Undergraduate Mathematics Research Symposium@UF
University of Florida, Department of Mathematics
April 23, 2021
The first annual Undergraduate Mathematics Research Symposium (UMRS) at UF took place on April 23, 2021 over Zoom.
It was organized by Konstantina Christodoulopoulou and Sara Pollock.
Schedule
Welcome from Chair
8:55–9:00
Kevin Knudson
Session 1
| Time | Presenter(s) | Presentation |
|---|---|---|
| 9:00–9:15 | Parker Knight | Data-driven adaptive penalties for high-dimensional regression |
| 9:20–9:35 | Elaine Danielson | Strategies for the Explorer-Director Game |
| 9:40–9:55 | Christian Chase | Shift Significance in Major League Baseball |
| 10:00–10:15 | Anthony Matos | On Sobolev’s inequality |
Coffee Break
10:20–10:30
Session 2
| Time | Presenter(s) | Presentation |
|---|---|---|
| 10:30–10:45 | Sydney Edwards | Analysis of the effects of bile salts exposure on Shigella flexneri using a graph theoretic approach |
| 10:50–11:05 | Cooper O’Kuhn | Primes in Short Intervals and Narrow Sectors |
| 11:10–11:25 | Teegan Bailey | Undergraduates’ Covariational Reasoning across Function Representations |
| 11:30–11:45 | Michael Freedman | On Lower Bounds for Erdös-Szekeres Products |
Lunch
11:50–12:30
Session 3
| Time | Presenter(s) | Presentation |
|---|---|---|
| 12:30–12:45 | Julia Franck | Modeling Hematopoiesis in Chronic Kidney Disease Patients |
| 12:50–1:05 | Ankit Vishnubhotla | A description of the Universal Minimal Dynamical System for \(\mathbb{R}^n\) |
| 1:10–1:25 | Muhammad Abdulla | Dynamics of Ramping Bursts in a Respiratory Neuron Model |
| 1:30–1:45 | Jenesis Escobar Vishweshwar Ramanakumar Max Zhang | Topological Data Analysis: Introduction and Applications |
Closing remarks
1:50–1:55
Konstantina Christodoulopoulou and Sara Pollock
Abstracts
Data-driven adaptive penalties for high-dimensional regression
Presenter: Parker Knight
Mentor: Sara Pollock (UF, Mathematics)
High-dimensional datasets are often characterized by having correlated or structured variables, which is unaccounted for by many commonly used regression models such as the LASSO or ridge regression. Kernel-penalized regression (KPR) is an established extension of ridge regression that attempts to address this by incorporating sample- and variable-wise structural matrices into the model fitting procedure. However, KPR models are only able to account for one source of extrinsic data at a time, which is a disadvantage when the correlation structure in the data is co-informed by multiple sources. Here we present adaptive kernel-penalized regression (aKPR), an improvement to KPR that allows the inclusion of multiple structural matrices in model estimation. We also present a maximum likelihood framework for assigning an optimal weight to each structural matrix, which aids in model interpretation and improves numerical stability when one or more of the matrices are ill-conditioned. We then demonstrate the model’s utility with data-driven simulations and an analysis of a microbiome dataset. The software implementation of aKPR is publicly available through an easy-to-use R package, which can be installed from GitHub.
Strategies for the Explorer-Director Game
Presenter: Elaine Danielson
Mentor: Patrick Devlin (Yale University), Erin Meger (Université du Québec à Montréal and Loyola University Chicago), Abigail Raz (University of Nebraska-Lincoln)
The Explorer-Director game contains two players, Explorer and Director, who jointly move a token across the vertices of a graph. Each round, Explorer calls a distance for the token to move and then Director moves the token to a vertex that distance away. While Explorer seeks to maximize the number of vertices visited, Director’s goal is to minimize this number. We will discuss the results for this game on path graphs, in particular the use of non-adaptive Explorer strategies. In addition, we will talk about how these results for paths relate to trees, and how we can determine when a game is finished.
Shift Significance in Major League Baseball
Presenter: Christian Chase
Mentor: Aaron Molstad (UF, Statistics)
The shift in baseball is classified as the special positioning of defenders in an attempt to record more outs and effectively increase any given team’s winning percentage. In order to quantify the statistical strength of the shift for any given at-bat, we are running a hierarchical clustering system to cluster batters and hitters into specific groups based on their categorical similarities. Utilizing these clusters, we will model a multinomial regression in order to predict the beta values of each individual interaction of clusters in order to see whether or not the shift significantly impacts specific outcomes on the baseball field. The data will be taken from the modern StatCast Era (2015-2019) and will attempt to provide a more effective formula for MLB teams to utilize when shifting their respective defenses.
On Sobolev’s inequality
Presenter: Anthony Matos
Mentor: Lei Zhang (UF, Mathematics)
In mathematical analysis, a Sobolev inequality is an inequality which controls the magnitude of lower order derivatives of a function by the magnitude of higher order derivatives. The simplest Sobolev inequality is the one originally discovered by Sobolev in his seminal 1938 paper On a theorem in functional analysis, which controls an integral norm of a function by an integral norm of its gradient. In this talk, I will present a simple proof of Sobolev’s original inequality by taking the difficult Hardy-Littlewood maximal inequality as a given.
Analysis of the effects of bile salts exposure on Shigella flexneri using a graph theoretic approach
Presenter: Sydney Edwards
Mentors: Nate Veldt (Cornell University, Center for Applied Mathematics) and Christina S. Faherty (Harvard Medical School, Department of Pediatrics)
Shigella flexneri is a bacterial facultative pathogen that invades the colonic epithelium and causes shigellosis, a disease characterized by fever, vomiting, and watery diarrhea (Alves da Cruz Gouveia, Torres Camara Lins, and Alves Pontes da Silva, 2020). Shigellosis is associated with a high disease burden and over 180 million cases of shigellosis were reported in 2010 (Kotloff et al., 2018). Within Shigella research, there is a need to better understand the role of bile exposure in Shigella pathogenesis. Shigella resists bile salts during transit through the small intestine and forms a biofilm (Nickerson et al., 2017). However, lingering questions remain regarding how bile exposure alters differential gene expression. Past RNA-sequencing analysis identified genes induced and repressed in the presence of bile salts. In order to better understand changes in gene expression, we utilized a graph theoretic framework to elucidate patterns in the RNA-sequencing data. Specifically, we examined patterns of gene expression with LambdaCC, an optimization method that locates communities of genes within the dataset. LambdaCC identified a unique clustering pattern of genes in which genes of similar function clustered together alongside seemingly unrelated genes. Additionally, LambdaCC categorized clusters of genes involved in bile salts resistance, biofilm formation, and virulence gene expression. Future work will leverage the results from this clustering analysis to formulate hypotheses and experimentally examine associations between bile exposure and differential gene expression.
Primes in Short Intervals and Narrow Sectors
Presenter: Cooper O’Kuhn
Mentors: Jesse Thorner (University of Illinois at Urbana-Champaign, Mathematics) and Ken Ono (University of Virginia, Mathematics)
We discuss a result in prime number theory which enumerates the number of primes satisfying certain conditions defined by polynomials in several variables. In particular, we count the number of primes of the form \(a^2 + b^2\) with \(a\) considerably smaller than \(b\) and how they are distributed in arithmetic progressions analogously to the Bombieri-Vinogradov Theorem.
Undergraduates’ Covariational Reasoning across Function Representations
Presenter: Teegan Bailey
Mentor: Darryl Chamberlain (UF, Mathematics)
Covariational Reasoning is the is the mental actions, processes, and constructions that an individual uses to coordinate and interpret the varying change between multiple variables. This presentation will focus on the results of a study on how Calculus 1 students were able to model and solve a real-life problem by using covariational reasoning to construct a parametric representation. The presentation will include a brief introduction to the study structure and framework used, which will be followed up by an in-depth presentation of study results and implications for undergraduate Mathematics teaching.
On Lower Bounds for Erdös-Szekeres Products
Presenter: Michael Freedman
Mentor: Doron Lubinsky (Georgia Tech, Mathematics)
In this talk I introduce a problem first posed in 1959 joint paper by Paul Erdös and George Szekeres. In the paper they considered the function \(f(n) = \prod_{j=1}^{n} \big\lvert 1 – z^{s_j} \big\rvert\) for \(j = 1,\ldots, n\), \(s_j\) an integer, and z on the unit circle. Much work has been done on upper bounds for the function, but investigation of lower bounds had been neglected. I present work that I co-authored that gives an exponential lower bound for the product under certain conditions. An effort will be made to maintain accessibility and there is no prerequisite knowledge required.
Modeling Hematopoiesis in Chronic Kidney Disease Patients
Presenter: Julia Franck
Mentor: Maia Martcheva (UF, Mathematics)
This talk concerns a delay-differential equation model of red blood cell populations and the growth factor erythropoietin, the primary stimulant of red blood cell production. In this talk we consider a system of two delay-differential equations, one for red blood cell population, and one for erythropoietin. We will discuss Mackey-Glass terms, which are delay-differential expressions often used in modeling hematological diseases, and their role in the model. We also consider the feedback mechanism signaling erythropoietin production and its relationship to red blood cell and erythropoietin populations at a given time. Equilibrium analysis is then performed on the model; results show the existence of a free boundary equilibrium and zero or two coexistence interior equilibria, depending on conditions. Simulations are run to determine the efficacy of the model given set parameter values, which are determined from literature/data fitting, and later varied. Results show the existence of oscillatory solutions under certain conditions.
A description of the Universal Minimal Dynamical System for \(\mathbb{R}^n\)
Presenter: Ankit Vishnubhotla
Mentor: Dana Bartosova (UF, Mathematics)
Abstract topological dynamics provides characterizations of the dynamics of topological groups, which are mathematical objects equipped with both an algebraic and a topological structure. We introduce concepts in topological dynamics such as minimal flows produced on the spaces on which these topological groups act, ambits, and the connection to ultrafilter spaces. Then we study these notions in relation to the group \(\mathbb{R}^n\) with the Euclidean topology to give a description of \(S(\mathbb{R}^n)\), the phase space of the universal ambit, and \(M(\mathbb{R}^n)\), the phase space of the universal minimal dynamical system.
Dynamics of Ramping Bursts in a Respiratory Neuron Model
Presenter: Muhammad Abdulla
Mentor: Jonathan Rubin (University of Pittsburgh, Mathematics)
Intensive computational and theoretical work has led to the development of multiple mathematical models of bursting in respiratory neurons in the pre-Bötziner Complex (pre-BötC) of the mammalian brainstem. Nonetheless, these previous models have not captured the pre-inspiratory ramping aspects of these neurons’ activity patterns, in which relatively slow tonic spiking gradually progresses to faster spiking and a full-blown burst, with a corresponding gradual development of an underlying plateau potential. In this work, we show that the incorporation of the dynamics of the extracellular potassium ion concentration into an existing model for pre-BötC neuron bursting, along with some parameter updates, suffices to induce this ramping behavior. Using fast-slow decomposition, we show that this activity can be considered as a form of parabolic bursting, but with burst termination at a homoclinic bifurcation rather than as a SNIC bifurcation. We also investigate the parameter-dependence of these solutions and show that the proposed model yields a greater dynamic range of burst frequencies, durations, and duty cycles than those produced by other models in the literature.
Topological Data Analysis: Introduction and Applications
Presenters: Jenesis Escobar, Vishweshwar (Vishwa) Ramanakumar, and Max Zhang
Mentor: Peter Bubenik (UF, Mathematics)
In this talk, we give a brief introduction to Topological Data Analysis (TDA), discussing its derivation from algebraic topology as well as its applications to fields such as Deep Learning and Computer Vision. We first give a simple description of what TDA is, then introduce the notion of topological features before introducing simplicial complexes. We tie these concepts to persistence and its representations before discussing the subject’s applications to modern fields.