Virtual Math Research Circle — 2026 Summer Research Projects

Overview

Virtual Math Research Circle brings motivated high-school students together with university mentors to pursue authentic, publishable mathematics. Registration for Summer 2026 is now closed. Below are the research topics that ran during Session 1 and Session 2.

How the Program Works Public Archives

Session 1 · June 8, 2026 – June 27, 2026

Learning from Evidence: The Mathematics of Updating Beliefs

Research Proposal — Virtual Math Research Circle

Session
Session 1: June 8, 2026 – June 27, 2026
Mentor
Dr. Mario Stipčić
Preceptor in Mathematics
Department of Mathematics
Harvard University

Project Title
Learning from Evidence: The Mathematics of Updating Beliefs
Topic Area
Probability Theory / Bayesian Statistics

Background
Basic probability knowledge is recommended for better understanding of the topic. Still, we will provide a brief overview of probability in both discrete and geometric settings, along with an introduction to rigorous mathematical notation.

For this project, we will use RStudio to write code in R. Before the first meeting, students should install this program or another environment for working with R. However, no prior coding experience is required; students will work with prewritten code and learn how to modify it for their own purposes.

Apart from this, no particular background knowledge is required.

Abstract
Suppose that you want to determine whether you have a certain rare disease. You take a test with very high accuracy, which rarely misclassifies individuals as carriers or non-carriers. If your test result is positive, how likely is it that you actually have the disease?

Surprisingly, the answer is often much smaller than one might expect; in fact, the probability may be so small that you most likely do not carry the disease at all. This scenario illustrates the importance of Bayes’ Theorem, one of the most fundamental results in probability theory.

In this project, students will develop a strong understanding of Bayes’ rule in both theoretical and practical settings. They will build probabilistic intuition and mathematical modeling skills by studying both the motivation behind the theorem and its formal mathematical statement. We will then transition to more complex scenarios involving medical diagnosis, machine learning classification, and statistical inference.

Students will also gain hands-on experience using R to analyze data, visualize distributions, and draw conclusions based on numerical computation and statistical summaries.

Possible Extension
Students may compare Bayesian methods with other standard approaches in statistics and data science. Possible directions include studying Bayesian inference alongside maximum likelihood estimation and confidence interval methods, or exploring classification problems through Bayesian classifiers and logistic regression.

Outline/Timeline
This is a tentative schedule for the 3-week summer program and may be adjusted as the session progresses.

Week 1 (Introduction and Problem Solving).
The first week will focus on active learning. Students will explore the motivation behind Bayes’ rule, develop its formal mathematical statement, and practice applying it to a variety of problems. Emphasis will be placed on understanding conditional probability and modeling probabilistic events.

Week 2 (Modeling and Coding).
In the second week, students will study more complex real-world scenarios. Using R, they will perform statistical analyses and learn how to draw conclusions from data. We will also discuss data visualization, including how to choose appropriate visual representations and why visualization is important for interpretation.

Week 3 (Finalizing Results and Conclusion).
In the final week, students will synthesize their results and explore how Bayesian inference connects probability theory, statistics, and machine learning. They will develop visualizations, outline their findings, and prepare a final presentation using a tool of their choice (such as PowerPoint, Canva, or LaTeX Beamer).

References
  1. Blitzstein, J., and Hwang, J. Introduction to Probability, CRC Press, 2019.
  2. Rice, J. Mathematical Statistics and Data Analysis, 3rd ed., Cengage Learning, 2006.
  3. James, G., Witten, D., Hastie, T., and Tibshirani, R. An Introduction to Statistical Learning: with Applications in R, Springer, 2021.

Session 2 · July 13, 2026 – August 1, 2026

From Interacting Particle Systems on Graphs to Disease Spreading

Research Proposal — Virtual Math Research Circle

Session
Session 2: July 13, 2026 – August 1, 2026
Mentor
Dr. Yangrui Xiang
Postdoctoral Researcher
Department of Mathematics
Louisiana State University

Project Title
From Interacting Particle Systems on Graphs to Disease Spreading
Topic Area
Probability Theory, Graph Theory, Stochastic Processes, Mathematical Biology

Background
Familiarity with basic algebra, functions, and mathematical notation (sets, summations, and proof structure) is helpful. Prior exposure to probability, combinatorics, and graph theory is encouraged but not required; all necessary concepts will be introduced during the project.

During the development of the final presentation, we will use tools such as PowerPoint, Beamer, and Overleaf.

Abstract
How fast does a disease spread through a population, and does it eventually die out? How do specific types of genes within a species evolve? And how long does it take before a stochastic system “forgets” its initial condition?

These questions can be studied using interacting particle systems (IPS), mathematical models in which many individuals evolve randomly while interacting through a network. IPS play a central role in probability theory and are widely used to model disease spread, population dynamics, and genetic evolution.

In this project, students will investigate simple multi-particle stochastic models motivated by epidemiology. We will focus on infection–recovery dynamics on graphs and study how local update rules lead to global behavior. A central theme will be the time it takes for the system to approach equilibrium. Through theory and investigation, students will observe the striking cutoff phenomenon, where convergence happens abruptly rather than gradually.

By the end of the project, students will understand how randomness and network structure interact, and they will communicate their findings in a research-style presentation.

Possible Extension
As follow-on directions, students may investigate how infection rates that depend on the total number of infected individuals influence the time it takes for a disease to spread. In this setting, the infection rate increases as the number of infected individuals in the population grows, so the disease spreads faster when more people are already infected.

A second direction is to study how infection rates that depend on the structure of the underlying graph influence the speed at which a disease spreads and stabilizes. For example, students can compare identical infection–recovery rules on cycles, grids, and d-regular expander graphs and observe differences in convergence behavior.

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

Week 1 (Graphs, Probability, and Disease Models).
Students will be introduced to graph basics, discrete probability, and the foundational ideas needed to study stochastic processes on networks. We will discuss Markov chains and build intuition for convergence. The group will then define an infection–recovery model on a graph, study configurations of the system, and begin initial simulations.

Week 2 (Mixing Time Analysis).
In the second week, students will study how to measure distance from equilibrium, both conceptually and through observable quantities. The focus will be on building upper- and lower-bound intuition for convergence time and understanding how these estimates help explain the behavior of the system.

Week 3 (Cutoff Phenomenon and Final Presentation).
In the final week, students will investigate abrupt convergence through the cutoff phenomenon and use theory to refine their conjectures. They will finalize figures and conclusions, then prepare and rehearse a research-style presentation summarizing their results.

References
  1. Stewart, James, and Day, Troy. Biocalculus: Calculus for Life Sciences, 2nd ed., Cengage Learning, 2015.
  2. Levin, David A., and Peres, Yuval. Markov Chains and Mixing Times, 2nd ed., American Mathematical Society, 2017.
  3. Norris, J. R. Markov Chains, Cambridge University Press, 1997.

Simulating Flow in a Channel with Obstacles

Research Proposal — Virtual Math Research Circle

Session
Session 2: July 13, 2026 – August 1, 2026
Mentor
Irene Erazo
Ph.D. Student in Mathematics
Department of Mathematics
Tulane University

Project Title
Simulating Flow in a Channel with Obstacles
Topic Area
Applied Mathematics

Background
This project is self-contained and does not require prior exposure to high school calculus. Basic concepts from precalculus, linear algebra, and ordinary differential equations will be introduced throughout the program.

Students will also be introduced to basic coding as needed. This project provides an accessible introduction to the use of ordinary differential equations and computational methods to model fluid flow in a channel with obstacles.

Abstract
Simulating real-world phenomena is essential across many disciplines, particularly in fluid dynamics, where understanding flow behavior plays a key role in physical applications. This project focuses on situations in which the relevant length scales are very small or the fluid has high viscosity. Such conditions frequently arise in biological systems, making the study of fluid flow important in areas such as microbiology and biomedical engineering.

In this project, students will explore how fluids move through a channel containing obstacles using mathematical models and computer simulations. First, we will develop a numerical method to compute fluid flow when forces are prescribed. Once this method is established, we will simulate different scenarios—such as fluid moving around obstacles—to observe how these barriers affect the speed and direction of the flow.

By visualizing these effects, students will gain insight into real-world applications, including the design of efficient waterways and the prediction of how flows behave in constrained environments.

Possible Extension
In the base model, obstacles are represented as isolated points within a channel. A natural extension is to consider flow around fixed line obstacles. Students may also explore how different solid obstacle shapes influence flow patterns and system behavior.

Outline/Timeline
This is a tentative schedule for the 3-week summer program and may be adjusted as the session progresses.

Week 1 (Background and Experimentation).
Students will be introduced to key concepts from precalculus, ordinary differential equations, and linear algebra that are needed for the project. We will also introduce the Method of Regularized Stokeslets, which will be used to compute fluid velocity from applied forces. Hands-on exercises will help students build familiarity with these ideas.

Week 2 (Coding and Applications).
In the second week, students will learn basic coding using Python or SageMath. They will implement the Method of Regularized Stokeslets step by step, gaining experience with both the computational and mathematical aspects of the model. Students will also be introduced to LaTeX for collaborative mathematical writing using Overleaf.

Week 3 (Flow with Obstacles and Final Presentation).
In the final week, students will extend their implementation to include flow with a point obstacle and visualize how the flow evolves in the channel. They will then synthesize their findings, contribute to a collaborative research summary, and prepare a final presentation using LaTeX Beamer in Overleaf.

References
  1. Cortez, R. The method of regularized Stokeslets, SIAM Journal on Scientific Computing, 23(4), 1204–1225, 2001.
  2. Cortez, R., Fauci, L., and Medovikov, A. The method of regularized Stokeslets in three dimensions: analysis, validation, and application to helical swimming, Physics of Fluids, 17(3), 2005.

How Math Explains Medicine: Modeling Drug Movement in the Body

Research Proposal — Virtual Math Research Circle

Session
Session 2: July 13, 2026 – August 1, 2026
Mentor
Roshana Sandadini Pitiyegedara
PhD Student in Applied Mathematics
Department of Mathematics and Statistics
Washington State University

Project Title
How Math Explains Medicine: Modeling Drug Movement in the Body
Topic Area
Mathematical Modeling / Differential Equations

Background
This project requires only basic algebra and a willingness to explore new ideas. No prior knowledge of differential equations is required, as all necessary concepts will be introduced during the project.

Familiarity with functions and graphs is helpful, but not required. Concepts from calculus, such as rates of change, will be introduced intuitively as needed.

Students will use simple computational tools such as graphing software or Python to visualize solutions and explore model behavior. For the final presentation, students may use PowerPoint or LaTeX-based tools depending on their comfort level.

Abstract
This project begins with a simple question: what happens to a drug after it enters the body?

If a person takes a dose of medicine, how does its concentration change over time? Why does it decrease, and what factors influence how long it remains effective? Such questions are central in medicine, where understanding how drugs behave in the body helps determine safe and effective treatments.

Students will begin by studying a basic model of exponential decay using the equation
dC/dt = −kC,
and interpreting its meaning in a real-world context. They will then extend this idea to a two-compartment model, where a drug moves between different parts of the body, described by a system of differential equations.

The project combines analytical reasoning and computational exploration. Students will solve and interpret equations and use computational tools to visualize how model behavior changes with different parameters. The emphasis will be on building intuition and understanding how mathematical models describe real-world systems.

By the end of the project, students will synthesize their findings into a written report and present their results, demonstrating both mathematical understanding and the ability to communicate ideas clearly.

Possible Extension
Students who wish to continue the project may explore how modifying the assumptions of the model changes the behavior of the system. For example, what happens if the rate of drug transfer is not constant?

They may investigate models with time-dependent parameters or explore how introducing additional compartments affects the dynamics. This leads naturally to questions about model complexity and how well different models capture real-world behavior.

For students interested in further exploration, we may introduce the idea of continuous models, where the distribution of a drug is described more smoothly across space or time. This provides a conceptual bridge to more advanced topics such as partial differential equations.

Outline/Timeline
This project follows a three-week intensive format, with regular meetings focused on building intuition, exploring models, and developing a final presentation. Students will gradually move from simple models to more realistic descriptions of how drugs behave in the body.

Week 1 (Introduction and Basic Models).
Students will explore how quantities change over time and how this can be described using simple equations. We will study exponential decay through the model
dC/dt = −kC,
and interpret its meaning in the context of drug concentration. Students will graph solutions and focus on developing an intuitive understanding of how quantities change over time.

Week 2 (Two-Compartment Models).
We will introduce a system of equations describing how a drug moves between different parts of the body:
dC1/dt = −k12C1 + k21C2 − k10C1,
dC2/dt = k12C1 − k21C2.
Students will explore how parameters affect system behavior and use computational tools to simulate and visualize the model.

Week 3 (Interpretation and Presentation).
Students will analyze their results and reflect on the strengths and limitations of the models they developed. They will prepare a final report and a presentation, explaining their work clearly and connecting mathematical ideas to real-world applications.

References
  1. Boyce, W. E., and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 10th ed., Wiley, 2012.
  2. Edelstein-Keshet, L. Mathematical Models in Biology. SIAM, 2005.
  3. Rowland, M., and Tozer, T. N. Clinical Pharmacokinetics and Pharmacodynamics, 4th ed., Lippincott Williams & Wilkins, 2011.

Exploring Waddington’s Epigenetic Landscape: Modeling Stem Cell Differentiation

Research Proposal — Virtual Math Research Circle

Session
Session 2: July 13, 2026 – August 1, 2026
Mentor
Kaylie Green
PhD Candidate
Department of Biomathematics
Florida State University

Project Title
Exploring Waddington’s Epigenetic Landscape: Modeling Stem Cell Differentiation
Topic Area
Mathematics in Biology and Probability Theory

Background
High school calculus is not required, but students should have a basic understanding of algebra, graphing, and functions. Calculus concepts such as derivatives may be introduced as an extension for advanced students interested in analyzing stability in Waddington’s landscape.

Students should also have a foundational understanding of cell biology, including the cell cycle, stem cell types, and differentiation. A background in algebra and probability will be useful for modeling cell fate decisions, while pre-calculus concepts such as exponential and logistic growth may be helpful but are not required.

Basic data analysis and computational skills using tools such as Excel, MATLAB, or other graphing software may enhance the project experience, but guidance will be provided.

Abstract
This three-week interdisciplinary project explores stem cell differentiation through the lens of Waddington’s epigenetic landscape, integrating ideas from biology, mathematics, and computational modeling.

Students will investigate how stem cells transition into specialized cell types, using the epigenetic landscape as a metaphor for gene regulation and cell fate decisions. Through hands-on activities, mathematical modeling, and computer simulations using tools such as MATLAB or Excel, students will analyze how different factors influence differentiation pathways.

The project begins with an introduction to stem cell biology, including key concepts such as pluripotency, self-renewal, and differentiation. Students will then explore mathematical representations of Waddington’s landscape, using potential energy functions, probability models, and phase diagrams to simulate cell fate decisions. By adjusting parameters, they will predict how environmental signals and genetic regulation shape differentiation outcomes.

The final stage of the project is dedicated to independent research, where students apply these models to real-world scenarios such as regenerative medicine or disease modeling. Interested students may also continue the work beyond the summer session toward a more formal project, potentially culminating in a poster presentation at LSU Discover Day.

Possible Extension
Students may extend the project by developing simple stochastic models of cell fate decisions using probability trees or dice-roll simulations to represent differentiation pathways.

Another possible extension is to explore optimization ideas by adjusting parameters in graphing tools such as Desmos or MATLAB to study how external factors influence cell fate.

Students interested in a real-world data component may also investigate bioinformatics questions by analyzing preprocessed gene expression data to identify patterns in differentiation.

Outline/Timeline
Week 1 (Stem Cells and Waddington’s Landscape).
Students will be introduced to stem cells, their types, and their role in tissue regeneration, leading to Waddington’s epigenetic landscape. Through discussion, multimedia activities, and mind mapping, students will explore factors that influence differentiation. The week concludes with an introduction to mathematical models, including potential functions and probability, used to describe landscape dynamics.

Week 2 (Mathematical and Computational Modeling).
Students will focus on mathematical and computational models of differentiation. They will create and manipulate landscape models, then use MATLAB, Excel, or online simulations to analyze how parameter changes affect cell fate. The week concludes with real-world case studies and ethical discussions related to stem cell research.

Week 3 (Independent Research and Presentation).
Students will work on independent research projects integrating biology and mathematical modeling. They will refine their models, create visual representations, and prepare presentations. The session concludes with group presentations and discussion of the strengths, limitations, and future applications of modeling in stem cell research.

References
  1. Murray, J. D. Mathematical Biology, 3rd ed., Springer, 2002.
  2. Waddington, C. H. Organisers and Genes. Cambridge University Press, 1940.
  3. Waddington, C. H., and Editors. Computational Biology of Stem Cells. Springer, 2006.

Predicting Student Assessment Scores from AI Interaction Patterns

Research Proposal — Virtual Math Research Circle

Session
Session 2: July 13, 2026 – August 1, 2026
Mentor
Dr. Ibrahem Aljabea
Instructor of Data Science and Mathematics
Department of Mathematics
Louisiana State University

Project Title
Predicting Student Assessment Scores from AI Interaction Patterns
Topic Area
Education Technology, Applied Machine Learning, and Statistical Modeling

Background
This project is designed for students interested in how artificial intelligence can be used as a learning tool and how data science can help measure student learning. Students should have basic familiarity with algebra, averages, percentages, and the idea that two variables can be related. No advanced statistics or machine learning background is required.

Students will be introduced to core ideas from exploratory data analysis, correlation, simple linear regression, and multiple linear regression. They will also learn how to interpret model outputs such as regression coefficients, R2, and root mean squared error in the context of a real educational question.

The computational portion of the project will use Python, with libraries such as pandas, scikit-learn, seaborn, and matplotlib. Students may use Google Colab or a local Python installation. Since the project involves collecting data from participants, students will also discuss basic research ethics, informed consent, and responsible handling of educational data.

Abstract
Artificial intelligence tools are increasingly used by students as study aids, but educators still have limited evidence about what effective AI-assisted studying actually looks like. A student may spend a long time with an AI chatbot and learn very little, while another student may ask a smaller number of thoughtful follow-up questions and develop a much deeper understanding. This project asks whether measurable patterns in student-AI interaction can predict learning outcomes.

In this study, approximately 100 students will each study a standardized STEM topic using an AI-based learning tool during a fixed session. For each participant, several behavioral features will be recorded, including the number of questions asked, average question length, session time, question diversity, follow-up ratio, and self-rated confidence. After the study session, students will complete a short comprehension assessment, which will serve as the outcome variable for the statistical model.

Students will then use exploratory data analysis and multiple linear regression to investigate which interaction features are most strongly associated with assessment scores. The central hypothesis is that active engagement behaviors, especially asking more questions and asking follow-up questions, will be stronger predictors of learning than passive indicators such as total session time.

The broader goal of the project is to move the conversation about AI in education from whether students should use AI to how students should use AI in order to learn effectively. The results may provide teachers with more concrete guidance about productive AI study habits and may help AI platform designers build tools that encourage deeper, more reflective engagement.

Possible Extension
Students may extend the project by comparing multiple linear regression with other predictive models, such as decision trees, random forests, or regularized regression methods. Another possible direction is to compare different groups of students, such as students who receive prompts encouraging follow-up questions versus students who use AI without guidance.

A further extension would be to study whether student confidence is well aligned with actual performance. This could lead to a discussion of the “illusion of learning,” where students may feel that they understand a topic after reading an AI explanation even if their assessment results suggest otherwise.

Outline/Timeline
This is a tentative schedule for a 3-week research project and may be adjusted as the project develops.

Week 1 (Research Design and Statistical Foundations).
Students will begin by studying the main research question: whether AI interaction patterns can predict student assessment scores. They will review the meaning of variables, correlation, averages, and simple regression. The group will also discuss the study design, including the standardized learning task, the 10-question comprehension assessment, the six behavioral features to be recorded, and the importance of consent and consistent data collection.

Week 2 (Data Collection, Exploratory Analysis, and Regression Modeling).
Students will organize the collected data in a spreadsheet and import it into Python. They will use pandas to clean the dataset and create exploratory visualizations, including scatterplots, histograms, and correlation heatmaps. Students will then fit simple linear regression models for individual features and a multiple linear regression model using all interaction features together. They will evaluate the model using R2, RMSE, regression coefficients, and residual plots.

Week 3 (Interpretation, Visualization, and Presentation).
In the final week, students will interpret their results and identify which AI interaction behaviors appear to be most strongly associated with learning outcomes. They will create visualizations such as a feature-importance chart, scatterplots with regression lines, and residual plots. The project will conclude with a written report and presentation explaining the research question, methods, findings, limitations, and implications for teachers, students, and AI learning platforms.

References
  1. Slamecka, N. J., and Graf, P. The Generation Effect: Delineation of a Phenomenon, Journal of Experimental Psychology: Human Learning and Memory, 4(6), pp. 592–604, 1978.
  2. Pressley, M., Symons, S., McDaniel, M. A., Snyder, B. L., and Turnure, J. E. Elaborative Interrogation Facilitates Acquisition of Confusing Facts, Journal of Educational Psychology, 80(3), pp. 268–278, 1988.
  3. Sweller, J. Cognitive Load During Problem Solving: Effects on Learning, Cognitive Science, 12(2), pp. 257–285, 1988.
  4. Roediger, H. L., and Karpicke, J. D. The Power of Testing Memory: Basic Research and Implications for Educational Practice, Perspectives on Psychological Science, 1(3), pp. 181–210, 2006.
  5. Kasneci, E., et al. ChatGPT for Good? On Opportunities and Challenges of Large Language Models for Education, Learning and Individual Differences, 103, 102274, 2023.
  6. James, G., Witten, D., Hastie, T., Tibshirani, R., and Taylor, J. An Introduction to Statistical Learning with Applications in Python, Springer, 2023.
  7. Pedregosa, F., et al. Scikit-learn: Machine Learning in Python, Journal of Machine Learning Research, 12, pp. 2825–2830, 2011.