MEGL ran an eight project program with 38 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 ran an Outreach program, which exposes K-12 students to important mathematical concepts through visualizations and hands-on activities.
The purpose of this project was to investigate the SL(2,C) character varieties of finitely presentable groups. In particular, we used Mathematica code which allowed us to input a finitely presentable group, and then output the polynomials defining the corresponding character variety. In doing so, we were able to completely classify the SL(2,C) character variety of all cyclic groups, as well as in the case that the group has two generators and one relation of the form a^n. We found that for the cyclic group of order n, the corresponding character variety was a finite collection of points of cardinality Floor(n/2)+1. In the case of two generators one relation of the form a^n where n is even, we found that the character variety was the disjoin union of two copies of C disjoint union with Floor(n/2) +1 disjoint copies of C^2. In the case of two generators one relation of the form a^n where n is odd, we found that the character variety was C disjoint union with Floor(n/2)+1 disjoint copies of C^2.
Faculty Mentor: Dr. Sean Lawton
Graduate Mentor: Gabe Lumpkin
Researchers:
- Dylan D Evans
- Nick Lear
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.
Team Eckley, consisting of Adam Metwally, Emma Burke, and Maxwell Laraia, proceeded by establishing 4 basic requirements that any legitimate utility function must follow.
The team analyzed the well-known and often-cited Cobb-Douglas utility function and proved that a Giffen good cannot exist in that case. This was well-known already, but served as a starting point.
The team then analyzed the modern theoretical utility function known as Wald-Jureen. This function appeared in a 2022 paper among others, and does display Giffen behavior over a specific region of consumption points. We showed that this utility violates 2 of the 4 basic requirements previously noted.
Finally, the team developed a proof that any utility function that meets the 4 basic requirements cannot exhibit Giffen behavior in any region. The proof used calculus and graphics.
We believe this is a contribution to economic theory that is publishable.
Faculty Mentor: Dr. Douglas Eckley
Researchers:
- Emma Burke
- Kirby Steiner
- Maxwell J Laraia
This semester, we focused on showing rigorous bounds and evidence for parallel tempering’s superiority over simulated annealing. Parallel tempering is a heuristic process for finding a global optimum without getting stuck in local extrema, which can happen with standard algorithms like simulated annealing. Although parallel tempering can do better than simulated annealing in specific contexts, research has not shown it to be a general improvement.
To support the theory of parallel tempering being a superior algorithm, we created a generalized parallel tempering library to solve the same problems as simulated annealing. We also began coding example problems to investigate the performance of simulated annealing relative to parallel tempering. We are using the library to develop state-of-the-art tools for studying logic synthesis.
We also studied ways to separate the class of problems solvable efficiently by simulated annealing from those solvable efficiently by parallel tempering. We outlined a method to show an exponential separation between the mixing times of simulated annealing and parallel tempering.
In the future, we plan to formalize and extend these results to wider classes of problems. We also plan to expand the range of example problems in our library. We currently have majority gate minimization and the Ising model as problems in the library, and plan to add protein folding as well.
Faculty Mentor: Dr. Michael Jarret
Graduate Mentor: Anthony Pizzimenti
Researchers:
- Mark Dubynskyi
- Raghu Guggilam
- Kyle Hess
The motivation is to study population dynamics within habitat patches. This semester, we focused on steady state solutions of a non-linear RDE (Reaction Diffusion Equation) where the reaction term follows the Allee Growth Model. This equation accounts for the limited resources in a large population density and for under-crowding at lower population density. Through the method of reduction of order and analyzing the Hamiltonians functions derived from our piecewise definition, we established a framework that characterizes the behavior of populations within the habitat.
Once obtaining these Hamiltonians, we examined behaviors of solutions on the zero-energy level stemming from the origin. Additionally, we identified three fixed points for the inside habitat with specific energy values defining each point. Furthermore, we implemented the finite difference method to compute numerical solutions for the standing wave problem using our derived parameters. Finally, we established a critical length formula for the habitat and formulated a theorem for the existence of a minimum length required for population persistence.
In future work, we aim to show that the length function implies two standing waves in the system–one stable and one unstable–and further investigate the unstable wave. We will also explore the traveling wave problem by adopting a moving-frame coordinate system, redefining the habitat function with a speed parameter. Using a similar approach, we will determine a critical speed at which a population can remain stable while moving.
Faculty Mentors: Dr. Emmanuel Fleurantin, Dr. Matt Holzer
Graduate Mentor: Julia Seay
Researchers:
- Enayah Rahman
- Ivan Chan
- Nicholas Maranto
The Luke team worked to produce probabilistic models to identify patterns in the immune response variability of convalescent COVID-19 patients over time. Utilizing SARS-CoV-2 antibody datasets, they applied Maximum Likelihood Estimation to determine optimal parameter values and created visualizations using MATLAB and Python.
The team’s work resulted in time-dependent probabilistic models for anti-RBD (receptor binding domain) and anti-N (nucleocapsid) IgG (immunoglobulin G) and IgA antibodies. These models were based on a modified gamma distribution, adapted to account for time-dependence. Additionally, they constructed a two-dimensional model comparing anti-N and anti-RBD IgG antibodies.
To enhance the accuracy of the two-dimensional model, further refinements are necessary to better account for the correlation between antibody targets. Future research will focus on expanding the models’ dimensionality by incorporating additional biologically significant variables, potentially revealing new structures and separations within the data.
Faculty Mentor: Dr. Rayanne Luke
Graduate Mentor: Kelsey Ellis
Researchers:
- Arpan Das
- Layan S Wahdan
Overview:
A polyform is a plane figure constructed by joining together identical basic polygons. In Euclidean geometry, there are three types of polyforms: polyominoes, polyiamonds, and polyhexes, made out of squares, equilateral triangles, and regular hexagons respectively. In this project we explored hyperbolic polyforms, of which there are infinite types. These are constructed by joining together tiles of a regular hyperbolic tessellation. In these polyforms, a hole is a bounded component of the complement of the polyform.
Our primary goal was to answer this question: Given a positive integer h, what is the minimum number of tiles required for a polyform to have h holes? Additionally, we aimed to find a way to effectively visualize hyperbolic polyforms with large numbers of tiles, find a way to algebraically represent hyperbolic polyforms, and to create examples of holey hyperbolic polyforms.
Key Outcomes:
1. Theoretical Advances: Established that the function for the minimum number of tiles g(h) required to form a polyform with h holes has linear asymptotic growth. The upper bound for this is was achieved by constructing polyforms with h separate single-hole units, each needing pq − 2p tiles. The lower bound for this was achieved by examining the number of edges each hole must have and the maximum number of edges a tile can add to the polyform.
2. Coding Developments: Developed an algorithm using DFS without backtracking to compute the minimum tiles for polyforms with a given number of holes across various tessellations. While effective for polyforms with few tiles, the algorithm has O(pn) time complexity where n is the number of tiles in the resulting polyform, so it quickly becomes infeasible for larger polyforms.
3. Representations: Produced detailed visualizations of holey polyforms with minimal tiles. These can seen at the end of the document, with tiles colored red and holes colored gray. We thank Malin Christersson for his website that generates hyperbolic tilings, which were then recolored to show the polyforms: https://www.malinc.se/noneuclidean/en/poincaretiling.php
Conclusions and Future Work:
We have demonstrated linear bounds for g(h) and provided computed minima for small h in various tessellations. Moving forward, our goal is to explore algebraic representations of polyforms, improve algorithmic efficiency, and seek a closed-form expression for g(h). Recently, we have also been pursuing a representation using group theory.
Acknowledgments:
We thank our faculty mentors Dr. Ros Toala and Dr. Erika Roldan, as well as our graduate mentor Summer Eldridge for their support and guidance. We also thank Dr. Peter Kagey for helping us better understand polyform enumeration.
Example Polyforms:
Faculty Mentors: Dr. Ros Toala, Dr. Erika Roldan
Graduate Mentor: Summer Eldridge
Researchers:
- Aiden Roger
- Cooper Roger
- Adithya Prabha
The ancient Egyptians expressed proper fractions as finite sums of distinct unit fractions. A unit fraction is one with numerator 1 and a positive integer denominator. For example, 5/8 may be represented as 1/2 + 1/8. We explore the application of this unique design on rational functions of the form f/g where the degree of f is less than or equal to the degree of g, decomposing f/g into a sum of polynomial reciprocals. For example, x/(x^2+1) = 1/x + 1/(-x-x^3).
The algorithms we found to be most adaptable to rational function decomposition were the Pierce-Engel, Fibonacci, and Golomb algorithms.
– The Pierce and Engel decompositions express fractions as a sum (or alternating sum) of reciprocals where the denominators are multiples of the original denominator.
– Fibonacci’s Greedy Algorithm iteratively subtracts the largest unit fraction from the given fraction until the remainder itself becomes a unit fraction.
– Golomb’s Algorithm repeats the process of taking the multiplicative inverse of the denominator modulo the numerator, continuing until only a unit fraction remains.
By imitating these three methods on polynomial fractions, we analyze the limitations and patterns of each algorithm. Beginning with the Pierce-Engel algorithm for rational functions documented by Dr. Epstein, we successfully implemented these three algorithms using Mathematica, after abortive attempts to do so in Matlab. This ultimately allowed us to quickly compute these decompositions, avoiding lengthy hand computations. The efficiency of these computer scripts enables us to focus on establishing bounds on the number of terms and degrees of denominators in the decompositions, as well as comparing the effectiveness and elegance of the decompositions. A complex hand computation has been reduced to seconds using the code we have developed this semester, making the work of potential future semesters much more streamlined, allowing all time to be dedicated to analysis.
Faculty Mentor: Dr. Neil Epstein
Graduate Mentor: Tim Banks
Researchers:
- Kaitlyn Sullivan
- Kalkedan Malefia
Although the method of training a neural network (NN) is well understood, how the data is transformed within the NN is not. Prior research by [1] assumes that the data is sampled from a manifold and suggests that a properly trained NN simplifies the shape of the data manifold by reducing its topological complexity (TC). Using this framework, the authors of [ 2 ] study the activation data at each layer of the NN using persistent landscapes. Surprisingly, the results of their study indicate that the TC does not necessarily decrease as the data passes through the NN and might even increase. We expand upon their work by examining how the homology of the data at each dimension contributes to the TC. Our study indicates that the main contributor to the increase in TC is from the 0th dimensional homology.
Faculty Mentor: Dr. Ben Schweinhart
Graduate Mentors: Shrunal Pothagoni and Justin Cox
Researchers:
- Diane T Hamilton
- Finn Brennan
- Joseph A Jung
Faculty Mentor: Ros Toala
Description: This semester, we reached 931 students of all ages across 10 different organizations. We were able to run 5 activities, with our primary focus being on “You Can Count on Monsters,” “The Shapes That Make Us,” and our most recent activity, “Living in a Math Bubble”. We received the MENSA Youth group at GMU to perform the bubbles session, and we were present at the Spooky Mad Science Expo and GMU’s Open House and STEM Access camp.
Graduate Assistants: Kelsi Listman, Suzie Castro-Tarabulsi
Undergraduate Interns:
- Oluwatomisin Badmus
- Mark Dubynskyi
- Khang Ly
- Joanne Romo