Undergraduate Mathematics Research Symposium@UF
University of Florida, Department of Mathematics
April 26, 2024
The fourth annual Undergraduate Mathematics Research Symposium (UMRS) at UF took place on April 26, 2024 in Little Hall 225.
It was organized by Konstantina Christodoulopoulou and Sara Pollock.
Schedule
Session 1
| Time | Presenter(s) | Presentation |
|---|---|---|
| 9:00–9:15 | Erik Shute | Surface Theory and Soliton Equations |
| 9:20–9:35 | Connor Panish | Coalescing Ballistic Annihilation |
| 9:40–9:55 | Jianda Du Armando Albornoz | Exploring the Hausdorff and Gromov–Hausdorff Distance |
| 10:00–10:15 | Iman Kazmi Diego Espinosa Youssef Kamel | Lamprey’s True Enemy Might Be Their Own Biology |
Coffee Break
10:20–10:35
Session 2
| Time | Presenter(s) | Presentation |
|---|---|---|
| 10:35–10:50 | Michael Poole | Analysis of Entities Less Than Any Real Magnitude |
| 10:55–11:10 | Audrey Edwards John McDonald Olutimilehin Sobanjo | Irrational Behavior: A Mathematical Approach to Modeling Aggression and Responses to Aggression |
| 11:15–11:30 | Martin Wall | The Gromov–Hausdorff Distance Between Quotient Metric Spaces |
| 11:35–11:50 | Oscar Camargo David Castellanos Sharan Majumder | Modeling Environmentally Triggered Sex Determination in Sea Lamprey Populations |
| 11:55–12:10 | Kevin Jin | Exploring K-Factor in Elo Rating Systems |
Lunch
12:15–1:10
Session 3
| Time | Presenter(s) | Presentation |
|---|---|---|
| 1:10–1:25 | Saisudharshan “Sai” Sivakumar | Free and Virtual Resolutions |
| 1:30–1:45 | Tuyen Truong Ziwen Zhu | Modeling Kangaroo Mother Care (KMC): Results from a Model Analysis and Health Outcomes |
| 1:50–2:05 | Alejandro Leon | Bounds on the Gromov–Hausdorff Distance Between the Unit Disk and Its Boundary Unit Circle |
Coffee Break
2:10–2:25
Session 4
| Time | Presenter(s) | Presentation |
|---|---|---|
| 2:25–2:40 | Samuel Kim | Minimal Flows of the Automorphism Group of the Random Graph |
| 2:45–3:00 | Natalie Nicole Perez Rong Fang Megan Sin | Using Differential Equations Modeling to Predict How People’s Likelihood to Prefer Retributive Punishment over Restorative Punishment Will Change Over Time |
| 3:05–3:20 | Ryan Campisi | Burnside’s Lemma for Analysis of Error-Correcting Codes |
Abstracts
Surface Theory and Soliton Equations
Presenter: Erik Shute
Mentor: Luca Di Cerbo
We will discuss the relationship between surfaces of constant negative curvature and the sine-Gordon equation, a nonlinear partial differential equation with applications in condensed matter physics. We describe how this relationship arises naturally from geometrical considerations and how classical transformations of surfaces generate families of sine-Gordon solutions, including so-called soliton solutions. Lastly, we will briefly mention how the surface theory discussed is broadly applicable to many soliton equations.
Coalescing Ballistic Annihilation
Presenter: Connor Panish
Mentor: Matt Junge, Baruch College
During the 1980s, the statistical physics literature introduced a model called ballistic annihilation that mimics chemical reactions. Particles in this model are placed randomly throughout the real line and then proceed to move at predetermined discrete velocities. Upon collision, the particles involved are annihilated—that is, removed from the system. In 2019, Haslegrave, Sidoravicius, and Tournier characterized initial conditions for which all particles are eventually annihilated. Expanding upon HST, we study a more general version of this system in which particles can survive annihilation with fixed probabilities that depend on the velocities of the colliding particles. We prove a formula for the initial conditions necessary for all particles to be annihilated based on the probabilities of survival. Our arguments make use of recursion and hidden symmetries within infinite sums.
Exploring the Hausdorff and Gromov–Hausdorff Distance
Presenters: Jianda Du and Armando Albornoz
Mentor: Henry Adams
Metric spaces are ubiquitous and critical in many fields of mathematical study, including topological data analysis, topology, and numerical analysis. We will discuss different notions of similarity between metric spaces and demonstrate the importance of considering Hausdorff distance. Finally, we will introduce the Gromov–Hausdorff distance, elucidate its connection with the Hausdorff distance through examples from our research, and show why it is a useful notion of similarity.
Lamprey’s True Enemy Might Be Their Own Biology
Presenters: Diego T. Espinosa, Youssef M. Kamel, and Iman Kazmi
Mentors: Tracy Stepien and Hemaho Taboe
Sea lampreys are a type of jawless parasitic fish. This project and its mathematical model, developed through an extensive literature review, examine how adaptive variation in the sea lamprey population’s sex ratio can be monitored and controlled based on its advantages and disadvantages. The parasitic behavior of lampreys often poses a challenge to fisheries and ecosystem stability because of their invasive nature. Focusing on the Great Lakes, we synthesize data and information from local monitoring agencies and scholarly papers. Our model incorporates variables including population sizes, the ratio of each sex, resource availability in the ecosystem, and reproductive success for each sex. By carefully considering certain parameters and purposefully neglecting others, our analysis provides a realistic model with potential long-term applications.
Analysis of Entities Less Than Any Real Magnitude
Presenter: Michael Poole
Mentor: Michael Jury
Throughout its history, calculus has been treated heuristically using infinitesimals and infinites, quantities smaller and larger than any real number, respectively. However, no infinitesimals or infinites exist within the real numbers. Seventeenth-century mathematicians knew that infinitesimals and infinites were fictitious and discouraged the use of these quantities in formulating the standard theory of analysis. In this presentation, we will discuss how to construct a non-Archimedean field that contains the real numbers and the desired infinitesimals and infinites. This field is known as the hyperreal numbers, and it can be used to recast elementary calculus and analysis in a more intuitive light.
Irrational Behavior: A Mathematical Approach to Modeling Aggression and Responses to Aggression
Presenters: Audrey Edwards, John McDonald, and Olutimilehin Sobanjo
Mentor: Tracy Stepien
This study investigates the dynamics of different human subpopulations based on the manner in which they react when faced with aggressive behavior, as well as the stability of these subpopulations over time in different societal environments. A multivariate Lotka–Volterra model was used to run simulations with parameters adjusted to represent environments in which retributive or meditative punishment is the dominant societal response to aggressive behavior.
The Gromov–Hausdorff Distance Between Quotient Metric Spaces
Presenter: Martin Wall
Mentor: Henry Adams
The Hausdorff distance measures how far apart two sets are in a common metric space. By contrast, the Gromov–Hausdorff distance provides a notion of distance between two arbitrary metric spaces. Fix a group G, and suppose that two metric spaces X and Y are each equipped with an action of G by isometries. For the Hausdorff distance, if X and Y are G-invariant subspaces of a common metric space, then dH(X, Y) = dH(X/G, Y/G). We will show, however, that the ratio dGH(X/G, Y/G) / dGH(X, Y) can be made both arbitrarily large and arbitrarily small. This demonstrates that, while the Hausdorff distance is inflexible under quotients, the Gromov–Hausdorff distance is inconstant.
Modeling Environmentally Triggered Sex Determination in Sea Lamprey Populations
Presenters: Oscar Camargo, David Castellanos, and Sharan Majumder
Mentor: Tracy Stepien
When modeling the population dynamics of lampreys, it is crucial to consider that their sex ratio fluctuates with the resources available in their environment. We adapted a nonlinear system of differential equations originally used to model sex determination in green sea turtles. Our results led us to develop hypotheses about the significance of environmental sex determination in lampreys and its potential effects on the broader ecosystem.
Exploring K-Factor in Elo Rating Systems
Presenter: Kevin Jin
Mentor: Miklós Bóna
The Elo rating system assigns players ratings that measure their relative skill levels. In this talk, we investigate the K-factor, a variable responsible for determining the sensitivity of ratings to change. In practice, it is set arbitrarily based on heuristics. We seek to optimize the K-factor by minimizing the expected error between a player’s true skill and the skill estimated by the player’s rating. Using calculus and convolutions, we attempt to obtain a closed-form expression for error as a function of the K-factor. With this expression, we can study how different K values affect the accuracy of ratings over time. We also present evidence for our theoretical results using Monte Carlo computer simulations.
Free and Virtual Resolutions
Presenter: Saisudharshan “Sai” Sivakumar
Mentors: Christine Berkesch and Michael Perlman, University of Minnesota Twin Cities
A syzygy, or free resolution, is a particular sequence of homomorphisms of free modules terminating at a module under investigation. Free resolutions provide useful information about the module, such as its structure, or geometric information when the module is the coordinate ring of a variety. Virtual resolutions provide similar information for Cox rings of smooth toric varieties.
We will briefly discuss free resolutions obtained for certain stable submodules of polynomial rings under the action of the general linear group, as well as short virtual resolutions obtained for Cox rings of finite sets of points in a product of two projective spaces. These are joint works with Dr. Michael Perlman, Bjorn Cattell-Ravdal, Erin Delargy, Akash Ganguly, and Sean Guan, and separately with Dr. Christine Berkesch, Isidora Bailly-Hall, Karina Dovgodko, Sean Guan, and Jishi Sun.
Modeling Kangaroo Mother Care (KMC): Results from a Model Analysis and Health Outcomes
Presenters: Tuyen Truong and Ziwen Zhu
Mentor: Tracy Stepien
We develop a differential model to analyze the impact of Kangaroo Mother Care (KMC) treatment on infant mortality rates. Following the standardized definition of KMC provided by the World Health Organization, we establish two factors in our model: the duration of skin-to-skin contact and birth weight. We improve the existing logistic differential model by accounting for diminishing returns and developing a separate equation for skin-to-skin contact. Using 2022 data from the World Data Bank, we calibrate our model’s constants accordingly. Our analysis suggests that increasing the duration of skin-to-skin contact can reduce the risk of infant mortality under KMC treatment.
Bounds on the Gromov–Hausdorff Distance Between the Unit Disk and Its Boundary Unit Circle
Presenter: Alejandro Leon
Mentor: Henry Adams
The Gromov–Hausdorff distance is a way of quantifying the dissimilarity between metric spaces. We investigate the Gromov–Hausdorff distance between the unit disk and its boundary, the unit circle, under the Euclidean metric. We derive upper and lower bounds on this quantity.
Minimal Flows of the Automorphism Group of the Random Graph
Presenter: Samuel Kim
Mentor: Dana Bartošová
It is known that the orbit of the universal minimal flow of the automorphism group of the random graph, Aut(G), consists of all linear orderings of the random graph. This space has a comeagre orbit of the order type of the random ordered graph, and the random ordered graph has exactly 42 reducts. We aim to find the minimal flows of Aut(G) using the reducts of the random ordered graph and invariant, closed equivalence relations on the linear orderings of the random graph.
Using Differential Equations Modeling to Predict How People’s Likelihood to Prefer Retributive Punishment over Restorative Punishment Will Change Over Time
Presenters: Natalie Nicole Perez, Rong Fang, and Megan Sin
Mentor: Tracy Stepien
People from different backgrounds, lifestyles, and experiences have differing propensities for wanting to punish others for crimes or misdeeds. Based on these experiences, some will prefer retributive punishment over restorative punishment, and vice versa. A model was created to examine how populations of people with different propensities to act against transgressors change over time. This model focuses on an individual’s wealth as a defining factor because of the ways it influences both lifestyle and experience. The model provides a better understanding of how these groups of people constitute and affect society.
Burnside’s Lemma for Analysis of Error-Correcting Codes
Presenter: Ryan Campisi
Mentor: Libin Rong
This research investigates the structural symmetries and equivalences of error-correcting codes, which are crucial for maintaining data integrity during transmission and storage. We will examine the use of Burnside’s Lemma from group theory to analyze relationships among code words under symmetry operations. By focusing on the formation of code-word orbits under group actions, this research demonstrates the utility of these concepts through illustrative examples.
Acknowledgments
Special thanks to Kevin Knudson and the Department of Mathematics for their support and for providing coffee and lunch.
Special thanks to Margaret Somers for all her help in organizing the symposium.