# Fall 2017

During Fall 2017 MEGL ran a program with 15 participants (faculty, graduate, and undergraduate students). There were four research/visualization groups (Arithmetic Orbits, Geometric Flows, Hyperbolic Soccer, Erdös-Szekeres Problems), and one public engagement group (Outreach). The research/visualization groups engaged in experimental explorations 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 progress reports from all teams at an end of term symposium.

The goal of the geometric flows project is to explore the evolution of geometric flows on complicated manifolds that are difficult to study analytically. Our goal for Fall 2017 was to approximate the geometric heat flow, which evolves a manifold according to its instantaneous heat equation. To accomplish this, we first implemented the Diffusion Maps algorithm, which provable recovers the Laplacian operator on a manifold from data. We then implemented a very stable spectral solver which is able to approximate the solution to the heat equation using the spectral decomposition of the Laplacian. Before attempting geometric flows, we first applied our algorithm to the unit circle and unit sphere in order to verify stable performance. These manifolds are natural starting points since the heat equation can be solved analytically and they are also fixed points of the geometric heat flow (up to volume rescaling).

In order to approximate the geometric flow on a manifold, we alternate between learning the Laplacian and then evolving the embedded data using the heat equation for a very short time step. After each short time step the manifold has changed so we must then repeat the Diffusion Maps algorithm in order to learn the Laplacian on the new manifold. We successfully applied this method to data on an ellipse and the geometric heat flow evolved this manifold into a circle as expected. However, we also found that this approach can develop numerical instabilities when iterated over long time scales. Our goal for next semester is to apply our approach to more complicated manifolds, including various standard surfaces and perturbations of these manifolds. We will search for steady state solution of the geometric heat flow. We will also attempt to understand the long-time numerical instability that we have discovered in the hopes of finding an improved algorithm.

In order to visualize the group $\mathrm{SL}_2\mathbb{R}$, we looked at its action on the hyperbolic plane. Subgroups are also related to motions on the hyperbolic plane, such as circle inversion. In particular the subgroup $\mathrm{SL}_2\mathbb{Z}$, also known as the modular group, is related to a particular tiling of hyperbolic space by triangles with an ideal vertex. Taking the quotient of the hyperbolic plane with respect to this tiling gives us the so-called modular surface $\mathbb{H}^2/ \text{SL}_2\mathbb{Z}$.

The surface $\mathbb{H}^2/\text{SL}_2\mathbb{Z}$ is an orbifold; It has two points at which it fails to be smooth, which correspond to points of hyperbolic space that are fixed by a finite subgroup of $\mathrm{SL}_2\mathbb{Z}$. Amazingly, this space is also the moduli space of a certain class of Riemann surfaces called flat tori. Colloquially, this means that each point of the modular surface corresponds to a particular torus and vice versa, continuously. Not only that, but geodesics in $\mathbb{H}^2$ viewed in the quotient $\mathbb{H}^2/ \text{SL}_2\mathbb{Z}$ correspond to geodesics in the moduli space.

A computer application created during the semester allows one to visualize a path in $\mathbb{H}^2/\text{SL}_2\mathbb{Z}$ as an animation of a torus. This program also visualizes geodesic flow on the modular surface as an animation of a soccer ball being hit towards a goal, which shows how a generic geodesic eventually reaches any height.

PLAY THE APP!

This project is devoted to the study of the famous Ersos-Szekeres Problem on convexly independent points in 3D. The main goal was to look from a new prospective on proof of the following statement: any set of nine points in general position in 3-dimensional space contains six points which are the vertices of a convex polytope. Another achievement of the project is a new MatLab program that verifies whether a given set of eight points in 3-space contains no convexly independent subset of six points.

The goal of the project is to understand the dynamics of the action of $\mathrm{Out}(F_2)$ on specific character varieties. In particular, we look at the induced action of the outer automorphism group $\Gamma$ which can be thought of as the automorphism group of the polynomial $\kappa(x,y,z)=x^2+y^2+z^2-xyz -2-\lambda$ on the solution set of $\kappa$ over the affine 3-space of finite field $\mathbb{F}_q$.

We hope to classify the elements based on the growth rate of the orbit as we increase $q$, which appears to be constant, linear or $q \log q$ from the experimental data we gathered. We also expect the action of $\Gamma$ or a subgroup of $\Gamma$ to be arithmetically ergodic. Most of the work in Fall 2017 was devoted to collecting more data which could confirm the above conjectures or suggest new ones which can then be generalized to higher cases. We also tried to look for tools that could help develop the theory which can prove some of the suggested conjectures by looking at approaches used in solving similar problems. One particular method involves using conic sections of the variety obtained by fixing the value of a specific coordinate and trying to look at the action on the conics instead of the points in the variety. Another approach was to use Pisano periods to get a bound for specific orbits in the specific case where $\lambda=2$ which proved to be difficult since it is hard to find the periods if we are not using mod prime $p$. We also managed to get some visualizations of the varieties for few $q$, orbits under the action of certain elements in $\Gamma$ and some conic sections.

The Mason Experimental Geometry Lab continued to grow its outreach program this semester. We made more school and library visits than any semester prior, and started development of our newest outreach activity.

We made a total of 26 visits to local schools and libraries. In particular we ran our Snowflake Symmetry activity, our most recent offering, at a middle school and an elementary school which allowed us to fine tune the activity. This process of real-time trial and error is central to making our activities strong and successful.

We also began development of our newest activity, Playground of the Infinite. Designed as a sequel to Really Big Numbers, but also to stand on its own, this activity challenges our intuition when thinking about collections of infinite size. We learn how to compare the sizes of sets without counting them, and how this comparison leads to unsuspected results when we allow the sets to be infinite in size.