Undergraduate Mathematics Research Symposium@UF
University of Florida, Department of Mathematics
April 22, 2022
The second annual Undergraduate Mathematics Research Symposium (UMRS) at UF took place on April 22, 2022 in Little Hall 225.
It was organized by Konstantina Christodoulopoulou and Sara Pollock.
Schedule
Session 1
| Time | Presenter(s) | Presentation |
|---|---|---|
| 9:00–9:15 | Hugh Dennin | Principal specializations of Schubert polynomials for 1243-avoiding permutations |
| 9:20–9:35 | Xinyi (Emily) Zhang | Mathematical Assessment of the Impact Insecticide-Treated Nets on Malaria Dynamics |
| 9:40–9:55 | Jiun Cho | An Introduction to Topological Data Analysis |
| 10:00–10:15 | Alisha Bhatia Ben Sherwin Greeshma Avaradi | SCUDEM Problem B: Throw the Bike or Throw the Race |
Coffee Break
10:20–10:40
Session 2
| Time | Presenter(s) | Presentation |
|---|---|---|
| 10:40–10:55 | Elaine Danielson | Translation Surfaces from Hecke Eigenforms |
| 11:00–11:15 | Simon Kato | Extrapolated Restarted Arnoldi for Solving the PageRank Problem |
| 11:20–11:35 | Munir Bilal Ben Jemaa | Automorphisms of a Free Spectrahedron |
| 11:40–11:55 | Shane Gladson | An Agent-Based Model of Biting Midge Dynamics to Understand Bluetongue Outbreaks |
Lunch
12:00–12:40
Session 3
| Time | Presenter(s) | Presentation |
|---|---|---|
| 12:40–12:55 | Jake Shannin Alexandria Teter Xinyi Zhang | Bitcoin Daily Trading Strategy |
| 1:00–1:15 | Christopher Vairogs | Hilbert’s Theorem in Differential Geometry |
| 1:20–1:35 | Muhammad Abdulla | Modeling studies of calcium dynamics generating periodic sigh activity in respiratory networks |
| 1:40–1:55 | Nathan Wood | Bicycle Races: Modeling a Cyclist on a Predetermined Route |
| 2:00–2:15 | Cooper O’Kuhn | The Mondrian Puzzle |
Closing remarks
2:20–2:25
Konstantina Christodoulopoulou and Sara Pollock
Abstracts
Principal specializations of Schubert polynomials for 1243-avoiding permutations
Presenter: Hugh Dennin
Mentor: Zachary Hamaker (UF, Mathematics)
Schubert polynomials \(\{\mathfrak S_w\}\) are a family of polynomials indexed by permutations that form a basis for \(\mathbb Z[x_1,x_2,\dots]\). A Schubert polynomial’s principal specialization \(\mathfrak S_w(1,\dots,1)\) can be computed combinatorially as the number of bumpless pipe dreams \(|\text{BPD}(w)|\) corresponding to the permutation \(w\). In a recent paper, Yibo Gao conjectured a lower bound on \(\mathfrak S_w(1,\dots,1)\) in terms of the patterns contained in \(w\). We show that this bound holds for Schubert polynomials of \(1243\)-avoiding permutations \(w\) via compatible embeddings \(\text{BPD}(u)\hookrightarrow \text{BPD}(w)\) for each occurence of each pattern \(u\) contained in \(w\).
Mathematical Assessment of the Impact Insecticide-Treated Nets on Malaria Dynamics
Presenter: Xinyi (Emily) Zhang
Mentor: Calistus Ngonghala (UF, Mathematics)
Malaria is a life-threatening disease caused by Plasmodium parasites and transmitted from human-to-human by female Anopheles mosquitoes. Insecticide-treated nets (ITNs) have proven to be environmentally friendly, less costly, easy to use, and highly effective in preventing mosquitoes from biting humans, as well as killing mosquitoes that land on the nets. We develop and use a novel dynamic model to assess the impact of ITNs and human behavior. We will investigate the impact of using different biting rates for protected and unprotected humans. Human behavior will be introduced through the proportion of susceptible humans who choose to use ITNs properly. To analyze the model, we use the basic reproduction number and nonlinear dynamical systems to assess the existence and stability of equilibria. Well-formulated and parameterized models can provide useful insights to disease dynamics and are useful for public health decision-making.
An Introduction to Topological Data Analysis
Presenter: Jiun Cho
Mentor: Peter Bubenik (UF, Mathematics)
Data are often equipped with meaningful geometry that are difficult to study with traditional means. Topology is a field with a long tradition of studying geometric properties and have developed many tools to analyze them. In this talk I will introduce a subbranch of applied topology that explores the geometry of data called Topological Data Analysis. It is a promising and vibrant field active in both pure and applied mathematics. I will begin the talk by first briefly introduce some topological notions, then show how we can use these notions to derive geometric features from data. Lastly I will give examples of successful applications of Topological Data Analysis in various fields.
SCUDEM Problem B: Throw the Bike or Throw the Race
Presenters: Alisha Bhatia, Ben Sherwin, and Greeshma Avaradi
Mentor: Tracy Stepien (UF, Mathematics)
Our main goal was to come up with a mathematical model to determine the most optimal timing and position for carrying out a bike throw. A “bike throw” is a maneuver that bicycle racers use to help gain speed in the final moments of a race, when they stop pedaling and stretch their arms and legs out in front in a sudden push. Using an initial model of a bicycle-rider system and incorporating different factors that could influence the effectiveness of the bike throw, we came up with an analytical differential equations model. Fixing some basic values (such as mass, initial time, initial velocity, final velocity, etc.), we used the model to determine the best time to carry out the bike throw relative to the start of the race, the optimal time interval of the throw, the maximum amount of time that can be made up, and the best position of the bicyclist.
Translation Surfaces from Hecke Eigenforms
Presenter: Elaine Danielson
Mentor: Paul Apisa (University of Michigan)
A translation surface can be represented both as a collection of polygons with sides identified and as a Riemann surface with a holomorphic one-form. Examples of the latter representation can be obtained by equipping a modular curve for the congruence subgroup \(\Gamma_0(N)\) with a weight two Hecke eigenform. These translation surfaces in particular are of interest because we suspect that some of these surfaces have novel \(\mathrm{GL}_2(\mathbb{R})^+\) orbit closures. However, these orbit closures are computed from the polygonal representation of the translation surface, which requires building a flat atlas from the eigenform as an intermediate step. We discuss our work towards an algorithm in SageMath to produce the flat atlas for a given eigenform.
Extrapolated Restarted Arnoldi for Solving the PageRank Problem
Presenter: Simon Kato
Mentor: Sara Pollock (UF, Mathematics)
This project investigates how extrapolation of the Arnoldi algorithm can accelerate the computation of the dominant eigenvector of the PageRank algorithm. The PageRank algorithm has famously been used by Google to rank web pages by assigning a reputation which is based on the quantity and quality of links to said web page. The underlying process in ranking the web pages is solving an eigenvalue problem with a Markov matrix. The focus will be on tuning the key parameter for an extrapolated version of the Restarted Arnoldi algorithm. The PageRank problem provides knowledge of the two dominant eigenvalues and much is known about the behavior of the PageRank convergence. The knowledge will be used to find an extrapolation parameter which works well in practice for web matrices, a subset of the stochastic class of matrices.
Automorphisms of a Free Spectrahedron
Presenter: Munir Bilal Ben Jemaa
Mentor: Scott Mccullough (UF, Mathematics)
The purpose of this project was to investigate the automorphisms of what is arguably the simplest example of a free spectrahedron whose automorphisms have not yet been classified. A spectrahedron is the scalar solution set of a Linear Matrix Inequality (LMI), generalizing the notion of a polytope from linear programming. They are the basis for semidefinite programming (SDP) within convex optimization. Free spectrahedra are their matricial analogs, and have more structure and are hence more tractable. Free spectrahedra have connections to quantum information theory and systems engineering, and the study of their automorphisms has close ties to noncommutative algebra.
An Agent-Based Model of Biting Midge Dynamics to Understand Bluetongue Outbreaks
Presenter: Shane Gladson
Mentor: Tracy Stepien (UF, Mathematics)
Bluetongue (BT) is a well-known vector-borne disease that infects ruminants such as sheep, cattle, and deer with high mortality rates. Recent outbreaks in Europe highlight the importance of understanding vector-host dynamics and potential courses of action to mitigate the damage that can done by BT. We present an agent-based model (ABM), entitled MidgePy, that considers the actions of individual Culicoides spp. biting midges and attempts to understand their role as vectors in BT outbreaks. Sensitivity analysis is performed and results indicate that midge survival rate has a significant impact on the probability of a BTV outbreak as well as its severity. This suggests that future methods to control BT spread could combine large-scale vaccination programs with biting midge population control measures such as the use of pesticides.
Bitcoin Daily Trading Strategy
Presenters: Jake Shannin, Alexandria Teter, and Xinyi (Emily) Zhang
Mentor: Tracy Stepien (UF, Mathematics)
Humans have used gold as a store of value for millennia, whereas Bitcoin (BTC) served as a currency for the first time on 22 May 2010 when Laszlo Hanyecz bought two pizzas for 10,000 BTC. We found that gold’s maturity and bitcoin’s immaturity translated to their market behavior. Our best predictions, in terms of mean absolute percentage error (MAPE), for Bitcoin price came from a short-sighted ARIMA(1,1,1) [Autoregressive(1 day back) integrated(1 day back) moving average(1 day back)] model. Predictions for gold preferred the less volatile ARIMA(4,1,0) model. We preferred ARIMA models to black-box alternatives, such as LSTM (Long short-term memory) models, since our chosen trading strategy depends on the ARIMA models’ 30-day forecasts and standard errors rather than just the predictions themselves. While we found bitcoin an asset worthy of investment, its notorious volatility merits a risk-aware trading strategy. We balance performance and risk level in selecting an agent-based approach, in which our portfolio is divided among 49 algorithmic agents evenly each day. Agents measure risk in terms of our predictions’ standard errors, so when an asset becomes more volatile, all but the most aggressive agents treat it more cautiously. Trading fees initially stiffened agents’ willingness to make trades, so we allowed agents to use the ARIMA models’ 30-day forecasts to justify the fees. Our risk-aware strategy paid off, building a starting portfolio of $1000 up to a final portfolio of net worth $63,147 while never falling below $929. However, results were sensitive to changes in fees.
Hilbert’s Theorem in Differential Geometry
Presenter: Christopher Vairogs
Mentor: Luca Di Cerbo (UF, Mathematics)
The Gaussian curvature of a surface is an intrinsic geometric quantity that corresponds the intuitive notion of how a surface curves at a point. Many key results in the theory of surfaces are concerned with the existence or nonexistence of surfaces with certain restrictions on their Gaussian curvature. An important result in this area is Hilbert’s theorem. Hilbert’s theorem states that there does not exist a complete surface of constant negative Gaussian curvature that may be immersed in \(\mathbb{R}^3\). In this talk, we first review basic concepts from differential geometry, including the first and second fundamental forms, geodesics and the exponential map, and immersions. We then sketch a proof of Hilbert’s theorem using the concepts of asymptotic curves and Chebyshev nets.
Modeling studies of calcium dynamics generating periodic sigh activity in respiratory networks
Presenter: Muhammad Abdulla
Mentor: Dr. Jeffrey C. Smith, Senior Investigator, National Institute of Neurological Disorders and Stroke
The pre-Bötzinger Complex (preBötC) is a region of the brainstem that plays a vital role in the generation of mammalian respiratory rhythms. Neurons in the preBötC with pacemaker-like properties undergo bursting behavior, ultimately inducing periodic spikes of synchronized activity across the circuit. One such respiratory rhythm is sighing, an augmented breath associated with a lower frequency and greater neural activity than standard breathing. While sighing is a fundamental mechanism in controlling breath-variability, preventing atelectasis, and prompting cortical arousal, its exact biophysical mechanism is not fully understood. In this work, we develop a dynamical systems model of a network of respiratory neurons. By incorporating a framework for the build-up of endoplasmic reticulum calcium stores, and subsequent calcium-induced-calcium-release (CICR), the proposed model simulates observed physiological behaviors associated with sighing activity. Moreover, we also investigate the implications of this model with respect to the biophysical pathways that generate network sighs.
Bicycle Races: Modeling a Cyclist on a Predetermined Route
Presenter: Nathan Wood
Mentor: Tracy Stepien (UF, Mathematics)
The 2022 Consortium of Mathematics and its Applications (COMAP) Competition Problem A had outlined creating a mathematical means of modeling the power exerted by a time trial cyclist. The model must not only specify two different cyclist styles, but must also account for variations in weather and elevations on a predetermined course, specifically the 2021 Olympic Time Trial in Tokyo, the 2020 UCI World Championship in Flanders, Belgium, and a custom made course. To accomplish this, rectilinear kinematics was used to model bicyclist force, wind speed, changes in friction due to rain, as well as changes in accelerations to safely turn. Differences in cyclist force when exerting uphill and relaxing downhill was also accounted for when modeling the necessary force to maintain a constant velocity. While the implemented model can successfully explain changes in force when exerting uphill, when cycling in rain, and when turning, there is less confidence in the models ability to model exertion by the cyclist on negligible inclines and difficulty in verifying the model.
The Mondrian Puzzle
Presenter: Cooper O’Kuhn
Mentor: Krishnaswami Alladi (UF, Mathematics)
We provide a conditional result which shows conjecturally that the quantity in the Mondrian Puzzle should never equal zero. We do so by number theoretic means, in contrast to the latent geometric nature of the problem.
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!