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.

Movies: Heat Flow, Heat Map, Gaussian Heat