MEGL will run seven projects with an estimated 40 members, including undergraduate researchers, graduate mentors and faculty mentors. For project names and descriptions, please click the boxes below. To apply, click here.
In addition to the research projects, MEGL will run an Outreach program, which exposed K-12 students to important mathematical concepts through visualizations and hands-on activities.
A geometric manifold is a manifold whose geometry (metric) locally looks like a familiar space, i.e. Euclidean, hyperbolic, or real projective space. One can endow a given surface with uncountably many different local geometric structures. A natural question to ask is how one can distinguish between two different geometric structures on the same surface. Length spectrum rigidity asks “can we answer this question only knowing the length of certain curves on the surface?” There are some cases where this is true, one of the most celebrated being the 9g-9 Theorem.
The goal of this project is to determine length spectrum rigidity for a class of spaces modelled on real projective space. These spaces are constructed using the theory of linear representations of coxeter groups, which will allow us to use tools from linear algebra such as eigenvalues and characteristic polynomials to attempt to establish a result. We will use software, such as Mathematica, to do computations and create visualizations.
Faculty Mentor: Dr. Harrison Bray
Graduate Mentors: Anunoy Chakraborty & Gabe Lumpkin
Pre – Requisites: MATH 203
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
Graduate Mentor: Allison Kohne
Pre – Requisites: MATH 321
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 and Dr. Emmanuel Fleurantin
Pre – Requisites: 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
Graduate Mentor: Anthony Pizzimenti
Pre – Requisites: Linear algebra, some combinatorics and/or graph theory would be nice
If you pick a number between 0 and 1 at random and look at its first 1,000,000,000 base-10 digits, approximately 100,000,000 of them will be 6s. If you instead look at its continued fraction expansion, approximately 29,747,343 of the digits will be 6s. The underlying digit frequency was figured out by Gauss and can be calculated from the probability distribution 1/(ln 2) 1/(1+x). We will adjust the continued fraction algorithm, run extensive computer computations to approximate the Taylor series for the new probability distribution, and then search for a simple description like the one Gauss found – or prove that they don’t exist by tracking how precise our estimates are and therefore ruling out specific types of expressions.
Faculty Mentor: Dr. Anton Lukyanenko
Graduate Mentor: Nicole Savir
Pre – Requisites: Completion of MATH 213. Preferred: MATH 300 and some programming experience.
Continued fractions are a way to represent numbers as $a_0+\cfrac{\pm 1}{a_1+\cfrac{\pm 1}{a_2+\dots}}$, where the $a_i$ 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, $\dfrac{5}{3} = 2 \text{ – } \dfrac{1}{3}$ and $\dfrac{8}{5} = 1 + \dfrac{3}{5} = 1 + \dfrac{1}{2 \text{ – } \dfrac{1}{3}}$.
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 Mentor: Dr. Claire Merriman
Graduate Mentor: Tim Banks
Pre – Requisites: MATH 203 or 213
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
Graduate Mentors: Anthony Pizzimenti & Morgan Shuman