on the finite field points of the relative $SL_2(\mathbb{C})$ character varieties of $F_r$.  We are interested in two main problems (1) understanding the length of the largest orbit and which classes achieve it, and also (2) defining and proving the action is “ergodic” in this arithmetic setting. Additionally we are working on creating, in 3D print, visualizations of this family of dynamical systems.

There has been some recent interest in this problem from other researchers and we have also been investing time in reading the work and methods of these other researchers.

View our symposium Mathematica Notebook here: Mathematica

We study elements of the rank 2 free group, which are known as words. Each word has an associated $SL(n,C)$-trace function. We consider two words to be n-special in relation to each other if they are not conjugate to each other but have the same $SL(n,C)$-trace function. Previous research has shown that there are unboundedly many 2-special words and that a word, its inverse, and its reverse will always be 2-special. It is not known if 3-special words exist, but it is known that if words are 3-special then they are also 2-special and that a word will never be 3-special with its inverse. Our goal is to develop necessary and sufficient conditions for the existence of 3-special words. To study this, we wrote computer programs to generate all 2-special words of a specified length and search the data set for 3-special words. In the data set of over 20 million words, there are no 3-special words. We have proven that sum of the exponent values in words must be equal in order to be 3-special, decreasing the amount of words that can be 3-special.

In the 1950’s, Nash and Kuiper proved existence of $C^1$ isometric embeddings (or immersions) of Riemannian manifolds into higher dimensional Euclidean space.  Two decades later, Gromov invented the convex integration technique, providing the tool for the construction of $C^1$ isometric embeddings. From 2006 to 2012, three French mathematics institutions collaborated in the Hevea Project to produce a 3D model of an isometric embedding of a flat torus in 3D Euclidean space.  In previous semesters, we formulated an alternative and simplified approach to constructing such an isometric embedding of a flat torus or short sphere.  This semester we read and presented the original work by Nash and Kuiper to round-off and ground our prior work.

We learned about various models of hyperbolic 3-space.  We then figured out how to draw geodesics in the upper-half-space model in Unity (for VR) and MATLAB (for computation).  Then using Mobius transformations and quaterions we coded the action of the isometry group in the upper-half-space model in Unity (for VR).  Lastly, we added features for the Oculus Rift Touch.  We expect to build on our work over Summer 2017.

Given two integers n and m, we can look at the following polynomial ring: $\bigoplus_{r\geq 0} \overbrace{Sym^r (\mathbb{C}^m)\otimes Sym^r(\mathbb{C}^m)\otimes \dots \otimes Sym^r(\mathbb{C}^m)}^{n}$.

We want to look at the subring invariant under the action of SL(m,C). This group acts diagonally on separate factors of the tensor product in each part of the sum. In order to bound just how complex this ring is, we transfer the study of this ring to the study of another, easier to classify ring. We can find a presentation of this associated ring using polyhedral cones formed from trees with n leaves and SL(m,C) Berenstein-Zelevinsky triangles. We can then compute the Hilbert basis, which is the set of integer vectors which span the lattice of the cone, and the Markov basis, which is the minimal set of relations among the Hilbert basis vectors that generate all other relations. The Hilbert basis and Markov basis of this cone are the bounds for the number of generators and relations of the invariant ring.