Virtual Math Research Circle — 2026 Spring Research Projects

Virtual Math Research Circle — 2026 Spring Research Projects

Overview

Virtual Math Research Circle brings motivated high-school students together with university mentors to pursue authentic, publishable mathematics. Below you’ll find logistics for the current cycle and the complete set of research proposals.

How the Program Works Public Archives

Cost & Registration

Tuition: $1,200 per session. A $300 deposit is due at the time of registration.

Status: Registration is now open for the 2026 VMRC sessions. Please use the link below to submit your application.

Register FAQ & Contact

College Credit

Students who are invited to the optional research extension and who satisfactorily complete the final extension presentation may earn one hour of LSU credit (MATH 1999), subject to VMRC approvals and standard registrar processes.

Research Topic Offerings (Preliminary)

  • Topics listed for each session are preliminary and may change as mentors and enrollments finalize.
  • During registration you will rank your top 3–6 topics.
  • Minimum enrollment: a project runs with at least three students. If a project has two or fewer registrants, we will place you into your next ranked choice.
  • VMRC does not publish live enrollment numbers.
  • We prioritize your rankings, balance enrollments across projects, and consider scheduling; placement is not guaranteed.
  • Final topic lists are posted about four weeks before each session.

Session 1 · Jan 12, 2026 – Feb 21, 2026

Extensions of Colley's Matrix and Ranking Methods

Research Proposal — Virtual Math Research Circle

Session
Session 1: Jan 12, 2026 – Feb 21, 2026
Mentor
Jonathan Engle
PhD Student
Department of Mathematics
Florida State University

Project Title
Extensions of Colley's Matrix and Ranking Methods
Topic Area
Applied Linear Algebra

Background
This research project requires only basic algebra and arithmetic. All necessary linear algebra concepts will be taught by the instructor. This project serves as an excellent introduction to linear algebra, introductory computer science, statistics, or modeling courses. During the computational phase, we will use tools such as Microsoft Excel, Matlab, and Julia. For the final presentation, students will use PowerPoint and/or Beamer.

Abstract
Colley's matrix method for ranking college football teams is an exciting application of linear algebra. Traditional ranking methods rely on team records, margin of victory, or subjective committee decisions, which may not always produce accurate rankings. Colley's method improves upon this by adjusting team rankings based on strength of schedule without considering conference bias or margin of victory.

Over three weeks, we will replicate Colley's Matrix for a small example using recent college football data. We will analyze these rankings to determine whether Colley's method predicted upsets that the College Football Committee overlooked. This implementation requires solving a linear system:

C * r = b, where C is Colley's matrix, r is the ranking vector, and b is one plus the average win rate for a team. This leads to the research question: how can Colley's Matrix be improved with additional information?

Possible Extension
Beyond sports, students will explore applications of ranking methods in other fields, such as resource distribution, natural disaster response, and stock rankings.

Outline/Timeline
Weeks 1-2: Introduction to traditional ranking methods and how linearization can improve rankings. Basic concepts of matrices, vectors, matrix operations, and inverses. Literature review on Colley's method.

Weeks 3-4: Computational experiments: implementing Colley's matrix in Matlab, Julia, or another preferred coding language. Data mining and extracting elements for Colley's matrix. Comparing Colley rankings with College Football Committee rankings and actual results. Students will modify Colley's method to include additional parameters and compare their results.

Weeks 5-6: Refining the modified ranking methods, analyzing differences, and preparing final results. Constructing and practicing presentations using PowerPoint, Google Slides, or Beamer.

References
  1. Boginski, V., Butenko, S., & Pardalos, P. M. (2004). Matrix-based methods for sports rankings: A survey.
  2. Colley, W. N. (2002). Colley's Bias-Free College Football Ranking Method.

A Mathematical Model to Mitigate the Spread of Environmentally Transmissible Diseases

Research Proposal — Virtual Math Research Circle

Session
Session 1: Jan 12, 2026 – Feb 21, 2026
Mentor
Hemaho B. Taboe
Ph.D. candidate in Mathematical Biology
Department of Mathematics
University of Florida

Project Title
A Mathematical Model to Mitigate the Spread of Environmentally Transmissible Diseases
Topic Area
Mathematical Biology; Epidemiology

Background
High-school calculus and first-order differential equations will be useful for this project. These concepts will be taught or reviewed at the beginning, so strong interest and readiness to learn are more important than prior coursework.

Abstract
Mathematical models provide guidance on controlling communicable diseases. In recent years, many epidemic models have focused on COVID-19, often overlooking transmission from contaminated surfaces (the “environment”).

In this project, we will develop a basic compartmental model with logistic recruitment into the susceptible (uninfected) class that explicitly incorporates transmission from the environment to humans. As a case study, we will consider Lassa fever, a well-known environmentally transmissible disease, using data from one of the mentor’s prior publications. Certain parameters will be drawn from the literature, while the remaining parameters will be estimated by calibrating the model to weekly confirmed case data.

We will then analyze several control strategies—such as surface disinfection, mask wearing, and treatment of infectious individuals—to identify scenarios in which the overall burden of disease can be significantly reduced.

Possible Extension
A natural extension of the model is to include a class of individuals who have recovered from infection but lose immunity after a finite “immunity period” and can be reinfected. We may also extend the framework to study the impact of vaccination in the presence of environmental contamination.

Depending on group progress and interest, students may help formulate and analyze these extended models.

Outline/Timeline
Weeks 1-2: Meet with students to introduce the project and provide background materials. Review calculus and ordinary differential equations as needed. Give tutorials on using MATLAB (or a similar software package) to solve ODEs and on Overleaf for writing the research report and presentation. Students attend sessions, complete short assignments, ask questions as needed, and keep in regular contact by email.

Weeks 3-4: Develop a compartmental model for environmentally transmissible diseases, motivated by Lassa fever epidemiology. Together, we polish the model proposed by the students. We introduce the basic reproduction number and herd immunity, and discuss how to estimate parameters from the literature and data. Students formulate a first full version of the model, derive key formulas, collect data, and begin estimating parameters.

Weeks 5-6: Use numerical simulations to address the main research questions. Students run and interpret simulations, refine model scenarios, and work on the manuscript in Overleaf. The group reviews and revises the draft, preparing it for final presentation and possible submission.

References
  1. Taboe, H. B., Pilyugin, S. S., & Ngonghala, C. N. (2024). “Resolve Lassa Fever Persistence: A Compartmental Model with Environmental Virus–Host–Vector Interaction.” https://doi.org/10.21203/rs.3.rs-5248593/v1
  2. Nagle, R. K., et al. Fundamentals of Differential Equations and Boundary Value Problems. New York: Addison-Wesley, 1996.
  3. Ngonghala, C. N., Taboe, H. B., Safdar, S., & Gumel, A. B. (2023). “Unraveling the dynamics of the Omicron and Delta variants of the 2019 coronavirus in the presence of vaccination, mask usage, and antiviral treatment.” Applied Mathematical Modelling, 114, 447–465. Article link

Graphs of Commutative Rings

Research Proposal — Virtual Math Research Circle

Session
Session 1: Jan 12, 2026 – Feb 21, 2026
Mentor
James Branca
Ph.D. Student in Biomathematics
Department of Mathematics
Florida State University

Project Title
Graphs of Commutative Rings
Topic Area
Algebra, Graph Theory

Background
Knowledge of calculus is not required for this project. Some familiarity with algebra and mathematical notation is helpful, but all necessary concepts will be taught during the first week.

Abstract
Algebra is a very rich subject, and one large area of interest is the study of commutative rings. In such a ring, it can happen that the product of two nonzero elements is zero; elements that do this are called zero-divisors.

A recent application of commutative rings within graph theory is the construction of graphs whose vertices come from elements (or special subsets) of the ring and where adjacency is determined by a zero-divisor relationship. These constructions link algebraic properties of the ring to structural properties of the resulting graph.

In this project, students will study graphs associated with commutative rings, explore examples, prove basic results from the literature, and apply computation to build and analyze specific zero-divisor graphs.

Possible Extension
A possible extension of this research is to apply these ideas to other families of rings, or to related constructions such as graphs built from ideals or other special elements. Depending on interest and time, students may explore variations that appear in current research papers and compare how different graph constructions capture the algebraic structure of a ring.

Outline/Timeline
Weeks 1-2: Introduction to rings and basic ring theory. We will review operations in rings, standard examples, and key terminology. All new terms will be thoroughly explained and illustrated with many examples.

Weeks 3-4: The concept of the zero-divisor and the zero-divisor graph. We will introduce standard definitions, look at important examples, and read portions of the research literature, discussing how algebraic conditions show up in the associated graphs.

Weeks 5-6: Computing graphs by hand for small or familiar rings (such as \(\mathbb{Z}_n\)) and, if time permits, beginning to create or adapt an algorithm to generate zero-divisor graphs more systematically. Students will analyze the resulting graphs and relate their observations back to the ring structure.

References
  1. Beck, I. Coloring of commutative rings, J. Algebra 116 (1988), 208–226.
  2. Anderson, D., Axtell, M., & Stickles, J. Zero-divisor graphs in commutative rings, in Commutative Algebra, Springer.

Lattice paths under a line: from ballot numbers to Catalan and beyond

Research Proposal — Virtual Math Research Circle

Session
Session 1: Jan 12, 2026 – Feb 21, 2026
Mentor
Fernando Heidercheidt
PhD Student
Department of Mathematics
Louisiana State University

Project Title
Lattice paths under a line: from ballot numbers to Catalan and beyond
Topic Area
Enumerative Combinatorics

Background
This project is aimed at motivated high-school students who know basic combinatorics (binomial coefficients), simple counting arguments, and elementary algebra. No prior knowledge of Catalan numbers is required, these will be developed during the project. The work mixes rigorous counting, short proofs, and computational experimentation (Python on Google Colab), so students will see how theory and experiment complement each other. The investigation leads naturally to algorithmic implementations and conjecture-making.

Abstract
Consider lattice paths on the integer grid starting at \((0,0)\) and ending at \((a,b)\), where \(a>0\), \(b>0\), consisting only of steps \(E=(1,0)\) and \(N=(0,1)\). The classical total number of such paths is just some binomial term depending on \(a\) and \(b\). We restrict attention to those paths that stay at or below a prescribed function \(y=f(x)\); in the simplest linear case the boundary is a straight line.

When the boundary is the identity line \(y=x\) and \(a=b=n\), the number of paths from \((0,0)\) to \((n,n)\) that never rise above the diagonal is the Catalan number \(C_n\). More generally, if the function is just the diagonal line from \((0,0)\) to \((a,b)\) the number of paths from \((0,0)\) to \((a,b)\) that never cross the diagonal is the Catalan rational number \(C_{a,b}\).

The primary goal of the project is to guide students to derive these combinatorial formulas (using simple bijection constructions between finite sets), verify them computationally, and then attempt to generalize to other slopes/lines. The final challenge is to formulate and test conjectured formulas or asymptotics for these generalized counting problems.

Possible Extension
After mastering the linear case, students may explore several natural extensions:
  • Study boundaries given by simple polynomials \(y = p(x)\) and generalized nonnegative functions.
  • Investigate random versions: sample uniformly from all paths to \((a,b)\) and estimate the probability a random path stays below the given boundary; compare experimental frequencies to exact formulas.
  • Explore bijective proofs and connections to Young diagrams, plane partitions, or Dyck-path–like encodings when appropriate.

Outline/Timeline
The schedule below fits a three-week mini-research program. It is flexible: students who progress quickly may move to extensions earlier.

Weeks 1-2 (Foundations & Examples).
• Introduce lattice paths, binomial coefficients, and the basic enumeration.
• Work small concrete examples by hand and visualize allowed/disallowed paths.
• Introduce the diagonal restriction \(y=x\) and derive the Catalan formula for \(a=b\) using standard counting arguments.

Weeks 3-4 (General linear case).
• Translate the problem to other rational slopes via coordinate changes or lattice scalings and attempt to obtain analogous formulas.
• Students write Python code (in Google Colab) to enumerate paths and confirm formulas for small \(a,b\).

Weeks 5-6 (Generalizations, conjectures, and presentation).
• Formulate conjectures about exact formulas and recurrences. Attempt short proofs or partial results.
• Prepare a final presentation (slides or short paper) that explains the main results, includes computational evidence, and proposes next steps.

References
  1. R. P. Stanley, Catalan Numbers, Cambridge University Press, 2015.
  2. R. P. Stanley, Enumerative Combinatorics, Vol. 1, 2nd ed., 2011.

Exploring RSA Cryptography: How Numbers Keep Secrets

Research Proposal — Virtual Math Research Circle

Session
Session 1 & 2: Jan 2026 – Feb 2026
Mentor
Boluwatife Aderinto
3rd year PhD student
Department of Mathematics
Florida State University

Project Title
Exploring RSA Cryptography: How Numbers Keep Secrets
Topic Area
Number Theory, Cryptography

Background
This research project requires students to be comfortable with basic algebra and arithmetic operations. No prior knowledge of cryptography or programming is necessary. All required cryptographic concepts, modular arithmetic, and elementary number theory topics such as prime numbers, factorization, and exponents will be introduced and thoroughly explained. This project provides an exciting introduction to the fields of number theory and cryptography.

During the research, we will utilize computational tools, mainly Python. For the final presentation, students will learn and use tools such as PowerPoint or LaTeX with Beamer.

Abstract
Have you ever wondered how secret messages and confidential information (such as Social Security numbers, credit card details, and other financial data) remain hidden from hackers, ensuring online security and privacy? The answer lies in the fascinating world of cryptography, and one of the most important cryptographic methods is RSA encryption.

RSA cryptography is one of the most widely used encryption schemes for securing digital information. At its core, RSA leverages simple but powerful mathematical concepts involving prime numbers, modular arithmetic, and factoring large integers. But how exactly does RSA keep our digital secrets safe?

In this research project, students will uncover the mathematical foundations behind RSA cryptography. We will explore the mechanics of encryption and decryption using RSA, including the ways prime numbers serve as “guardians of secrecy.” Students will actively experiment with encrypting and decrypting messages, learning firsthand how mathematical principles protect privacy.

Additionally, we will investigate the security strength of RSA by discussing potential vulnerabilities related to factorization techniques. Through these activities, students will not only grasp essential cryptographic concepts but also strengthen their skills in mathematical reasoning, problem-solving, and computational experimentation.

Possible Extension
A natural extension of this research would involve investigating the impact of quantum computing on RSA cryptography. Specifically, students can explore Shor’s algorithm, a quantum algorithm that efficiently factors large integers in polynomial time and poses a serious threat to the long-term security of RSA.

Students will have the opportunity to assess and discuss modern cryptographic approaches that are believed to withstand quantum-based attacks, positioning their work at the intersection of classical cryptography and quantum computing.

Outline/Timeline
This is a general (tentative) outline of how the research project will progress. We will adjust the schedule as necessary throughout the 3-week period.

Weeks 1-2 (Foundations & Basics).
• Review algebraic operations, modular arithmetic, prime numbers, factorization, and exponents.
• Introduce basic encryption/decryption ideas (for example, classical ciphers such as the Caesar and Vigenère ciphers). Students will practice with simple examples and numerical exercises.
• Develop the basics of mathematical reasoning and proof writing through examples and guided exercises. Computational tools (Python) will be introduced for practical demonstrations.

Weeks 3-4 (RSA Encryption & Experimentation).
• Give a detailed introduction to RSA encryption and decryption algorithms, ensuring a thorough understanding of how RSA works.
• Students learn RSA key generation, encryption, and decryption processes in detail and carry out practical computational experiments to encrypt and decrypt messages, verifying the principles learned.
• Analyze encryption strength by exploring security implications of prime-number sizes and common methods for attempting to break RSA, emphasizing the difficulty of factoring large integers, and discussing empirical results.

Weeks 5-6 (Finalizing Results & Presentation).
• Conclude experiments and refine understanding, finalizing mathematical explanations and computational experiments, and summarizing findings.
• Prepare the final presentation. Students will decide collectively whether to use PowerPoint or a LaTeX-based Beamer presentation, depending on their comfort level and preferences. We will also practice presenting, ensuring clarity and confidence in the final talk.

References
  1. W. Stallings, Cryptography and Network Security: Principles and Practice, 7th ed., Pearson Education, 2016.
  2. T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms, 3rd ed., MIT Press, 2009.
  3. N. D. Koblitz, A Course in Number Theory and Cryptography, 2nd ed., Springer-Verlag, 1994.
  4. D. Boneh, V. Shoup, A Graduate Course in Applied Cryptography, Stanford University, 2020. (Online text available at https://toc.cryptobook.us/.)

Numerical Shape Optimization

Research Proposal — Virtual Math Research Circle

Session
Session 1 & 2: Jan 2026 – Feb 2026
Mentor
Alvis Donghan Zahl
PhD Candidate (ABD)
Department of Mathematics
Rutgers University

Project Title
Numerical Shape Optimization
Topic Area
Numerical analysis, shape optimization, free boundary problems

Background
This research project requires only basic algebra and familiarity with high-school geometry. No prior experience with calculus, partial differential equations, or numerical analysis is required; all analytical and computational tools will be introduced during the program. The project connects geometry, optimization, and introductory numerical methods.

Students will be introduced to numerical experimentation using accessible tools such as Python, Matlab, or Mathematica. For the final presentation they may use PowerPoint or Beamer. Throughout the project, emphasis will be placed on exploring examples, formulating conjectures from observed patterns, and writing concise explanations.

Abstract
One of the oldest questions in mathematics is the isoperimetric problem: among all shapes with a given area, which one has the smallest perimeter? Modern shape optimization generalizes this question to more complicated functionals, often defined by solutions of partial differential equations, and leads naturally to free boundary problems, where the optimal boundary is part of the unknown.

This project will explore the classical isoperimetric problem using discrete and numerical methods and then apply these ideas to a model Bernoulli free-boundary problem, where we seek a domain that minimizes an energy involving solutions of the Laplace equation. Students will design and run numerical experiments that evolve shapes toward optimal configurations, track how boundaries move, and build intuition for how curvature, perimeter, and energy interact.

By the end of the project, students will have developed computational experiments illustrating both isoperimetric and Bernoulli-type free-boundary behavior, and they will see how these questions play an important role in modern applied and theoretical mathematics.

Possible Extension
Students who continue in a Research Extension may:
  • Compare several discrete energies (perimeter, a simple curvature proxy, and discrete Dirichlet energy) and study how each choice changes the “optimal’’ shapes.
  • Investigate stability and time-step constraints for a parabolic Bernoulli scheme, relating numerical behavior to intuitive ideas of stability.
  • Introduce randomness (noisy sources or random initial data) and observe how free boundaries fluctuate, then attempt to quantify these fluctuations numerically.
  • Design higher-resolution experiments that approximate continuous isoperimetric shapes and prepare a polished poster or expository paper suitable for undergraduate-friendly conferences or outreach events.
These extensions deepen both the numerical and conceptual understanding of shape optimization and free-boundary evolution.

Outline/Timeline
The project is designed for a 6-week Spring VMRC session with two meetings per week. We organize the work into three two-week phases combining guided instruction, experimentation, and writing.

Week 1 — Background & Free Boundaries.
• Introduction to free boundaries: what they are, where they arise (e.g., melting/solidification, fluid flow, electrostatics), and basic examples.
• Overview of the isoperimetric problem and simple perimeter/area comparisons in the plane.
• Students become familiar with the overall goals of shape optimization and with the software tools we will use.

Week 2 — Numerical Tools & the Isoperimetric Problem.
• Introduction to simple numerical methods (finite-difference approximations, discrete gradients, and gradient-descent–type updates).
• Implement basic schemes that evolve planar sets toward perimeter-minimizing shapes and observe how shapes evolve toward circles.

Weeks 3–4 — Implementation & Experiments.
• Students implement numerical routines in Python/Matlab/Mathematica to simulate gradient flows for different initial domains (ellipses, rectangles, noisy shapes, etc.).
• Run systematic experiments, record when and how circular shapes emerge, and compare rates of convergence and possible numerical instabilities.

Week 5 — Bernoulli Free-Boundary Problem.
• Apply the numerical methods to an elliptic Bernoulli free-boundary problem driven by a solution of the Laplace equation.
• Compare the resulting shapes with those from pure perimeter flows, highlighting similarities, differences, and the effect of the energy condition on the optimal boundary.

Week 6 — Write-Up & Presentation.
• Students finalize computations, summarize observations, and refine figures and plots.
• Prepare a written summary and a short Beamer/PowerPoint presentation outlining their methods, results, and conjectures; practice delivering the final talk.

References
  1. F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2012.
  2. J. Crank, Free and Moving Boundary Problems, Oxford University Press, 1984.

Using Minimum Spanning Trees to Approximate the Traveling Salesman Problem

Research Proposal — Virtual Math Research Circle

Session
Session 1: Jan 12, 2026 – Feb 21, 2026
Mentor
Mae Lee
Graduate Teaching Assistant, Bridge-to-Doctorate Affiliate graduate student
Department of Mathematics
University of Texas at Arlington

Project Title
Using Minimum Spanning Trees to Approximate the Traveling Salesman Problem
Topic Area
Graph Theory and Algorithms

Background
Students should be comfortable with basic algebra and arithmetic-style reasoning. Fundamentals of proof writing, graph theory, and algorithmic thinking will be introduced by the instructor during the program. No prior exposure to graph theory or algorithms is assumed.

Abstract
Imagine you want to visit five different cities and then return home while traveling as little as possible. How do you decide which route is best? This simple question is a small example of the famous Traveling Salesman Problem (TSP).

In this project, students will study how a simpler problem—the minimum spanning tree (MST)—can be used to build good approximate solutions to the TSP on weighted graphs. We will ask questions such as: How is an MST constructed? How can we turn an MST into a tour that visits every vertex exactly once? And how close can such tours be to the true optimal TSP tour?

Depending on interest and background, students may also implement MST-based approximation algorithms in Python code and present their findings in a research-style talk.

Possible Extension
Students may work toward formulating and proving general performance guarantees for their MST-based algorithms (for example, showing that the tour they construct is never more than a fixed factor longer than the optimal TSP tour). They may also explore other heuristic or approximation algorithms and compare their behavior on larger networks through computational simulations.

Outline/Timeline
This is a tentative schedule for the 6-week session that will be adapted as needed as the session progresses.

Week 1 (Graph Theory basics).
• Introduction to basic graph theory, the Traveling Salesman Problem, and minimum spanning trees; students experiment with simple greedy algorithms on small graphs.

Week 2 (MST Algorithms).
• Finding MSTs using Prim’s and Kruskal’s algorithms. We discuss why these algorithms always produce a minimum-cost spanning tree and why the MST provides a natural lower bound on TSP cost.

Week 3 (Bridging MSTs and TSPs).
• Developing an MST-based TSP algorithm. Students learn how to turn an MST into a TSP tour (via doubling edges, Eulerian walks, and shortcutting).

Week 4 (Enumerating examples).
• Applying the algorithm on small instances. Students generate small weighted graphs, run their algorithm, and compute optimal TSP tours by enumeration for comparison.

Week 5 (Data Analysis).
• Analyzing results. Students look for patterns and trends in their approximation ratios, generate plots and tables, and, if appropriate, carry out additional computational experiments on random or structured families of graphs.

Week 6 (Prepare presentation).
• Students organize their results, refine proofs and explanations, and prepare a final Beamer or PowerPoint presentation summarizing their methods and findings.

References
  1. Sariel Har-Peled, Chapter 12: Approximation Algorithms for the Traveling Salesman Problem, lecture notes for Algorithms, University of Illinois at Urbana–Champaign, 2023.

Machine Learning and How to Predict Everything

Research Proposal — Virtual Math Research Circle

Session
Session 1: Jan 12, 2026 – Feb 21, 2026
Mentor
Han Nguyen
PhD Student
Department of Mathematics
Louisiana State University

Project Title
Machine Learning and How to Predict Everything
Topic Area
Machine Learning, Data Science, and Basic Probability Theory

Background
Required: Basic arithmetic and algebra. Optional: Coding experience is not required but is helpful. Basic linear algebra will be taught during the program. This project will serve as a friendly introduction to data analysis and machine learning for motivated high-school students.

Abstract
As humans we use past experiences to determine our future choices and decisions, but how a computer does something similar is not always clear. Machine learning uses different algorithms to learn patterns from past data and then make predictions, classifications, and statistical assessments in many fields.

In this project, students will be introduced to a variety of basic machine learning models, such as k-nearest neighbors, linear regression, and decision trees. They will see how these models are trained on data, what their strengths and weaknesses are, and how to interpret their outputs. The project will also touch on the neural network model, which has grown popular in recent times. Throughout, students will work with example datasets and discuss which models are best suited to which kinds of problems.

Possible Extension
This project can extend into different types of data (for example, images, text, or time series) and into more advanced methods. Students who continue may study more sophisticated neural network architectures, experiment with tuning hyperparameters, and explore ways of augmenting a standard neural network for improved performance on particular datasets.

Outline/Timeline
This outline is designed for a 6-week VMRC session and may be adjusted as needed based on student progress and interests.

Week 1 (Learning about Basic Machine Learning). Students will learn about several foundational machine learning models (such as k-nearest neighbors, linear regression, and decision trees) and how they are used in practice. They will experiment with basic tools such as WEKA to run simple models on example datasets.

Week 2 (More Machine Learning with Emphasis on Neural Networks). We will finish any remaining introductory material and then focus on the basics of neural networks. Students will see how a simple neural network is structured and trained, and they will also begin learning the basics of LaTeX for writing up their work.

Weeks 3–5 (Tackling the Problem). Each week, students will work with a different dataset. For each dataset they will choose one or more models, train and evaluate them, and compare performance. At the end of each week, students will produce a short write-up, using LaTeX, summarizing their methods, results, and observations.

Week 6 (Synthesis). In the final week, students will combine the analyses from the previous weeks into a coherent final presentation. They will refine their LaTeX or slide decks, practice presenting, and deliver a polished talk explaining what they learned about machine learning and model selection.

References
  1. G. James, D. Witten, T. Hastie, and R. Tibshirani, An Introduction to Statistical Learning with Applications in Python, 1st Edition, Springer, 2023.
  2. C. Bishop, Pattern Recognition and Machine Learning, 1st Edition, Springer, 2007.

Predicting Future Events Using Markov Chains and Machine Learning

Research Proposal — Virtual Math Research Circle

Session
Session 1: Jan 12, 2026 – Feb 21, 2026
Mentor
Jacob Kapita
PhD Student
Department of Mathematics
Louisiana State University

Project Title
Predicting Future Events Using Markov Chains and Machine Learning
Topic Area
Stochastic Probability Theory, Machine Learning, Data Science

Background
Basic arithmetic and algebra skills are sufficient. Coding is not required, but it may be helpful. Necessary skills will be taught as the project progresses. This project serves as an excellent introduction to probability theory, stochastic analysis, data analysis, and machine learning. We will use Python in Google Colab and create the final presentation using LaTeX Beamer in Overleaf.

Abstract
Modeling real-life events is essential across many disciplines, especially when predicting the likelihood of future random outcomes. Markov Chains offer a powerful approximation tool because future states depend only on the present state, not on past events—making the process memory-less. In this project, we focus on discrete-time Markov Chains, using next-word prediction as our primary example. We will develop machine learning models grounded in the Markov property, where transition probabilities between words estimate the most likely next word. This demonstrates how Markovian behavior applies to real-world predictive tasks. For the final presentation, students will choose an application area in which to apply the concepts learned.

Possible Extension
The project may be extended to continuous-time Markov Chains, particularly in life insurance (actuarial mathematics). We may also explore Markov Chain inference, such as determining the probability of being in a particular state after a number of steps.

Outline/Timeline
A bi-weekly outline is provided below; adjustments may be made as necessary.

Weeks 1–2 (Background):
We will begin with the fundamentals of probability theory, focusing on probability laws, independence, and conditional probability. After building this foundation, we will introduce Markov Chains and the Markov property. These weeks will conclude with an exploration of matrix operations—particularly matrix multiplication—followed by the construction and analysis of transition matrices and introductory examples of Markov Chains.

Weeks 3–4 (Markov Chains in Machine Learning):
Students will be introduced to Python using Google Colab. We will then discuss key machine learning concepts and highlight how Markov Chain principles are applied in predictive modeling. Using real-world text data, students will build a next-word prediction model based on transition probabilities.

Weeks 5–6 (Finalizing Results and Presentation):
Students will learn to create professional slides using Beamer in Overleaf. During this period, we will finalize the results, refine visualizations, and assemble the final presentation for the research showcase.

References
  1. Fewster, Rachel. Chapter 8: Markov Chains, Lecture Notes for Stats 325, Stochastic Processes, 2014. Available at https://www.stat.auckland.ac.nz/~fewster/325/notes/ch8.pdf
  2. Patel, Vatsal. Markov Chain Explained, Built In, 2022. Available at https://builtin.com/machine-learning/markov-chain
  3. Verma, Yugesh. A Guide to Markov Chain and its Applications in Machine Learning, 2021. Available at https://analyticsindiamag.com/deep-tech/a-guide-to-markov-chain-and-its-applications-in-machine-learning/

Survey Weighting and Nonresponse Adjustment with GSS Data

Research Proposal — Virtual Math Research Circle

Session
Session 1: Jan 12, 2026 – Feb 21, 2026
Mentor
Isaac B. Michael, PhD
Gordon A. Cain Center for STEM Literacy
Louisiana State University

Project Title
Survey Weighting and Nonresponse Adjustment with GSS Data
Topic Area
Statistics / Data Science

Background
This project introduces survey methodology and nonresponse adjustment using social science data. The General Social Survey (GSS), conducted by NORC at the University of Chicago, is one of the most widely used surveys for studying U.S. public opinion and social trends.

Students will use a subset of the GSS with weights removed. They will reconstruct weights using Census demographic margins, apply decision tree models for nonresponse, and generate final weighted demographic reports. The project provides an introduction to applied statistics, social science research, and machine learning.

Abstract
Unweighted survey results can be misleading if the sample is not representative of the population. For example, if younger respondents are underrepresented, the survey may overestimate traditional views on social issues. By constructing and applying weights, statisticians correct for these imbalances.

In this project, students will analyze a subset of GSS data, learning how to adjust for nonresponse and produce representative estimates. They will use decision trees to model response patterns, reconstruct survey weights, and evaluate how weighted results differ from unweighted results. Students will then present their findings in the form of final demographic summaries.

Possible Extension
Students may test how different sets of population margins (e.g., age and sex vs. age, sex, and race) affect results. They may also explore whether more complex models (e.g., logistic regression or random forests) improve representativeness compared to simple weighting methods.

Outline/Timeline
The schedule below is designed for a six-week Spring VMRC session. It is flexible and may be adjusted based on student progress and interests.

Week 1: Introduction to survey bias and weighting.
• Discuss goals of surveys, sources of bias, and why representativeness matters.
• Students compute unweighted demographic distributions from the GSS subset and compare them with Census benchmarks for key variables (e.g., age, sex, region).

Week 2: Decision trees and nonresponse modeling.
• Introduce decision trees as simple, interpretable classification models.
• Students build classification trees to predict response vs. nonresponse and examine the resulting rule-based splits (e.g., by age, region, or education).

Week 3: Constructing and applying post-stratification weights.
• Walk through the steps of post-stratification using Census margins.
• Students generate weighted demographic reports, compare unweighted vs. weighted interpretations, and write short reflections on how conclusions change.

Week 4: Sensitivity to margin choices.
• Students construct alternative weighting schemes (e.g., using age & sex only vs. age, sex, and race or region).
• Compare how different sets of margins change key estimates and discuss trade-offs between complexity and stability.

Week 5: Advanced models and diagnostics.
• Introduce more complex models for response propensity (e.g., logistic regression or random forests) at an intuitive level.
• Students compare results from simple decision-tree–based adjustments with those from more complex models and examine basic diagnostics (such as weight distributions).

Week 6: Final summaries and presentation.
• Students refine their final weighted demographic summaries, tables, and graphics.
• Assemble a final presentation (slides or short paper) describing data, methods, weighting choices, and key findings, and practice delivering a clear research-style talk.

References
  1. Smith, T. W., Davern, M., Freese, J., & Hout, M. (2019). General Social Surveys, 1972–2018. NORC at the University of Chicago.
  2. GSS Data Explorer: https://gssdataexplorer.norc.org/
  3. Groves, R. M., Fowler Jr., F. J., Couper, M. P., Lepkowski, J. M., Singer, E., & Tourangeau, R. (2011). Survey Methodology. Wiley.

Session 2 · Feb 24, 2026 – Apr 4, 2026

Modeling COVID-19 Comorbidity and Control Measures

Research Proposal — Virtual Math Research Circle

Session
Session 2: Feb 24, 2026 – Apr 4, 2026
Mentor
Hemaho B. Taboe
PhD Candidate in Mathematical Biology
Department of Mathematics, University of Florida, Gainesville, FL 32611, USA

Project Title
Modeling COVID-19 Comorbidity and Control Measures
Topic Area
Mathematical Biology, Differential Equations, Epidemiological Modeling

Background
High school calculus and first-order differential equations skills will be useful. These concepts will be taught or revised at the beginning of the project as needed, so motivated students without prior exposure can still participate successfully.

Abstract
The unprecedented COVID-19 pandemic continues to unfold with unimaginable consequences, despite the implementation of stringent control measures since its initial outbreak in China in 2019. The burden of the disease, in terms of its associated death toll, is higher in the United States compared to other parts of the world. One of the factors cited in the literature to explain this phenomenon is the prevalence of preexisting health conditions among Americans.

The objective of this study is to develop a deterministic basic COVID-19 model that incorporates patients who concurrently have COVID-19 and either cancer or diabetes (comorbidity), as well as those who are solely positive for COVID-19. We will utilize this framework to assess the magnitude of COVID-19–related deaths during the primary wave of the delta variant in the US. Certain parameters of the model will be sourced from existing literature, while the remainder will be estimated by calibrating the model to the daily COVID-19 death data. Additionally, various control strategies (such as vaccination and mask-wearing) will be analyzed to determine the scenarios in which the burden may be alleviated.

Possible Extension
Possibly, this project will be extended to countries such as China, France, and Nigeria for comparison purposes. The model structure will remain the same, but the parameters may vary for each country. Students could compare outcomes across settings and explore how differences in demographics, health systems, or control measures affect the projected burden of disease.

Outline/Timeline
The outline below summarizes the original three-week research plan. It can be adapted to fit the VMRC Spring 2026 schedule as needed.

Weeks 1-2
• Meet with students, explain the project goals, and provide background materials.
• Review calculus and ordinary differential equations relevant to compartmental models.
• Provide a tutorial on using MATLAB to solve ordinary differential equations.
• Provide a tutorial on using Overleaf for writing the research report and presentation.
• Students attend all sessions, read the material, complete assignments, ask questions, and check email regularly for project updates.

Weeks 3-4
• Teach compartmental model formulation from a disease-epidemiology perspective.
• Work with students to polish the model they propose, refining compartments and flows.
• Introduce the notions of basic reproduction number and herd immunity.
• Teach parameter-estimation methods, from selecting values in the literature to fitting parameters to data.
• Students participate in the training, formulate the first draft of the model, determine expressions for key parameters, and begin collecting data and estimating parameters.

Weeks 5-6
• Teach model simulation techniques to address the main research questions/objectives.
• Assist students in running simulations and interpreting results in the context of COVID-19 comorbidity and control strategies.
• Support students with Overleaf and manuscript-writing questions.
• Students simulate the model, answer research questions, and write the first draft of the manuscript.
• The mentor reviews the manuscript for submission; students make the necessary corrections and refinements suggested.

References
  1. Chatterjee, Sayan, et al. “Association of COVID-19 with comorbidities: an update.” ACS Pharmacology & Translational Science 6.3 (2023): 334–354. https://doi.org/10.1021/acsptsci.2c00181
  2. Nagle, R. Kent, et al. Fundamentals of Differential Equations and Boundary Value Problems. New York: Addison-Wesley, 1996.
  3. Eikenberry, Steffen E., et al. “To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic.” Infectious Disease Modelling 5 (2020): 293–308.
  4. Ngonghala, C. N., Taboe, H. B., Safdar, S., & Gumel, A. B. (2023). “Unraveling the dynamics of the Omicron and Delta variants of the 2019 coronavirus in the presence of vaccination, mask usage, and antiviral treatment.” Applied Mathematical Modelling 114, 447–465. https://www.sciencedirect.com/science/article/pii/S0307904X22004358

Modeling Cooperation

Research Proposal — Virtual Math Research Circle

Session
Session 2: Feb 24, 2026 – Apr 4, 2026
Mentor
James Branca
Ph.D. Student in Biomathematics
Department of Mathematics
Florida State University

Project Title
Modeling Cooperation
Topic Area
Game Theory

Background
No calculus needed. Knowledge of set theory will be helpful. Necessary concepts will be taught along the way.

Abstract
In 1951, John Nash published one of his seminal papers on game theory, helping to formalize how rational players make decisions in competitive situations. A parallel line of work, cooperative game theory, focuses on how groups (coalitions) form and how the benefits of cooperation are divided among participants.

In this project we will study simple cooperative games, where a coalition either “wins” or “loses” depending on which players are included. A classical example is a voting game with 101 participants in which coalitions of 51 or more people win and coalitions with 50 or fewer people lose. Such models raise natural questions: Which players are most “powerful”? How should we measure influence or fairness in a voting system?

Goals for this project are to learn the basic ideas of cooperative game theory, work through concrete examples (especially simple games and voting games), and see how these tools are used to model real-world situations involving alliances, negotiations, and shared decision-making.

Possible Extension
There are many possible extensions of this research. An interested student may, for example, investigate more sophisticated power indices, explore applications to real voting bodies (such as councils or committees), or examine how changing the rules (such as the winning threshold) affects the distribution of power. Students who wish to pursue an extension should reach out to discuss options.

Outline/Timeline
The schedule below is designed for a 6-week Session 2 program and may be adjusted as needed based on student progress and interests.

Week 1: Introduction to game theory.
• Overview of game theory and the distinction between cooperative and noncooperative games.
• Work through small examples of strategic and cooperative situations and discuss basic terminology (players, coalitions, payoffs).

Week 2: Simple games and voting models.
• Introduce simple games formally and study majority voting games (e.g., 101 voters where coalitions of 51 or more win).
• Identify winning and losing coalitions in specific examples and discuss informal notions of power and fairness.

Week 3: Power and influence in cooperative games.
• Explore ideas of player “desirability” and relative influence using examples from the literature.
• Compute basic power measures for small voting games by hand and interpret the results.

Week 4: Applications and literature exploration.
• Read selected sections from introductory texts on cooperative game theory and simple games.
• Connect the theory to real-world voting systems or decision-making bodies chosen by the group.

Week 5: Computation and analysis.
• For selected games, systematically enumerate coalitions and compute power measures or related quantities (by hand and/or with light computational assistance).
• Begin drafting preliminary results, tables, and figures for the final presentation.

Week 6: Synthesis and presentation.
• Finalize calculations and interpretations; organize examples into a coherent narrative.
• Prepare and practice a Beamer or PowerPoint presentation explaining the model, examples, and what the group learned about cooperation and fairness.

References
  1. Curiel, I. Cooperative Game Theory and Applications, 1997.
  2. Taylor, A. and Zwicker, W. Simple Games: Desirability Relations, Trading, Pseudoweightings, 1999.

Exploring RSA Cryptography: How Numbers Keep Secrets

Research Proposal — Virtual Math Research Circle

Session
Session 1 & 2: Jan 2026 – Feb 2026
Mentor
Boluwatife Aderinto
3rd year PhD student
Department of Mathematics
Florida State University

Project Title
Exploring RSA Cryptography: How Numbers Keep Secrets
Topic Area
Number Theory, Cryptography

Background
This research project requires students to be comfortable with basic algebra and arithmetic operations. No prior knowledge of cryptography or programming is necessary. All required cryptographic concepts, modular arithmetic, and elementary number theory topics such as prime numbers, factorization, and exponents will be introduced and thoroughly explained. This project provides an exciting introduction to the fields of number theory and cryptography.

During the research, we will utilize computational tools, mainly Python. For the final presentation, students will learn and use tools such as PowerPoint or LaTeX with Beamer.

Abstract
Have you ever wondered how secret messages and confidential information (such as Social Security numbers, credit card details, and other financial data) remain hidden from hackers, ensuring online security and privacy? The answer lies in the fascinating world of cryptography, and one of the most important cryptographic methods is RSA encryption.

RSA cryptography is one of the most widely used encryption schemes for securing digital information. At its core, RSA leverages simple but powerful mathematical concepts involving prime numbers, modular arithmetic, and factoring large integers. But how exactly does RSA keep our digital secrets safe?

In this research project, students will uncover the mathematical foundations behind RSA cryptography. We will explore the mechanics of encryption and decryption using RSA, including the ways prime numbers serve as “guardians of secrecy.” Students will actively experiment with encrypting and decrypting messages, learning firsthand how mathematical principles protect privacy.

Additionally, we will investigate the security strength of RSA by discussing potential vulnerabilities related to factorization techniques. Through these activities, students will not only grasp essential cryptographic concepts but also strengthen their skills in mathematical reasoning, problem-solving, and computational experimentation.

Possible Extension
A natural extension of this research would involve investigating the impact of quantum computing on RSA cryptography. Specifically, students can explore Shor’s algorithm, a quantum algorithm that efficiently factors large integers in polynomial time and poses a serious threat to the long-term security of RSA.

Students will have the opportunity to assess and discuss modern cryptographic approaches that are believed to withstand quantum-based attacks, positioning their work at the intersection of classical cryptography and quantum computing.

Outline/Timeline
This is a general (tentative) outline of how the research project will progress. We will adjust the schedule as necessary throughout the 3-week period.

Weeks 1-2 (Foundations & Basics).
• Review algebraic operations, modular arithmetic, prime numbers, factorization, and exponents.
• Introduce basic encryption/decryption ideas (for example, classical ciphers such as the Caesar and Vigenère ciphers). Students will practice with simple examples and numerical exercises.
• Develop the basics of mathematical reasoning and proof writing through examples and guided exercises. Computational tools (Python) will be introduced for practical demonstrations.

Weeks 3-4 (RSA Encryption & Experimentation).
• Give a detailed introduction to RSA encryption and decryption algorithms, ensuring a thorough understanding of how RSA works.
• Students learn RSA key generation, encryption, and decryption processes in detail and carry out practical computational experiments to encrypt and decrypt messages, verifying the principles learned.
• Analyze encryption strength by exploring security implications of prime-number sizes and common methods for attempting to break RSA, emphasizing the difficulty of factoring large integers, and discussing empirical results.

Weeks 5-6 (Finalizing Results & Presentation).
• Conclude experiments and refine understanding, finalizing mathematical explanations and computational experiments, and summarizing findings.
• Prepare the final presentation. Students will decide collectively whether to use PowerPoint or a LaTeX-based Beamer presentation, depending on their comfort level and preferences. We will also practice presenting, ensuring clarity and confidence in the final talk.

References
  1. W. Stallings, Cryptography and Network Security: Principles and Practice, 7th ed., Pearson Education, 2016.
  2. T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms, 3rd ed., MIT Press, 2009.
  3. N. D. Koblitz, A Course in Number Theory and Cryptography, 2nd ed., Springer-Verlag, 1994.
  4. D. Boneh, V. Shoup, A Graduate Course in Applied Cryptography, Stanford University, 2020. (Online text available at https://toc.cryptobook.us/.)

Numerical Shape Optimization

Research Proposal — Virtual Math Research Circle

Session
Session 1 & 2: Jan 2026 – Feb 2026
Mentor
Alvis Donghan Zahl
PhD Candidate (ABD)
Department of Mathematics
Rutgers University

Project Title
Numerical Shape Optimization
Topic Area
Numerical analysis, shape optimization, free boundary problems

Background
This research project requires only basic algebra and familiarity with high-school geometry. No prior experience with calculus, partial differential equations, or numerical analysis is required; all analytical and computational tools will be introduced during the program. The project connects geometry, optimization, and introductory numerical methods.

Students will be introduced to numerical experimentation using accessible tools such as Python, Matlab, or Mathematica. For the final presentation they may use PowerPoint or Beamer. Throughout the project, emphasis will be placed on exploring examples, formulating conjectures from observed patterns, and writing concise explanations.

Abstract
One of the oldest questions in mathematics is the isoperimetric problem: among all shapes with a given area, which one has the smallest perimeter? Modern shape optimization generalizes this question to more complicated functionals, often defined by solutions of partial differential equations, and leads naturally to free boundary problems, where the optimal boundary is part of the unknown.

This project will explore the classical isoperimetric problem using discrete and numerical methods and then apply these ideas to a model Bernoulli free-boundary problem, where we seek a domain that minimizes an energy involving solutions of the Laplace equation. Students will design and run numerical experiments that evolve shapes toward optimal configurations, track how boundaries move, and build intuition for how curvature, perimeter, and energy interact.

By the end of the project, students will have developed computational experiments illustrating both isoperimetric and Bernoulli-type free-boundary behavior, and they will see how these questions play an important role in modern applied and theoretical mathematics.

Possible Extension
Students who continue in a Research Extension may:
  • Compare several discrete energies (perimeter, a simple curvature proxy, and discrete Dirichlet energy) and study how each choice changes the “optimal’’ shapes.
  • Investigate stability and time-step constraints for a parabolic Bernoulli scheme, relating numerical behavior to intuitive ideas of stability.
  • Introduce randomness (noisy sources or random initial data) and observe how free boundaries fluctuate, then attempt to quantify these fluctuations numerically.
  • Design higher-resolution experiments that approximate continuous isoperimetric shapes and prepare a polished poster or expository paper suitable for undergraduate-friendly conferences or outreach events.
These extensions deepen both the numerical and conceptual understanding of shape optimization and free-boundary evolution.

Outline/Timeline
The project is designed for a 6-week Spring VMRC session with two meetings per week. We organize the work into three two-week phases combining guided instruction, experimentation, and writing.

Week 1 — Background & Free Boundaries.
• Introduction to free boundaries: what they are, where they arise (e.g., melting/solidification, fluid flow, electrostatics), and basic examples.
• Overview of the isoperimetric problem and simple perimeter/area comparisons in the plane.
• Students become familiar with the overall goals of shape optimization and with the software tools we will use.

Week 2 — Numerical Tools & the Isoperimetric Problem.
• Introduction to simple numerical methods (finite-difference approximations, discrete gradients, and gradient-descent–type updates).
• Implement basic schemes that evolve planar sets toward perimeter-minimizing shapes and observe how shapes evolve toward circles.

Weeks 3–4 — Implementation & Experiments.
• Students implement numerical routines in Python/Matlab/Mathematica to simulate gradient flows for different initial domains (ellipses, rectangles, noisy shapes, etc.).
• Run systematic experiments, record when and how circular shapes emerge, and compare rates of convergence and possible numerical instabilities.

Week 5 — Bernoulli Free-Boundary Problem.
• Apply the numerical methods to an elliptic Bernoulli free-boundary problem driven by a solution of the Laplace equation.
• Compare the resulting shapes with those from pure perimeter flows, highlighting similarities, differences, and the effect of the energy condition on the optimal boundary.

Week 6 — Write-Up & Presentation.
• Students finalize computations, summarize observations, and refine figures and plots.
• Prepare a written summary and a short Beamer/PowerPoint presentation outlining their methods, results, and conjectures; practice delivering the final talk.

References
  1. F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2012.
  2. J. Crank, Free and Moving Boundary Problems, Oxford University Press, 1984.

How Random is a Shuffle?

Research Proposal — Virtual Math Research Circle

Session
Session 2: Feb 24, 2026 – Apr 4, 2026
Mentor
Mae Lee
Graduate Teaching Assistant, Bridge-to-Doctorate Affiliate graduate student
Mathematics Department
University of Texas at Arlington

Project Title
How Random is a Shuffle?
Topic Area
Combinatorics

Background
Students should be comfortable with basic algebra and arithmetic-style reasoning. All needed combinatorics concepts (permutations, simple counting arguments, etc.) will be introduced during the program. No prior exposure to combinatorics is assumed.

Abstract
It is your turn to shuffle and deal a deck of cards for a game. Everyone wants the deck to be “well shuffled,” but what does that actually mean? Is every possible order really “equally likely,” or are some orders still much more likely than others?

In this project, students will explore how to model a shuffle mathematically and how to measure “how random” the resulting deck order is. We will treat a deck as an ordered list (a permutation) and choose a specific shuffle rule—such as a simple riffle shuffle or a “top-in-at-random” shuffle—to study in detail.

Students will construct small examples, compute how a single shuffle moves the deck from one order to another, and then investigate what happens after many shuffles. They will design and analyze measures of “closeness to random” and use these to ask questions like: How many shuffles are needed before the deck looks “close to random”? The project will culminate in a research-style presentation explaining the models, experiments, and findings.

Possible Extension
As a possible extension, students may compare multiple shuffle rules (for example, different riffle-style shuffles or a “cut and interleave” rule), studying how fast each rule appears to mix the deck. They may also investigate how mixing behavior changes when the deck size changes (e.g., 10 cards vs. 52 cards), using a combination of exact calculations for very small decks and larger-scale computer simulations for bigger decks.

Outline/Timeline
This is a tentative schedule for the 6-week Session 2 program and may be adjusted as needed as the session progresses.

Week 1 (Cards and permutations).
• Introduce decks as ordered lists and review permutations and basic counting ideas.
• Discuss what “random” should mean in this context and brainstorm simple tests for randomness (for example, looking at positions of certain cards or simple statistics on the order).

Week 2 (Modeling a shuffle).
• Choose a specific shuffle rule and describe precisely how it acts on the deck.
• For a very small deck (e.g., 3–4 cards), build a transition table or diagram that shows how one shuffle moves between different deck orders.

Week 3 (Repeated shuffles and small-deck analysis).
• Use the transition table to study what happens after several shuffles of a small deck.
• Track how the distribution over deck orders changes with the number of shuffles and begin to define simple quantitative measures of “closeness to random.”

Week 4 (Scaling up: larger decks and simulations).
• Move from hand calculations to computer simulation for larger decks.
• Students write simple code (or use prepared notebooks) to simulate repeated shuffles, collect data, and visualize how quickly the deck appears to mix under their chosen rule.

Week 5 (Analyzing and comparing shuffle rules).
• Analyze simulation results, looking for patterns in how measures of randomness evolve with the number of shuffles.
• If time permits, compare two different shuffle rules and discuss which one appears to mix faster and why.

Week 6 (Prepare presentation).
• Students organize their definitions, models, and experimental results into a coherent story about “how random” their shuffle is.
• Prepare a final Beamer or PowerPoint presentation (and/or short write-up) explaining how many shuffles are needed before the deck looks “close to random” under their tests.

References
  1. Joy Morris, Combinatorics: An Upper-Level Introduction to Combinatorics and Design Theory, Version 2.1.1, University of Lethbridge, 2023.

Optimal Paths in Inhomogeneous Media: An Undergraduate Introduction to the Elvis Problem

Research Proposal — Virtual Math Research Circle

Session
Session 2: Feb 24, 2026 – Apr 4, 2026
Mentor
Safeyya Alyahia
PhD Candidate in Mathematics
Louisiana State University (LSU)

Project Title
Optimal Paths in Inhomogeneous Media: An Undergraduate Introduction to the Elvis Problem
Topic Area
Analysis / Optimization / Calculus of Variations / Differential Equations

Background
Prerequisites:
• Calculus I–III (single and multivariable calculus)
• Linear Algebra
• Familiarity with basic proofs (e.g., from an Intro to Proofs or Real Analysis I course) is helpful but not strictly required
• Recommended (but not required): an introductory course in Ordinary Differential Equations and some experience with a programming language (Python, MATLAB, or similar)

The project is structured so that students with strong calculus and linear algebra, but little prior exposure to analysis or optimization, can still participate meaningfully.

Abstract
This project studies a class of shortest-time path problems sometimes called “Elvis problems.” In these problems, a particle (or “traveler”) moves from a starting point to a target point in the plane, but its possible velocities at each point are constrained by the surrounding medium. Different regions of the plane may allow different maximum speeds or favor some directions over others.

Students will learn how to translate this situation into precise mathematics: describing allowed velocity sets, defining admissible paths, and formulating the time needed to traverse a path. They will investigate when a fastest path (an optimal path) exists and how it behaves in simple, low-dimensional examples. Special attention will be given to two-region models, which naturally connect to familiar physical examples such as refraction (light bending when it passes between media).

The project will combine theory and computation. Analytically, students will explore basic existence arguments and derive qualitative properties of optimal paths in simplified settings. Computationally, they will implement simple numerical methods to approximate optimal paths and visualize how they depend on the medium. The project is designed to be accessible to undergraduates with a strong calculus and linear algebra background and will introduce them to ideas from optimal control and the calculus of variations in a concrete and visual way.

Possible Extension
The mentor is available to support student work throughout the VMRC Research Extension period, including regular group meetings and office-hour–style sessions, and is also willing to supervise a short virtual follow-up project during the subsequent semester.

Beyond the main session, potential extensions include:
  • Developing additional numerical experiments and visualizations;
  • Refining and polishing a written report or poster;
  • Exploring preliminary generalizations (e.g., anisotropic media or more complicated geometries);
  • Advising students who wish to turn this into an honors thesis or independent study at their home institution.

Outline/Timeline
The original project is organized as a 4-week intensive research session. This outline can be adapted to the VMRC schedule as needed.

Week 1 – Introduction and Modeling.
• Introduce the Elvis problem in intuitive terms (fastest path under velocity constraints).
• Review key mathematical tools at an accessible level: norms and vectors in \(\mathbb{R}^2\); curves, parametrization, and basic notions of path length and speed.
• Define admissible paths \(x : [0,T] \to \mathbb{R}^2\) and Elvis velocity sets \(E(x) \subset \mathbb{R}^2\) (simple cases: disks and balls).
• Work through classical examples: straight-line motion with constant speed, and the “lifeguard problem” (running then swimming).
• Assign small exercises: students derive and compare travel times for different candidate paths in simple media.

Week 2 – Two-Medium Models and Basic Theory.
• Introduce two-region models: the plane divided into two media with different speed bounds \(v_1, v_2\).
• Formulate the shortest-time problem between two points separated by an interface (e.g., a straight line).
• Derive necessary conditions for optimality using geometric reasoning: straight segments inside each region; a “refraction” condition at the interface (analogue of Snell’s law).
• Discuss, at a gentle level, why optimal paths exist in simple settings (bounded speed, compactness ideas, convex velocity sets).
• Students begin writing up partial results and computing simple symbolic examples (e.g., minimizing total time as a function of crossing point on the interface).

Week 3 – Numerical Approximation and Visualization.
• Introduce a discrete model: represent the domain by a grid and assign local speed or cost at each node.
• Present simple algorithms for approximating shortest-time paths (e.g., weighted-graph viewpoint and shortest-path algorithms like Dijkstra’s algorithm).
• Students implement basic code (Python or MATLAB) to approximate optimal paths in two-region examples and visualize paths.
• Compare numerical paths to analytic predictions (angles, crossing points).
• Begin exploring variations: different shapes for interfaces (e.g., curved boundary); simple anisotropic examples (e.g., faster in horizontal than vertical direction).

Week 4 – Extensions, Synthesis, and Presentation.
• Students refine one or two main examples to study in depth (e.g., a two-region isotropic problem plus a simple anisotropic extension).
• Consolidate: clear statement of main models; theoretical properties they can justify; numerical experiments and figures.
• Draft a short research report: introduction and motivation; mathematical model and main definitions; main analytical results; numerical experiments, plots, and interpretation.
• Prepare a final presentation or poster for the program’s closing session.
• Discuss possible follow-up work for students interested in deeper theory (e.g., connections to calculus of variations, optimal control, or Hamilton–Jacobi equations) that could continue during the research extension.

References
  1. Evans, L. C. An Introduction to Mathematical Optimal Control Theory. Lecture notes (freely available online).
  2. Evans, L. C. Partial Differential Equations, 2nd ed., AMS, 2010. Selected sections on Hamilton–Jacobi equations and optimal control.
  3. Ekeland, I., & Temam, R. Convex Analysis and Variational Problems. SIAM, 1999.
  4. Giaquinta, M., & Hildebrandt, S. Calculus of Variations I. Springer, 1996.
  5. Additional expository sources on the lifeguard problem, Fermat’s principle, and Snell’s law (e.g., undergraduate-level articles in American Mathematical Monthly), and basic programming tutorials (Python or MATLAB) and documentation for numerical experiments.

Survey Weighting and Nonresponse Adjustment with NHANES Data

Research Proposal — Virtual Math Research Circle

Session
Fall 2025 (Virtual Math Circle)
Mentor
Isaac B. Michael, PhD
Gordon A. Cain Center for STEM Literacy
Louisiana State University

Project Title
Survey Weighting and Nonresponse Adjustment with NHANES Data
Topic Area
Statistics / Data Science

Background
This project introduces students to survey sampling, nonresponse bias, and statistical weighting using real-world public health data. The National Health and Nutrition Examination Survey (NHANES) is a nationally representative survey conducted by the CDC that collects demographic, health, and nutrition information.

Students will begin with a simplified dataset in which the official survey weights are removed. They will then reconstruct weights using population margins, analyze nonresponse using classification trees, and compare weighted vs. unweighted results. This project provides an introduction to applied statistics, survey methodology, and machine learning, with hands-on use of R or Python.

Abstract
Survey data often suffer from nonresponse bias: some groups of people are less likely to respond, and this can skew results. Statisticians correct for these imbalances by applying survey weights so that the sample better reflects the target population.

In this project, students will learn how survey weights are constructed and why they matter. Using a subset of NHANES data, they will model response patterns using decision trees and apply post-stratification techniques to align the survey sample with Census population margins. The final step is producing a weighted demographic report and comparing it with unweighted results. This investigation mirrors methods used in major public health surveillance systems, such as PRAMS and SOARS.

Possible Extension
Students may explore how different nonresponse models (logistic regression, random forests, and decision trees) change final weights and estimates. They may also investigate how survey weights influence key health indicators such as smoking prevalence or obesity rates, and how sensitive those indicators are to model and weighting choices.

Outline/Timeline
Weeks 1-2: Introduction to survey sampling, weights, and nonresponse bias.
• Discuss why representativeness matters and how nonresponse can distort estimates.
• Students compute unweighted demographic distributions from NHANES and compare them to Census margins for key variables (e.g., age, sex, race/ethnicity).

Weeks 3-4: Classification trees and nonresponse modeling.
• Introduce classification trees (decision trees) as an interpretable modeling tool.
• Students build tree-based models to predict response/nonresponse and explore how tree splits highlight differences across subgroups.

Weeks 5-6: Constructing weights and reporting results.
• Construct post-stratification weights using selected population margins.
• Students produce final weighted demographic reports, compare weighted vs. unweighted interpretations, and present their findings on how weighting changes the story told by the survey data.

References
  1. Korn, E. L., & Graubard, B. I. (1999). Analysis of Health Surveys. Wiley.
  2. CDC NHANES Website: https://www.cdc.gov/nchs/nhanes/index.htm
  3. Lohr, S. L. (2019). Sampling: Design and Analysis. CRC Press.

Questions

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