9 Math Majors Present Math Research at 2025 UMaine Student Symposium
Nine math majors presented posters on their math research projects at the University of Maine Student Symposium on Friday, April 11. Ryan Allen (pictured) won the top prize in the Physical and Mathematical Sciences category for his poster “Modeling the Mechanics of Avian Breeding and Migration”, based on his math capstone project under advisement of Prof. Jane Wang.
Math Capstone Projects
101. An Exploration in Bootstrapping (Poster)
Author(s): Connor Evans-Ralston
Faculty Mentor: Jane Wang
Abstract: The bootstrap method is a resampling technique in statistics that focuses on estimating the distribution of a sample statistic (sample mean, sample median, etc). The technique, introduced by Efron (1979), is an extension of the jackknife method, another way to measure the bias and standard error of a test statistic, but is more widely applicable under regular cases. However, there are several irregular cases where the traditional bootstrap fails, namely when bootstrapping a minimum or maximum which estimate endpoints of population distribution, bootstrapping the sample mean when the population variance is infinite, bootstrapping sample eigenvalues when population eigenvalues have multiplicity, and the case of sample median when the population is discontinuous at the population density. In these situations, the approximation of the limiting distribution between the bootstrap and the population are nonequivalent. Fortunately, a solution for such problems exist in the “m out of n” bootstrap. This project is structured as follows. I introduce the bootstrap method and the mathematics behind it before exploring some of the irregular cases in which the bootstrap fails and why it does not work. I close with an overview of the “m out of n” bootstrap and how it remedies these issues.
104. Topological Phenomena in Physical Systems (Poster)
Author(s): Maren Adnee
Faculty Mentor: Jane Wang
Abstract: In this project, we will examine the mathematical formalism behind topologically protected phenomena physical systems, focusing primarily on phenomena in magnetic materials. Topology is a mathematical framework in which objects are characterized by countable properties (How many holes are there? How many twists are there?), as opposed to measured properties (What is the diameter of the hole? To what degree is the object twisted?). Characterization of physical systems using their topological characteristics has become a vibrant area of research in recent decades. As topology focuses on global properties, it can provide an effective way to characterized highly complex systems, providing a new avenue for research in theoretical condensed matter physics. Furthermore, it has been shown experimentally that materials with non-trivial topology can allow for unique phenomena to exist, a state called ”topological protection”. We will we begin by looking at theoretical 1-dimensional (1D) magnetic materials and showing how the topology of magnetic moment configurations can affect the material’s properties. We will then expand into 2- and 3- dimensional systems, mathematically defining topologically protected phenomena including solitons, vortices, and skyrmions. Finally, we will qualitatively explore topological phenomena in other fields of physics, including fluid dynamics, quantum mechanics, and cosmology.
105. From Caesar to Cybersecurity: The Mathematical Journey of Cryptography (Poster)
Author(s): Anne Bostock
Faculty Mentor: Jane Wang
Abstract: Cryptography, the practice of concealing messages so that only intended recipients can understand them, has evolved alongside mathematical advancements throughout history. The earliest recorded instance, dating back to 1900B.C., featured Egyptian hieroglyphs modified in an unusual way to obscure their meaning from unintended viewers. As mathematics progressed, so did cryptographic methods. By 44B.C., the Romans had developed the Caesar Cipher– a cryptograph in which they utilized modular arithmetic to systematically shift letters in the alphabet.
The study explores the historical evolution of cryptography and its deep-rooted connection to mathematics. From early substitution ciphers to the increasingly sophisticated methods of World War I and II– such as the Enigma machine and Alan Turing’s codebreaking efforts– each cryptographic advancement reflects contemporary mathematical knowledge. We will also examine the transition into modern cryptographic techniques, including RSA and Elliptic Curve Cryptography, which rely on number theory and discrete mathematics.
By analyzing key historical milestones and their mathematical foundations, we will illustrate how cryptography has continually adapted to meet security challenges. As mathematics continues to evolve, so will cryptography, ensuring its role as a “key” component in cybersecurity in the modern digitalized age.
107. Rendering of Hyperbolic Spaces (Poster)
Author(s): Kieran Firkin
Faculty Mentor: Jane Wang
Abstract: Euclidean spaces conform to certain seemingly universal geometric intuitions: parallel lines remain equidistant at all points along their lengths, the interior angles of a triangle add up to 180°, and a shape consisting of all right angles is a quadrilateral. Non-Euclidean―e.g. hyperbolic, elliptical, and spherical―spaces violate these assumptions, often leading to surprising results. In this project, we will explore the process of efficiently rendering hyperbolic spaces onto a two-dimensional screen to allow interactive visualization.
This involves transforming hyperbolic objects between different reference frames using hyperbolic trigonometry, using math that is analogous to the Euclidean case, and then selecting and applying a useful projection method to embed the resulting objects into the Euclidean two-dimensional plane.
108. Modeling the Mechanics of Avian Breeding and Migration (Poster)
Author(s): Ryan Allen
Faculty Mentor: Jane Wang
Abstract: Since 1970, the global population of birds has dropped by roughly 30%. While some of this loss is well-documented as a result of habitat loss or collisions with buildings, some species like the Rusty Blackbird have experienced unprecedented loss from unknown causes. Conservation efforts have taken many forms, but a recent development is mathematical modeling of breeding and migration patterns. In particular, systems of ordinary differential equations are constructed using parameters that change instantaneously in order to roughly model the changing of seasons. These so-called time-switching hybrid systems allow a more accurate description of the behavior of bird species over the course of a year. In this project, we reconstruct one of these models, namely (Donohue, 2015), and provide an analysis of it versus other models. Using Cornell’s citizen science project eBird as a basis for parameters, we code the model and provide examples of the results.
109. What’s the Point? (Group): Application of Mathematical Crystallography (Poster)
Author(s): Henry Johnson
Faculty Mentor: Jane Wang
Abstract: The examination of physical crystals in order to study their chemical composition and structural properties leads to a natural derivation of repeating patterns and symmetries on a atomic level. Due to the scale of these repeating chemical units, an analogous infinite mathematical crystal model can be used to study the symmetrizes formed in crystals. Restricted to three-dimensions and euclidean space, mathematical lattices, point groups and space groups can constructed allowing the derivation of 14 Bravais Lattice Structures and 32 associated Crystal Groups.
However leaving the restrictions of three-dimensions and euclidean space the notion of a mathematical crystal structure can be extended. By examining the past literature on mathematical crystallography, as well as modern research into crystals structures in more abstract spaces a general definition of a mathematical crystal structure will be derived. With this definition naturally occurring structures will be examined with the aim to broaden what can be considered a “crystal” within a mathematical perspective.
609. The Math Proofs of Music Theory (Poster)
Author(s): Enny Lolar
Faculty Mentor: Jane Wang
Abstract: This project will focus on the technical side of music theory. Specifically, we will be looking into what makes good sounds in musical composition. A majority of music theory is based upon already established laws that we are implied to just accept. It is common knowledge that sound is produced by waves, so the goal is to go deeper into what makes a well-tuned sound by examining everything that goes into producing the sound, what happens when you blend sounds, and explaining overtones.
Other Math Projects
121. Applying Mathematical Methods for Verifying Authorship to Texts Attributed to Virgil (Poster)
Author(s): Zachary Scott
Faculty Mentor: Jane Wang
Abstract: Ancient texts have long presented a problem in authorship accountability. Many have used statistics to approach authorship problems – from the Pauline epistles to the Federalist Papers – and this practice has come to be called stylometry. Our interest is the investigation of pseudepigrapha, which refers to a text that is falsely attributed to a particular author, usually a famous one. This problem is very common in ancient texts. Horsfall, a classicist, says that “in the ancient world, (…) you could measure fame by the accumulation of forgeries.” Stylometry has enjoyed a surge in popularity in the digital age due to its potential to combat disinformation and plagiarism, but its roots are in the study of these ancient texts.
The targets for our analysis are two pieces of potential pseudepigrapha from the poet Virgil. The primary target is a potential interpolation within Virgil’s Aeneid dubbed the Helen Episode. The episode is a chunk of 22 lines in the midst of the second book. The transmission of the Aeneid is generally sound, but the Helen Episode only appears several centuries after Virgil’s death. There are classicists on either side of the question of authenticity. Our second target is a collection of juvenilia called the Appendix Vergiliana. These are poems attributed to a young Virgil, and are widely considered to be spurious.
We have identified and demonstrated effective stylometric methods for authorship verification on ancient Latin poetry, such as Mahalanobis distance analysis and SVM, and applied them to the Helen Episode and the Appendix Vergiliana.
127. Breakdown in Smoothness of a Fluid Interface As It Moves Towards a Change in Porosity (Poster)
Author(s): Andrii Obertas
Faculty Mentor: Neel Patel
Abstract: In this research, we study the evolution of the free boundary separating two fluids in porous media, known as a Muskat problem. The permeability of the medium is described as a step function across a fixed boundary below the interface of the two fluids. The fluids are in the stable regime, with the denser fluid below the lighter one. The physical principle governing the fluid velocity in the porous medium is Darcy’s law. Rigorous analysis is used to derive the partial differential equation (PDE) describing evolution of the boundary separating the two fluids. When the free boundary approaches the change in the porosity of the media, the kernel of the PDE becomes singular. We specifically study the breakdown of the smoothness of the free boundary during this process. To achieve this, we are trying to prove an inequality relating the minimal distance between the free boundary and the change in porosity graph.