MEGL will be running a nine project program with over 50 expected participants between faculty, graduate research assistants, and undergraduate research interns. For project names and descriptions, please click the boxes below.

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.

*Character varieties* are certain affine varieties (sets arising as the solutions to polynomial equations) that are constructed from equivalence classes of homomorphisms of groups. The collection of all affine varieties (up to natural equivalence) forms a group itself called the *Grothendieck group*. It turns out that these classes of character varieties generate the Grothendieck group. In this project, we will experiment with character varieties and use those experiments to understand relationships within the Grothendieck group and, conversely, reverse-engineer character varieties from classes in the Grothendieck group.

**Faculty Mentor: **Dr. Sean Lawton

**Graduate Mentor:** Gabe Lumpkin

**Recommended Courses or Experience: **

- MATH 321 with a B or better.

In economics and consumer theory, a *Giffen good* is a product that people consume more of as the price rises, thus violating the basic law of demand in microeconomics. For any other sort of good, as the price of the good rises, the substitution effect makes consumers purchase less of it, and more of substitute goods; for most goods, the income effect (due to the effective decline in available income due to more being spent on existing units of this good) reinforces this decline in demand for the good. But a Giffen good is so strongly an inferior good in the minds of consumers (being more in demand at lower incomes) that this contrary income effect more than offsets the substitution effect, and the net effect of the good’s price rise is to increase demand for it. This phenomenon is known as the Giffen paradox.

The most commonly cited example of a Giffen good is that of the Irish Potato Famine in the 19th century. This was a period of starvation and disease in Ireland lasting from 1845 to 1852 that constituted a historical social crisis and subsequently had a major impact on Irish society and history as a whole. During the famine, as the price of potatoes rose, impoverished consumers had little money left for more nutritious but expensive food items like meat (the income effect). The goal of the project is to prove that a Giffen good cannot exist.

**Faculty Mentor:** Dr. Douglas Eckley

**Recommended Courses or Experience:**

- Calculus I, II
- One semester of micro-economics

Though quantum computing has exploded in popularity, fundamental questions about its applications and efficacy remain unanswered: when are quantum tools or procedures the right tools for the job? What kinds of experiments benefit from their use? Recently, researchers have used these techniques to tackle *logic synthesis*, minimizing the complexity and cost of computer circuitry. This project will compare classical and quantum randomized procedures within the logic synthesis problem, connecting ideas in computer science, mathematics, and statistical physics to identify new classes of algorithms.

**Faculty Mentor:** Dr. Michael Jarret

**Graduate Mentor:** Anthony Pizzimenti

**Recommended Courses or Experience:**

- MATH 125
- Calculus I, II
- CS 310

The goal of this project is to build foundational work to study stability in problems involving tipping points in reaction-diffusion equations (RDEs) in one spatial dimension, where the reaction term decays in space (asymptotically homogeneous) and varies linearly with time (nonautonomous) due to an external input. Using a compactification argument and some visualization techniques needed for studying the geometry behind the dynamics of these complex systems, we compute heteroclinic orbits along intersections of stable and unstable invariant manifolds. This will allow us to obtain multiple coexisting pulse and front solutions for the RDEs. Our goal is to apply this framework in a model of a habitat patch that features an Allee effect in population growth and is geographically shrinking or shifting due to human activity or climate change. If time allows, we plan to extend this project by linearizing the system at the standing waves and deriving the eigenvalue problem to investigate their stability.

**Faculty Mentors:** Dr. Emmanuel Fleurantin, Dr. Matt Holzer

**Graduate Mentor:** Julia Seay

**Recommended Courses or Experience:**

- MATH 214
- Familiarity with or desire to learn MATLAB

The goal of this project is to analyze the antibody levels in individuals post severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection and vaccination using publicly available datasets. Students will design probabilistic models to fit to data to characterize the variability inherent in immune responses. We will build various models to consider factors such as time since infection or vaccination, vaccine manufacturer, disease severity, prior infections or vaccinations, age, and biological sex. Time permitting, students will apply statistical metrics to measure model similarity to determine which factors distinguish antibody response, and construct joint distributions that consider two or more factors.

**Faculty Mentor:** Dr. Rayanne Luke

**Graduate Mentor:** Kelsey Ellis

**Recommended Courses or Experience:**

- Basic programming
- Some knowledge of probability preferred

A polyform is a plane figure constructed by joining together identical basic polygons. In Euclidean geometry we have polyominoes (made out of squares), polyiamonds (out of equilateral triangles) and polyhexes (out of regular hexagons). In this project we will explore hyperbolic polyforms. These are constructed by joining together tiles of a regular hyperbolic tessellation. A hole is a bounded component of the complement of the polyform. We are interested in the question: Given a positive integer h, what is the minimum number of tiles required for a polyform to have h holes. Along the way we will pursue the following milestones:

- Find a way to effectively visualize hyperbolic polyforms with large numbers of tiles;
- Find a way to algebraically represent hyperbolic polyforms;
- Create examples of holey hyperbolic polyforms.

**Faculty Mentors:** Dr. Ros Toala, Dr. Erika Roldan

**Graduate Mentor: **Summer Eldridge

**Recommended Courses or Experience:**

- MATH 213

The ancient Egyptians expressed proper fractions as sums of reciprocals of distinct positive integers (i.e. of *unit* fractions). For instance, they might represent 2/15 as 1/10 + 1/30. It goes back to Fibonacci that this is always possible, and that given a proper (i.e. with m <= n) fraction of the form m/n, one may represent it as a sum of *at most m* distinct unit fractions. However, in many cases it seems one can do better. The Erdős-Straus (resp. Sierpinski) conjecture posits that any proper fraction of the form 4/n (resp. 5/n) is the sum of at most 3 unit fractions; both conjectures are open. Recently, Bloom has resolved the Erdős-Graham conjecture by showing that any set S of integers with “positive upper density” can be used to express the integer 1 as a sum of distinct reciprocals from S.

In another direction, let us consider fractions that arise from polynomials (let us call them “polynomial fractions”). Say a polynomial fraction f/g is “proper” if the degree of g is at least that of f. I have shown that any proper polynomial fraction can be written as a sum of reciprocals of distinct polynomials, and that we need at most deg(f) of them. But can we do better? This MEGL project consists of exploring breakdowns of polynomial fractions as sums of distinct unit fractions, both by finding bounds on the length of representations and winnowing down what polynomials one needs to express them, by analogy with the case of integer fractions.

**Faculty Mentor:** Dr. Neil Epstein

**Graduate Mentor:** Tim Banks

A Neural Network may be geometrically interpreted as a nonlinear function that stretches and pulls apart data between vector spaces. If a dataset has interesting geometric or topological structure, one might ask how the structure of the data will change when passed through a neural network. This is achieved by explicitly viewing the dataset as a manifold and observing how the topological complexity (i.e., the sum of the Betti numbers) of the manifold changes as it passes through the activation layers of a neural network. The goal of this project is to study how the topological complexity of the data changes by tuning the hyper-parameters of the network. This enables us to possibly understand the relationship between the structural mechanics of the network and its performance. Research interns will be expected to meet twice a week to learn the relevant literature, implement neural network architectures, visualize numerical results, and develop heuristics.

**Faculty Mentor:** Dr. Ben Schweinhart

**Graduate Mentors:** Shrunal Pothagoni and Justin Cox

**Recommended Courses or Experience:**

- Familiarity with principles of data science
- CS 112
- MATH 203
- MATH 321, MATH 431 (recommended)