Mason Experimental Geometry Lab

Fall 2025

MEGL is running a six project program with an expected 35 participants between faculty, graduate research assistants, and undergraduate research interns. For project names and descriptions, please click the boxes below. Apply here by 5pm on May 5: https://forms.office.com/r/6x9Xp352YB.

In addition to the research projects, MEGL is running an Outreach program, which exposes K-12 students to important mathematical concepts through visualizations and hands-on activities. We are recruiting one new Outreach member for Fall 2025; this is a paid position with an expected hourly compensation of $15-16. Please contact Ros Toala at [email protected] for more information. You may be on both a research project and outreach simultaneously.

Graphons are a tool in the large random networks that allow researchers to replace large complex networks with nonlocal operators which are sometimes easier to study. The goal of this project is to use Graphons to infer the dynamics of and make rigorous proofs regarding some dynamical systems on large random networks.

Faculty Mentors: Dr. Matt Holzer

Prerequisites: MATH 315 and some exposure to MATLAB or a similar programming language

In this project, we develop rigorous techniques and apply them to understand various models of computation. We will investigate real and theoretical devices, as well as heuristic algorithms, for understanding models such as reversible computing, quantum computing, cellular automata, probabilistic, and potentially analog computation.

Faculty Mentor: Dr. Michael Jarret

Prerequisites: Linear algebra, some combinatorics and/or graph theory would be nice

Special relativity holds that spacetime is a 4-dimensional geometric surface.

As one moves about, and passes time in doing so, he/she travels on a path in spacetime.

Spacetime can be analyzed, geometrically, in terms of geodesics. It’s a little more complicated than the surface of the earth, where one type of geodesic connects any two points on the spherical surface.

In spacetime, three types of geodesics exist.

The twin paradox might be stated as follows:

Two people of exactly the same age meet at the same time and place.

Then, one stays put while the other travels to Mars and returns safely.

They rejoin at the original place but one year later.

The twin who traveled to Mars is now younger than the twin who stayed.

The twin who stayed traveled through spacetime on a time-like geodesic. The twin who traveled to the moon did not follow a spacetime geodesic.

We will analyze this paradox and present a clear explanation using geometry.

Faculty Mentor: Dr. Douglas Eckley

Groups form an integral part of many cryptographic systems; mainly due to the presence of inverse elements that facilitate secure encryption and decryption processes. This project first explores the role of group theory in established cryptographic methods, such as RSA Public Key Encryption. Changing tack, we then examine semigroups (non-empty sets equipped with an associative binary operation) as alternative algebraic structures for cryptography. In doing so, we analyse how the lack of inverse elements, and potentially the absence of an identity element, impacts the design and security of the encryption methods that are possible. Students are encouraged to apply these concepts creatively by developing and assessing their own cryptographic systems using semigroups.

Faculty Mentor: Dr. Scott Carson

Prerequisites: MATH 321

Percolation is a model of fluid flow through a random medium. Take an N x N square grid and randomly declare squares open, one at a time, until there is an open path from one side of the square to the other. It turns out that this path behaves like a fractal! The fractal properties of random paths in percolation have been of great interest in both mathematics and physics. Recently, researchers have introduced higher-dimensional versions of these models for which the objects of interest are random surfaces rather than random paths. In this project, we will perform computational experiments to investigate the geometry of these random surfaces, and to see whether they have fractal properties.

Faculty Mentor: Dr. Ben Schweinhart

Returning project; not accepting new members

Continued fractions are a way to represent numbers as $latex a_0+\cfrac{\pm 1}{a_1+\cfrac{\pm 1}{a_2+\dots}}$, where the $latex a_i$ are positive integers. They appear in various areas of mathematics, from number theory to dynamical systems to geometry. We can use different rules or functions to “generate” different types of continued fraction. This project will explore geometric descriptions of the “nearest integer” continued fractions, which rounds to the nearest whole number at every step. For example, $latex 5/3=2-1/3$ and $latex 8/5=1+3/5=1+\cfrac{2-\cfrac{1}{3}}$.

The continued fraction expansions provide a nice description of paths on the geometric surfaces. The geometric properties of these pictures help us to describe patterns in the continued fraction expansions, and the continued fraction expansions provide a compact description of the geometry. This project will likely explore modular surfaces, but there are other directions based on student interest.

Faculty Mentors: Claire Merriman

Prerequisites: MATH 203 or 213