MEGL is running an 8 project program with 56 participants between faculty, graduate research assistants, and undergraduate research interns. The research projects are titled

- Universal Cycles in Higher Dimensions
- Embeddings of a semi-homogeneous tree in the hyperbolic disk.
- Monte Carlo, Las Vegas, and Quantum Algorithms for Quantum Systems
- Kelly criterion
- Biological Locomotion via Surface Tension
- Topology of Neural Networks
- Finite 2-groups
- Hyperbolic Knots in Thickened Surfaces

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.

The research teams are meeting weekly to generate empirical data which helps them form and explore mathematical conjectures based on observed patterns.

At the end of the semester, the research teams will have the opportunity to present their findings at a poster session followed by a Symposium. For project descriptions and names of the researchers involved, click the boxes below.

**Faculty Mentor: **Rachel Kirsch

**Description:** Description: Building on last semester’s exploration of universal cycles, we found a way to construct uptori from upcycles and De Bruijn sequences. An uptorus is an array of characters (including a wildcard in at least one position) that is viewed cyclically and covers every rectangular matrix with a specified height and width exactly once (like a De Bruijn torus, which does not have wildcards, does). Provided that an upcycle exists for word length n, this construction produces uptori for sub-matrices of size n × m for all integers m > 2. We also modified our construction so that it constructs uptori using up-families and alternating De Bruijn sequences. A universal partial family (or “upfamily”) is a set of cyclic strings that covers every substring of a specified length exactly once (similar to a De Bruijn family, which does not contain wild- card characters). We also have conjectured a construction of upfamilies, as well as found different types of upfamilies.

**Graduate Research Assistant: **Matthew Kearney

**Research Interns:**

- Stefan Popescu
- William Carey

**Faculty Mentor:** Flavia Colonna

**Description:** This semester we worked on embedding infinite homogeneous trees in the complex disk. We mainly worked on learning the math behind embedding these trees (Hyperbolic Space, geodesics, mobius transformation, etc.) then implementing it in a code project. We were able to embed both even and odd degree trees in the code, where you input the initial vertex length and number of branches then it outputs the embedding. We plan on working on this project next semester, and one goal that we know of now is being able to implement alternating degree trees. We also look forward to learning more of the math behind this.

**Graduate Research Assistant: **Madeline Horton

**Research Interns:**

- Arpan Das
- Connor Poulton
- Tate Fitzmaurice

**Faculty mentor:** Michael Jarret

**Description: **In the Fall 2023 semester, we studied Quantum Markov Chain Monte Carlo methods. We learned about using Metropolis-Hastings Algorithms with the Ising Model. We began researching the Negative-Sign Problem, which is known as the biggest bottleneck in Quantum in numerical simulations of quantum models, on the basis that it seems to have been mischaracterized.

Next semester, we hope to find ways to aid physics researchers in curing or easing the Negative-Sign Problem, such as categorizing occurrences of the problem and pairing them with optimal methods for curing and easing the problem.

Here is visualization of a known Monte Carlo method that we implemented during the semester:

**Graduate Research Assistants: **John Kent and Anthony Pizzimenti

**Research Interns:**

- Andy Miller
- Mark Dubynskyi
- Raghu Guggilam

**Faculty:** Douglas Eckley

**Description: **The Kelly Criterion is a formula that is used to maximize the geometric return from playing a favorable game or making a favorable investment.

The Kelly formula says that you should put into play, or invest, this proportion of your wealth:

𝑝/𝑎 − (1 − 𝑝)/𝑏

Where p is the implied probability, a is the outcome of a win, and b is the outcome of a loss. The formula tells you to play only when you have an expected gain. To see that, set p = 0.5 and a = b, in the case of our research, both are also 0.5.

In previous semesters the Kelly Criterion was used in real-life examples, such as stock market trading and insurance. This Semester we worked with two well-known situations; the famous “two-envelope” problem and the less famous, but to-date not convincingly resolved, “Siegel’s paradox”.

There appeared to be an expected gain in both cases, yet the apparent gains have paradoxical aspects. In both, the player has a 50% chance of doubling their money and a 50% chance of losing only half of his/her money. Thus, an expectation of a 25% gain. The Envelope Problem was analyzed using a stochastic game simulation where two envelopes were stuffed with an “x” and “2x” amount of money. A player would select one envelope and be given the option to switch. It was found that regardless of whether the player switched the gain was the same, where the paradox lies but the problem was expanded and a strategy for winning every time was explored. Seigel’s Paradox is a foreign currency trading paradox. If two currencies with an equal chance of increasing and decreasing in value are traded against each other then the paradox arises because there appears to be an expected gain for a trader whose primary currency is of either currency to trade to the other although only one can go up in value. As the trades affect the rate of exchange, when the value of one currency peaks the players cannot cash out their winnings as the rate of exchange is affected by the amount of the currency available to the other side. An equation was developed which described the relationship between the currencies and the rate of exchange and this was used in an R simulation to begin to simulate this paradox and show graphically what is occurring to lay a foundation to attempt to solve it in the future.

**Research Interns:**

- Bemnet Bekele
- James O’Hanlon
- Tiffany Sun

**Faculty Mentors:** Evelyn Sander and Dan Anderson

**Description: **In this study, an extended application of Archimedes Principle is utilized to incorporate the force of surface tension into the force balance equation that describes a uniformly dense object floating at the air-water interface. In particular, this model describes a floating object with a uniform cross-sectional convex shape and area in the xy-plane and which is sufficiently larger in its length on the z-axis than the shape in the xy-plane. This problem is depicted in the figure below.

The model is adapted to a computational description via the extended Archimedes principle, which is constituted of a system of three equations and three unknowns:

The inputs to this model are H, the height of the waterline on the object at infinity, and the rise/fall of the meniscus against this waterline height on the left, ∆y_L, and right, ∆y_R, side of the object. This semester, this model was constructed to determine how deep the object floats in the water by observing a cylinder. In the future, this model will be extended to describe the potential energy of the floating object at the determined submersion depth as a function of the rotation of the polygon that constitutes the cross-sectional shape. This model’s adherence to experiment is determined by 3D printing an object and comparing experimental and theoretical results.

**Graduate Research Assistants: **Brandon Barreto-Rosa and Patrick Bishop

**Research Interns:**

- Daniel Horvath
- Mariah Tammera
- Max Werkheiser
- Sarah Wendt

**Faculty:** Benjamin Schweinhart

**Description: **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.

**Graduate Research Assistants: **Shrunal Pothagoni and Justin Cox

**Research Interns:**

- Alex Martinez
- David Wigginton
- Eugenie Ahn
- Finn Brennan

**Faculty:** Casey Blacker

**Description:** In this project, we went over the intuition of a 2-group, which can be thought of as a category G with a binary operation ⊗ between two objects or two morphisms, such that (G, ⊗) has a group structure for both its objects and morphisms.

a 2-group describes multiplication of both objects and arrows in certain categories.

To find and verify analogues of theorems in group theory for the 2-group setting, we worked with crossed modules for strict 2-groups. A crossed module consists of two groups H and G, a group homomorphism t, and a group action α, such that t, α satisfies equivariance (t(αg (h)) = gt(h)g−1) and the Peiffer identity (αt(h)(h′) = hh′h−1). Crossed modules induce 2-groups, and 2-groups induce crossed modules.

Using crossed modules, we were able to define and verify notions of homomorphisms, kernel, images, and quotients in the 2-group setting, which leads us to the first isomorphism theorem for crossed modules. Progress has also been made for finding a 2-group analogue to the fundamental theorem of finite abelian groups, but it remains an open research question.

**Graduate Research Assistants: **Ethan Clelland and Michael Merkle

**Research Interns:**

- Anthony Vu
- Morgan Schuman
- Nick Lear

**Faculty: **Rose Kaplan-Kelly

**Description: **In this project, we will be investigating tilings, knots, links, and surfaces. Imagine taking a few pieces of string, tying them up, and then gluing the loose ends together in pairs. The result is a mathematical link(except that we think of the string as having no thickness, so its cross section is a point). Each individual piece of string that we started with is a component of the link and when a link has only one component we call it a knot. We can build links from tilings by turning the edges of the tiling into strands of the link and the vertices of the tiling into crossings.

A thickened surface is a surface crossed with an interval. For example, to imagine a thickened torus,we could picture a pool noodle with its ends glued to each other. The complement of a link in a thickened surface is the space left over after we remove the components of the link. We say that a link is hyperbolic if its complement admits a complete metric of constant curvature -1. A way to think of negative curvature is to consider lines of sight. Here our lines of sight diverge quickly. We can see this on the hyperbolic pentagons in Figure 2. Near a vertex, geodesic segments have a small angle between them, but as we move along the sides of the pentagons these segments quickly bend away from each other.

In this project, we will explore questions such as:

1. Can we construct knots in any genus thickened surface from tilings consisting of one or two types of polygons?

2. How many links built from tilings with one or two types of polygons are there with diagrams on a genus 2 surface?

**Graduate Research Assistants: **Gabe Lumpkin and Kiefer Green

**Research Interns:**

- Antonio Alt
- Haboon Yusuf
- Nicholas Maranto