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. 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 Summer 2026 VMRC sessions. Please use the link below to submit your application.

Register FAQ & Contact

Financial Assistance

VMRC has applied for several grants to help support student fee assistance. While funding decisions are still pending, we are optimistic that some support will be available. Families may indicate interest in financial assistance during registration, and any available awards will be shared once funding is confirmed.

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

  • 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 2–4 weeks before each session.

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.

Information Diffusion on Networks: How Fast Do Opinions Reach Consensus?

Research Proposal — Virtual Math Research Circle

Session
Session 1: June 8, 2026 – June 27, 2026
Mentor
Sixian Liu
Graduate Research, Ph.D. Student
Department of Applied Mathematics and Statistics
Stony Brook University

Project Title
Information Diffusion on Networks: How Fast Do Opinions Reach Consensus?
Topic Area
Graph Theory, Dynamical Systems, Multi-agent Optimization

Background
This project assumes only basic familiarity with algebra and arithmetic. All additional concepts and tools required to study the problem will be introduced during the program. This project serves as an excellent introduction to graph theory, optimization, differential equations, dynamical systems, and mathematical programming. During the experimental stage, we will implement computational tools such as Python and/or Matlab. During the development of the final presentation, we will use tools such as PowerPoint, Jupyter Notebook, and/or Overleaf.

Abstract
Social networks describe how people are connected and how information spreads among them. In mathematics, a social network can be represented using a graph, where each person is a node and each connection between two people is an edge. This representation allows us to study how local interactions between individuals influence the behavior of the whole group.

In this project, we study a simple model for how opinions or information spread across a network. Each node holds a number representing the opinion or information value of that individual. At each step, people update their values by interacting with their neighbors. Over time, these updates can cause the values in the network to become similar.

A common mathematical model for this process is the diffusion equation on a graph,
dc/dt = −Lc,

where c represents the values at each node and L is the graph Laplacian that describes the structure of the network. This system models the idea that each person adjusts their value toward the values of their neighbors. Intuitively, each node repeatedly averages its value with its neighbors, which gradually smooths differences across the network.

Using this model, we will study how the structure of the network affects how quickly agreement is reached. In particular, we will explore whether different network shapes lead to faster or slower spreading of information. Students will simulate these dynamics using simple computational tools such as Python networkx and observe how values evolve over time.

Possible Extension
First, instead of assigning a single number to each node, we may allow each node to store a vector representing multiple opinions or pieces of information. Second, we may investigate how the diffusion process changes when the network contains weighted edges, meaning that some connections between individuals are stronger than others. Finally, we may explore ideas from sheaf theory. This framework allows information to be stored not only at nodes but also along the connections of the network, with compatibility conditions between neighboring nodes. In this setting, diffusion can be described using a generalized operator called the sheaf Laplacian.

Outline/Timeline
This is a tentative timeline.

Week 1 (Introduction to Graphs and Diffusion Models).
During the first week, students will be introduced to the basic concepts of graphs, such as nodes, edges, and node degrees, as well as the graph Laplacian. We will learn how nodes and edges can represent relationships in real-world systems such as social networks. Students will explore simple examples of networks and visualize them using diagrams. We will also discuss how information or opinions might spread through these connections. Each node will hold a value representing an opinion or piece of information, and we will study how these values change through interactions with neighbors. The graph Laplacian matrix will be introduced as a mathematical way to describe the network structure. Finally, we will examine the diffusion equation
dc/dt = −Lc,

and interpret how this model describes the spreading of information across a network.

Week 2 (Simulation and Computational Experiments).
In the second week, students will implement simulations of the diffusion process using computational tools such as Python or MATLAB. They will construct small graphs and assign initial values to the nodes. By running simulations, students will observe how the values change over time and gradually approach consensus. Different network structures will be tested, such as chains, grids, and more connected networks. Students will compare how quickly agreement is reached in each network. These experiments will help illustrate how the structure of a network influences the diffusion process. Students will record and analyze the results from their simulations. Students can be creative and explore their own ideas related to this topic as well.

Week 3 (Analysis, Extensions, and Final Presentation).
In the final week, students will analyze the results from their experiments and summarize their findings. We will discuss what patterns were observed and what conclusions can be drawn about information diffusion on networks. Students will explore possible extensions of the model. We will also briefly introduce the idea of a sheaf-based network model. Finally, students will prepare a short written report and presentation describing their methods, simulations, and conclusions.

References
  1. West, D. B. Introduction to Graph Theory, 2nd ed., Pearson Education, Inc., 2001.
  2. Sayama, H. Diffusion on Networks, in Introduction to the Modeling and Analysis of Complex Systems, LibreTexts.
  3. Hansen, J. and Ghrist, R. Opinion Dynamics on Discourse Sheaves, SIAM Journal on Applied Mathematics, 81(5), pp. 2033–2060, 2021.

Learning from Uncertainty: A Bayesian View of Probability and Data

Research Proposal — Virtual Math Research Circle

Session
Session 1: June 08, 2026 – June 27, 2026
Mentor
Mohammad Rafiqul Islam
Ph.D. Candidate in Mathematics
Department of Mathematics
Florida State University

Project Title
Learning from Uncertainty: A Bayesian View of Probability and Data
Topic Area
Probability, Bayesian Statistics, and Uncertainty Quantification

Background
This project assumes familiarity with basic algebra, graphs, and averages. No prior knowledge of probability or statistics is required. All probabilistic ideas will be introduced from first principles using concrete examples and visual intuition. Calculus is not required.

Students will engage with hands-on experiments and simulations using simple computational tools such as Excel or Python (no prior programming experience assumed). Emphasis will be placed on conceptual understanding, interpretation, and communication rather than technical formalism.

Abstract
In many real-world situations, we must make decisions based on limited or noisy information. Bayesian probability provides a mathematical framework for describing uncertainty, updating beliefs, and learning from data.

In this project, students will explore the Bayesian view of probability, where probability represents a degree of belief rather than a fixed long-run frequency. We will begin with intuitive examples such as coin flips, guessing unknown quantities, and predicting outcomes with incomplete information. Students will learn how initial beliefs (called priors) can be updated using observed data to form improved beliefs (called posteriors).

Through simulations and experiments, students will investigate how uncertainty changes as more data is collected, how prior assumptions influence conclusions, and how Bayesian reasoning differs from traditional deterministic thinking. Visual tools such as probability distributions, histograms, and simulation plots will be used throughout.

The project emphasizes intuition, experimentation, and explanation. By the end of the program, students will understand how Bayesian ideas help quantify uncertainty and support rational decision-making in science, data analysis, and everyday life.

Possible Extension
Interested students may explore how different prior beliefs lead to different posterior conclusions, even when observing the same data. Additional extensions include comparing Bayesian predictions with simple averages, investigating how uncertainty shrinks as sample size increases, or exploring basic Bayesian decision-making under uncertainty.

Outline/Timeline
Brief overview of cadence and expectations for the chosen session format (e.g., 3-week intensive vs. semester).

Week 1 (Foundations of Uncertainty and Belief).
Introduction to randomness, probability, and uncertainty. Discussion of everyday uncertainty and belief-based reasoning. Introduction to Bayesian probability through simple coin-flip and guessing experiments. Students will learn how to represent beliefs using probability distributions.

Week 2 (Bayesian Updating and Experimentation).
Students will explore how beliefs change after observing data. We introduce priors, likelihoods, and posteriors through simulations and visual examples. Computational experiments will be used to study how uncertainty evolves as more data is collected.

Week 3 (Interpretation and Communication).
Students will analyze and interpret results from their experiments, focusing on explaining uncertainty clearly. The final week will be devoted to preparing presentations and discussing broader applications of Bayesian thinking in science, data analysis, and decision-making.

References
  1. Martin O. Bayesian Analysis with Python. Birmingham, UK: Packt Publishing; 2016 Nov 25.
  2. Gelman, A., et al. Bayesian Data Analysis, 3rd ed., CRC Press, 2013.

When Can We Trust Simple Decisions?: Binary Classification with Imbalanced Data

Research Proposal — Virtual Math Research Circle

Session
Session 1: June 8, 2026 – June 27, 2026
Mentor
Hyelim Jung
Ph.D. Candidate
Department of Mathematics and Statistics
Auburn University

Project Title
When Can We Trust Simple Decisions?: Binary Classification with Imbalanced Data
Topic Area
Applied Mathematics; Mathematical Modeling; Data-Driven Decision Rules

Background
This project assumes only a standard high-school background in algebra. Students are expected to be comfortable with basic inequalities, simple functions, proportions, and percentages, as well as reading tables and graphs. No prior coursework in statistics, probability, or calculus is assumed.

All necessary concepts—such as decision rules, evaluation metrics, and experimental comparison—will be introduced within the context of the research questions. Computational experiments will be conducted using Excel, R, or Python, with code templates provided as needed.

Abstract
Many real-world problems require binary decisions, such as determining whether an email is spam or deciding whether an alert should be issued. A common way to evaluate such decisions is accuracy—the proportion of correct outcomes. But is accuracy always a reliable measure of performance?

In this project, students investigate simple decision rules for binary classification and explore how these rules behave when data are highly imbalanced, meaning that one outcome is much rarer than the other. Through hands-on experiments, students discover that a rule with very high accuracy can nevertheless perform poorly in practice by systematically missing rare but important cases.

Rather than introducing complex models, students focus on understanding why accuracy can be misleading and how evaluation criteria shape our conclusions. Using tables, graphs, and basic arithmetic, students compare different decision rules under varying levels of imbalance.

In the final stage of the project, students propose a simple remedy by redefining what it means for a rule to perform well. By examining multiple error rates and incorporating basic cost considerations, students develop a more nuanced framework for evaluating decisions. The project emphasizes the process of mathematical research—posing questions, designing experiments, interpreting results, and communicating findings—using tools accessible to high-school students.

Possible Extension
Students who continue in the VMRC Research Extension may explore how different levels of data imbalance affect conclusions, or how varying the relative cost of different types of errors changes which decision rules are preferred. Additional extensions may include introducing label noise or examining real-world scenarios such as medical screening or fraud detection. These extensions emphasize deeper interpretation and more refined experimental design rather than additional technical machinery.

Outline/Timeline
This project is designed for a 3-week intensive format with near-daily meetings. The emphasis is on experimentation, interpretation, and research communication rather than technical depth.

Week 1 (Introduction and Background).
Introduction to binary decision problems and simple decision rules. Students construct small, synthetic datasets and evaluate rules using accuracy. Comparisons between balanced and imbalanced datasets lead to the initial observation that accuracy may behave counterintuitively. Students articulate preliminary research questions and hypotheses.

Week 2 (Problem Setup and Experiments).
Systematic experimentation with varying levels of data imbalance and different decision thresholds. Students organize results using tables and visualizations, and compare multiple rules under identical conditions. The focus is on identifying patterns and understanding why certain rules appear to succeed or fail.

Week 3 (Conclusion and Presentation).
Students revisit their earlier conclusions and consider alternative ways to evaluate decision rules. By examining multiple error rates and introducing simple cost considerations, students propose and justify a remedy to the shortcomings of accuracy. The final week is devoted to synthesizing results, drafting a written report, and preparing a 30–45-minute colloquium-style presentation.

References
  1. He, H. and Garcia, E. A. Learning from Imbalanced Data. IEEE Transactions on Knowledge and Data Engineering, 21(9), 2009.
  2. Saito, T. and Rehmsmeier, M. The Precision-Recall Plot Is More Informative than the ROC Plot When Evaluating Binary Classifiers on Imbalanced Datasets. PLoS ONE, 10(3), 2015.

The Life Cycle of Products: How People Adopt and Abandon Them

Research Proposal — Virtual Math Research Circle

Session
Session 1: June 8, 2026 – June 27, 2026
Mentor
Dr. Lingju Kong
Professor of Mathematics
University of Tennessee at Chattanooga

Project Title
The Life Cycle of Products: How People Adopt and Abandon Them
Topic Area
Mathematical Modeling

Background
Familiarity with high school calculus and differential equations will enhance comprehension of the research. Key concepts will be introduced and reviewed during the first week of the project.

Abstract
Understanding how people start using new products and stop using them is an important question in marketing. In this study, we create a simple model to study how users choose between two competing products. We identify key points that determine whether a product becomes popular or fades away. We also look at different possible outcomes: no one uses either product, one product becomes dominant, or both products are used by some people. Finally, we run computer simulations to show that our predictions match what the model suggests.

Possible Extension
There are several ways this project could be taken further:

(1) Exploring a similar control problem — looking at ways to influence or guide the system in a related situation.
(2) Testing how sensitive the model is to changes — checking how small changes in the model's parameters affect the results.
(3) Studying real-world data — using historical data on daily active users of Facebook and LinkedIn to see how well the model matches reality. This involves adjusting the model's parameters to make predictions about future user trends on both platforms.

Outline/Timeline
Week 1: Go over the important ideas from calculus and differential equations that we will need, and learn about the compartmental model.

Week 2: Explore the model to find key points where behavior changes, and look at different steady states the system can reach.

Week 3: Run computer simulations using Python to see the model in action, and finish preparing the project presentation.

References
  1. Bass, F. M. A new product growth model for consumer durables, Management Science, 15 (1969), 215–227.
  2. Kong, L. and Wang, M. Optimal control for an ordinary differential equation online social network model, Differential Equations and Applications, 14 (2022), 205–214.
  3. Chen, R., Kong, L., and Wang, M. Modeling the dynamics of adoption and abandonment of multiple products, Mathematics and Computers in Simulation, 241 (2026), 868–889.

Extensions of Colley’s Matrix and Ranking Methods

Research Proposal — Virtual Math Research Circle

Session
Session 1: June 8, 2026 – June 27, 2026
Mentor
Jonathan Engle
PhD Candidate (ABD)
Department of Mathematics
Florida State University

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

Background
This project requires little background knowledge beyond basic algebra and arithmetic. High school calculus is not a prerequisite. All required linear algebra concepts needed to approach the problem will be introduced during the project.

This project serves as a strong introduction to ideas from linear algebra, introductory computer science, statistics, and mathematical modeling. During the experimental stage, students will implement the mathematical theory computationally using tools such as Excel, Matlab, and/or Julia. During the final week, when results are compiled and presented, students may use tools such as PowerPoint or Beamer.

Abstract
Colley’s matrix method for ranking college football teams is an exciting application of linear algebra. While many traditional ranking methods are based primarily on a team’s record, margin of victory, or subjective committee decisions, such approaches can still fall short in producing accurate rankings. Colley addressed this issue by developing an objective ranking method that adjusts a team’s rating according to strength of schedule, without taking conference affiliation or margin of victory into account.

In this project, students will work toward replicating Colley’s Matrix for a small “toy” example using college football teams from the previous season. They will analyze these replications and investigate whether Colley’s method was able to identify potential major upsets that the College Football Committee did not anticipate. This implementation will require solving a linear system of the form:

Cr = b,   C ∈ ℝn×n,   b, r ∈ ℝn

where r is the ranking vector, b is one plus the average win rate for a team, and C is Colley’s matrix. This framework leads naturally to the central research question of the project: how can Colley’s Matrix be improved when additional information about the system is available?

The project emphasizes mathematical structure, computational experimentation, and interpretation. By the end of the program, students will understand how a linear algebra model can be used to construct rankings and how modifying that model may lead to more informative decision-making tools in sports and beyond.

Possible Extension
Once students explore the sports-ranking example, they will be encouraged to extend the same ideas to settings beyond athletics. Possible directions include, but are not limited to, resource distribution, natural disaster response, stock rankings, and other systems in which competing entities must be ranked using incomplete or structured information.

Outline/Timeline
Below is a general tentative outline of how the research project will progress. Minor adjustments may be made based on student interests and the direction of the group’s investigations.

Week 1 (Background and Examples).
The first week will introduce the problem and review traditional ranking methods. Students will discuss ways to linearize the ranking system and examine how linearization can be used to help re-rank teams. Basic linear algebra ideas such as matrices, vectors, matrix-vector operations, matrix-matrix operations, and inverses will be introduced as needed. Students will also read and analyze selected literature to learn more about the construction of the method and its mathematical motivation. The week will also include discussion of why linearization is useful for computational purposes.

Week 2 (Computational Experiments and Coding).
In the second week, students will implement the standard Colley matrix method using Matlab, Julia, or another coding language of their choice. This stage will involve the necessary data mining and extraction required to build the entries of Colley’s matrix. Once the implementation is complete, students will compare the Colley rankings to the College Football Committee rankings and to actual outcomes from the previous season in order to identify possible errors or limitations of the method. Students will then modify the system to incorporate additional parameters or information they consider meaningful, and compare their revised method to both the original Colley method and the committee rankings.

Week 3 (Finalizing Results and Presentation).
During the first half of the final week, students will refine their modified method and compile results comparing the three approaches. The second half of the week will be devoted to preparing and practicing the final presentation. The final presentation may be constructed using PowerPoint, Google Slides, or Beamer.

References
  1. Boginski, V., Butenko, S., and Pardalos, P. M. Matrix-based methods for college football rankings. Economics, Management and Optimization in Sports, 2004, pp. 1–13.
  2. Colley, Wesley N. Colley’s bias free college football ranking method: The Colley matrix explained, 2002.

The Brachistochrone Problem and the Calculus of Variations

Research Proposal — Virtual Math Research Circle

Session
Session 1: June 8, 2026 – June 27, 2026
Mentor
Philip Olapade
PhD Student, Mathematics
Georgia Institute of Technology

Project Title
The Brachistochrone Problem and the Calculus of Variations
Topic Area
Calculus of Variations

Background
This research project is designed for motivated students with a background in algebra and single-variable calculus. No prior exposure to multivariable calculus, physics, or the calculus of variations is assumed. All mathematical and physical concepts required for the project will be introduced during the program.

The project serves as an accessible introduction to advanced mathematical ideas through a classical and historically significant problem. Students will be introduced to multivariable calculus concepts, variational thinking, and physical intuition through guided instruction, interactive experimentation, and mathematical reasoning. Computational tools such as interactive simulations and basic numerical experiments will be used to support intuition and exploration. During the development of the final presentation, students will use PowerPoint.

Abstract
The Brachistochrone Problem

Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time?

This was the challenge problem that Johann Bernoulli set to the thinkers of his time in 1696. Calculus had only just been invented, yet the solution to this problem uses a fairly advanced technique called the calculus of variations. The problem was solved independently by several mathematicians, including Isaac Newton, who approached the problem using physical reasoning and geometric arguments.

Students may already be familiar with using calculus to compute quantities such as the area under a curve. Using the calculus of variations, however, one instead seeks to optimize an entire function. In the case of the brachistochrone problem, the unknown is the curve itself rather than a single numerical quantity like area or time. This leads naturally to the study of infinitesimal variations of a function, in analogy with the infinitesimal changes studied in single-variable calculus.

In this project, we approach the brachistochrone problem primarily from Newton’s perspective, emphasizing first principles, physical intuition, and geometric reasoning. Through experimentation and mathematical analysis, we show why the curve of fastest descent is a cycloid.

The name brachistochrone comes from the Greek words brachistos, meaning shortest, and chronos, meaning time.

Possible Extension
Fermat’s Principle

As a possible extension, we will explore Fermat’s principle, which states that the path taken by a beam of light between two points is the one that requires the least time. This principle leads to Snell’s law of refraction in geometrical optics.

In 1697, Johann Bernoulli used Fermat’s principle to derive the brachistochrone curve by considering the trajectory of a beam of light traveling through a medium in which the speed of light increases with depth. In this analogy, the varying speed of light plays a role analogous to the constant vertical acceleration due to gravity. This extension highlights deep connections between mechanics, optics, and variational principles.

Outline/Timeline
This is a general tentative outline of how the research project will progress. The schedule may be adjusted as needed throughout the three-week period.

Week 1 (Background and Mathematical Foundations).
The first week will be primarily instruction-based. We will introduce the physical and mathematical background needed for the brachistochrone problem, including motion under gravity, speed, time, and acceleration. Students will be guided through the relevant ideas from multivariable calculus, such as parametrized curves and how small changes in a curve affect global quantities like travel time. All concepts will be developed from first principles.

Week 2 (Experimentation and Mathematical Solution).
The second week will focus on interactive experimentation and the development of the mathematical solution. Students will use interactive simulations to compare descent times along straight lines, circular arcs, and other candidate curves. Guided by these experiments, we will begin studying the mathematical reasoning behind the solution, showing from first principles why the curve of fastest descent is a cycloid.

Week 3 (Finalizing Results and Presentation).
The first half of Week 3 will be spent finalizing the empirical results of the experiments and completing the mathematical arguments developed in Week 2. The second half of the final week will be devoted to creating the final presentation, which will be prepared using PowerPoint and focused on clearly communicating the key ideas, results, and historical context of the project.

References
  1. Brachistochrone Curve. Wikipedia.
  2. Brachistochrone Problem — Interactive Simulation. myPhysicsLab.

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

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.

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.

Dynamics of a Single Species Metapopulation Model with Dispersal

Research Proposal — Virtual Math Research Circle

Session
Session 2: July 13, 2026 – Aug 1, 2026
Mentor
Sajan Bhandari
PhD Candidate (ABD)
Department of Mathematics
University of Louisiana at Lafayette

Project Title
Dynamics of a Single Species Metapopulation Model with Dispersal
Topic Area
Discrete Dynamical Systems and Mathematical Biology

Background
This research project requires a background knowledge of basic calculus and linear algebra, mainly the calculation of eigenvalues of matrices. All other necessary knowledge related to discrete dynamical systems and model formulation and analysis will be taught by the instructor. The project will introduce students to fundamental techniques used in biological research. MATLAB or Python will be used for numerical simulations. For the final writing, we will use tools such as Overleaf/LaTeX, and the presentation will be developed using Beamer.

Abstract
Let x(t) denote the population size of a species at time t. If we want to predict the population size at time t + 1, we need to create a model that takes in the knowledge of the population size x(t) at time t and predicts how that population will grow from time t to time t + 1. A general population model is given by the equation:

x(t + 1) = f(x(t))

where f is the function that provides x(t + 1) if x(t) is known. While formulating a discrete model, modelers need to find this function f by observing the mechanisms of the phenomena they are trying to model. In other words, they need to know the order of events taking place between time t and time t + 1.

We will develop a simple two-dimensional model describing the dynamics of a single species distributed across two habitat patches. To this end, students will first explore and investigate different dispersal mechanisms by studying biological and ecological examples from nature. Based on these mechanisms, discrete-time models will be formulated. The resulting models are then thoroughly analyzed by determining their equilibria and stability. This will help answer questions such as: under what conditions will this species go extinct? How does dispersal stabilize or destabilize the system? What will be the long-term fate of the species? Finally, numerical simulations will be performed to verify theoretical results.

Possible Extension
The model presented above can be extended in several ways. First, we can change the order of events, such as dispersal first and then reproduction, which can generate completely different dynamics. Further, we can add more species in the patches, capturing more complex dynamics.

Outline/Timeline
The timeline for the three-week session is summarized below:

Week 1: Introduction to difference equations and modeling. We will also cover a summary of matrices and eigenvalues and their importance in the stability of a dynamical system. Students will be given examples and assignments to better understand these concepts.

Week 2: Develop a single species competition model with dispersal. Analyze the model to find the equilibria and determine their stability and interpret the results. We will then introduce MATLAB/Python for the numerical simulation of the model. Students will then simulate the model and interpret the results.

Week 3: Introduction to Beamer for presentation. The group will discuss the analytical and numerical results and begin writing the draft. The manuscript will be reviewed by the mentor, and students will make the necessary changes before submission.

References
  1. Allen, Linda J. S. An Introduction to Mathematical Biology. Pearson Prentice Hall, 2007.
  2. Elaydi, Saber N., and Jim M. Cushing. Discrete Mathematical Models in Population Biology: Ecological, Epidemic, and Evolutionary Dynamics. Springer Nature, 2025.

Digit Representations of Fractals and Self-Affine Carpets

Research Proposal — Virtual Math Research Circle

Session
Session 2: July 13, 2026 – Aug 1, 2026
Mentor
Manisha Garg
6th Year Graduate Student
Department of Mathematics
University of Illinois, Urbana-Champaign

Project Title
Digit Representations of Fractals and Self-Affine Carpets
Topic Area
Fractal Geometry, Number Theory, Discrete Geometry

Background
This project assumes familiarity with algebra, basic geometry, and mathematical reasoning. Knowledge of sequences and basic exponents is helpful but not required. All necessary background on iterative constructions, geometric series, and proof techniques will be introduced during the program.

Optional computational components will use Python; no prior programming experience is required.

Abstract
Many geometric objects in nature—coastlines, snowflakes, and branching trees—appear too irregular to be described using traditional notions of length and area. Fractals provide a mathematical framework for understanding such shapes. Even more surprisingly, many fractals can be described using simple rules about digits in number systems.

For instance, the classical Cantor set consists precisely of those real numbers whose base-3 expansion avoids the digit 1. The Sierpiński carpet can be described using restrictions on pairs of base-3 digits. In this project, we will uncover how seemingly intricate geometric objects arise from surprisingly simple arithmetic rules.

We will begin by constructing the Cantor set, Sierpiński carpet, and Menger sponge through iterative processes, and then translate these constructions into precise digit-based characterizations. Students will derive and prove criteria determining exactly which points belong to these sets, and compute how length, area, or volume evolves at each stage of iteration. Through computational experiments, we will develop algorithms that measure the decay rate of area up to a given iteration n, leading students to formulate and test conjectures.

Finally, we will extend these ideas to a broader class of self-affine carpets. Depending on interest and time, we may explore how combinatorial digit restrictions influence geometric and topological properties such as area, local connectedness, and fractal dimension.

The project combines proof-writing, number theory, geometry, and computational experimentation, culminating in a colloquium-style presentation (and possibly, a short paper).

Possible Extension
After deriving the decay rates of classical and self-affine carpets, students can explore additional geometric and metric properties. Possible directions include estimating box-counting or Hausdorff dimension, studying connectivity and symmetry properties, analyzing intersections of fractals with lines, or comparing different carpets using computational invariants.

Students will combine theoretical reasoning with computational experiments to formulate and justify new conjectures. They may choose either to focus on a single class of carpets and investigate multiple properties, or to fix a particular property and compare how it behaves across different families of carpets.

Outline/Timeline
Week 1 (Foundations: Self-Similarity and Iteration).
We introduce fractals and the concept of self-similarity through the construction of the chosen fractals. Students will study the base-3 digit representation of the Cantor set and investigate properties such as connectedness. We will derive formulas for length, area, and volume at stage n, connecting combinatorial growth to geometric decay. Students will learn basic LaTeX typesetting; on the first Monday we will collaboratively draft a project plan document (including individual goals and meeting times), and each Friday students will submit a brief written progress report.

Week 2 (Digit Criteria and Generalization).
Students will prove digit-based membership criteria for the Cantor set and Sierpiński carpet using inductive arguments from their iterative constructions. We will generalize these ideas to digit-restricted carpets in an m × n grid and derive formulas for area at stage n, analyzing exponential decay rates. Students will begin implementing Python algorithms to generate stage-n approximations and verify theoretical results. Computational experiments will guide conjectures relating digit patterns to geometric properties such as decay rate and connectedness.

Week 3 (Synthesis and Research Exploration).
We synthesize theoretical and computational results, refine conjectures, and complete selected proofs. Students will compare different digit-restricted constructions and analyze how combinatorial parameters influence geometric behavior. Depending on progress, we will introduce self-affine carpets as a preview of further research directions. The final week focuses on drafting a polished written report and preparing a 30–45 minute Beamer presentation so that students have ample time to rehearse and discuss future directions.

References
  1. Falconer, Kenneth. Fractal Geometry: Mathematical Foundations and Applications, 2nd ed., Wiley, 2003.
  2. Barnsley, Michael. Fractals Everywhere, 2nd ed., Academic Press, 1993.

Comparative Analysis of SIR Models for Epidemic Spread and Containment Strategies

Research Proposal — Virtual Math Research Circle

Session
Session 2: July 13, 2026 – August 1, 2026
Mentor
Jacob Kapita
Ph.D. Student
Department of Mathematics
Louisiana State University

Project Title
Comparative Analysis of SIR Models for Epidemic Spread and Containment Strategies
Topic Area
Biostatistics, Probability Theory, Data Science

Background
Basic arithmetic and algebra skills are sufficient for this project. Differential calculus and coding are not required, though they may be helpful. Necessary skills will be introduced as the project progresses.

This project provides an introduction to probability theory, biostatistics, and data analysis. Python in Google Colab will be used to generate results, and LaTeX Beamer in Overleaf will be used to prepare the final presentation.

Abstract
Mathematical models play a crucial role in understanding the dynamics of infectious disease transmission and in guiding public health interventions. Among these models, the Susceptible–Infected–Recovered (SIR) model is one of the most widely used frameworks for studying epidemic spread.

The classical SIR model divides the population into three compartments—susceptible, infected, and recovered—and describes the transitions between these groups using systems of differential equations. However, real-world epidemics often involve additional complexities such as vaccination, quarantine measures, behavioral changes, and time-varying transmission rates.

In this project, students will conduct a comparative analysis of different SIR-based models to evaluate their effectiveness in describing epidemic spread and assessing the impact of containment strategies. Through mathematical analysis and numerical simulations, students will investigate how different model structures influence epidemic dynamics and the predicted outcomes of intervention strategies.

Possible Extension
The project may be extended to investigate stochastic SIR models, allowing for random variations in infection and recovery processes that occur in real epidemic scenarios.

Outline/Timeline
Week 1 (Background and Model Formulation).
Review the basic concepts of infectious disease modeling and study the classical SIR model. This includes examining the governing system of differential equations, understanding its assumptions, and introducing key epidemiological concepts such as the basic reproduction number. Relevant literature on SIR models will also be reviewed.

Week 2 (Model Extensions and Simulations).
Study extensions of the classical SIR model that incorporate containment strategies such as vaccination, quarantine, and time-varying transmission rates. Implement numerical simulations in Python and explore epidemic dynamics under different parameter values.

Week 3 (Analysis, Results, and Presentation).
Analyze and compare results obtained from the different SIR models. Investigate how containment strategies influence epidemic characteristics such as peak infection levels and epidemic duration. Prepare the final report and presentation using LaTeX Beamer.

References
  1. Kermack, W. O., and McKendrick, A. G. A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society A, Vol. 115, 1927.
  2. Hethcote, H. W. The Mathematics of Infectious Diseases. SIAM Review, 42(4), 2000.
  3. Keeling, M. J., and Rohani, P. Modeling Infectious Diseases in Humans and Animals. Princeton University Press, 2008.
  4. Bird, A. A Simple Introduction to Epidemiological Modelling. King’s College London, 2020.
  5. Misra, S. Mathematical Modeling of Infectious Disease Spread Using the SIR Model. Biomedical Journal of Scientific & Technical Research, 2024.

Coupling Times of Random Walks on Graphs: A Computational Study

Research Proposal — Virtual Math Research Circle

Session
Session 2: July 13, 2026 – August 1, 2026
Mentor
Sixian Liu
Graduate Research, Ph.D. Student
Department of Applied Mathematics and Statistics
Stony Brook University

Project Title
Coupling Times of Random Walks on Graphs: A Computational Study
Topic Area
Probability Theory, Markov Chains, Graph Theory

Background
This project assumes only basic familiarity with algebra and arithmetic. All additional concepts and tools required to study the problem will be introduced during the program. The project provides an accessible introduction to probability theory, graph theory, and Markov chains. During the experimental stage, students will implement simulations using computational tools such as Python. In preparing the final presentation, students may use tools such as PowerPoint, Jupyter Notebook, and/or Overleaf.

Abstract
We study random walks on graphs, which provide a simple example of a Markov chain. A graph consists of nodes connected by edges. A random walk describes a process in which a walker moves from its current node to one of its neighboring nodes at random.

This process forms a Markov chain, meaning that the next position of the walker depends only on its current location and not on the past path it has taken. At each step, the walker selects its next move by choosing uniformly among the neighboring nodes. In other words, if a node has d neighbors, each neighbor is chosen with probability 1/d. The randomness can be generated by drawing a number from the uniform distribution
U ∼ Uniform(0, 1).

An interesting phenomenon occurs when two walkers move on the same graph using the same sequence of random numbers: although they may start at different nodes, they may eventually meet and then follow the same path thereafter.

A central question is how the structure of the graph influences the behavior of the random walk over time. In this project we will compare random walks on several types of graphs, such as line graphs, cycle graphs, and complete graphs. Students will simulate these random walks using computational tools such as Python and observe how the structure of the graph influences the time until the walkers meet.

Possible Extension
First, we may consider variations of the random walk, such as a lazy random walk, in which the walker stays at its current node with some probability before moving to a neighboring node. Second, we may simulate multiple walkers moving simultaneously on the graph and study their interaction over time. Finally, we may explore how the meeting time changes as the size of the graph increases.

Outline/Timeline
This is a tentative timeline.

Week 1 (Introduction to Graphs and Random Walks).
During the first week, students will be introduced to the basic concepts of graphs, such as nodes, edges, and node degrees. We will learn how graphs can represent networks and how movement on a graph can be modeled mathematically. Students will explore simple examples of graphs and visualize them using diagrams. We will then introduce the idea of a random walk, where a walker moves from its current node to one of its neighboring nodes at random. This process provides a simple example of a Markov chain. At each step, the walker chooses a neighboring node with equal probability, which can be generated using a random number from the uniform distribution
U ∼ Uniform(0, 1).

We will also discuss the interesting phenomenon that two walkers starting at different nodes may eventually meet and then follow the same path when driven by the same sequence of random numbers.

Week 2 (Simulation and Computational Experiments).
In the second week, students will implement simulations of random walks on graphs using computational tools such as Python. They will construct several graphs, such as line graphs, cycle graphs, and complete graphs. Students will simulate two walkers starting from different nodes and use the same sequence of random numbers to generate their steps. By running multiple trials, students will record how long it takes for the two walkers to meet. These experiments will help illustrate how the structure of a graph influences the behavior of the Markov chain and the time until coupling occurs. Students will analyze and visualize their results using simple plots and summary statistics.

Week 3 (Analysis, Extensions, and Final Presentation).
In the final week, students will analyze the results from their simulations and summarize their findings. We will discuss how the structure of a graph influences the meeting time of two random walkers and how this relates to the mixing behavior of the Markov chain. Students will explore possible extensions of the model, such as lazy random walks or multiple walkers on the same graph. Finally, students will prepare a short written report and presentation describing their simulations, results, and observations.

References
  1. West, D. B. Introduction to Graph Theory, 2nd ed., Pearson Education, Inc., 2001.
  2. Williamson, D. P. ORIE 6334 Spectral Graph Theory: Lecture 11 – Random Walks on Graphs, Cornell University, 2016.
  3. Klappenecker, A. Coupling of Markov Chains, Texas A&M University, Lecture Notes, 2018.

Exploring Balanced Incomplete Block Designs: Properties, Symmetry, and Applications

Research Proposal — Virtual Math Research Circle

Session
Session 2: July 13, 2026 – Aug 1, 2026
Mentor
Lázaro Aguiar
Math PhD Student
Tulane University

Project Title
Exploring Balanced Incomplete Block Designs: Properties, Symmetry, and Applications
Topic Area
Combinatorics

Background
This project is designed to be self-contained; all necessary material will be introduced as we progress. High school mathematics is more than enough.

Coding experience in Python is helpful but not required; basic coding concepts will be introduced if needed.

Abstract
Combinatorial design theory focuses on organizing elements of a finite set into smaller groups that satisfy specific conditions. A particular case is balanced incomplete block designs (BIBDs), whose existence has been studied since the 18th century. A well-known example is Kirkman’s Schoolgirl Problem, which asks how to arrange a group of students into rows each day so that no two students walk together more than once in a week. This problem is, in fact, a specific instance of BIBDs, structures with applications in various fields.

The formal study of design theory emerged in the 20th century, driven by its use in statistical experimental design, particularly by Ronald Fisher. Today, combinatorial designs have applications in cryptography, tournament scheduling, network design, mathematical biology, and more. This project is structured into three weeks: the first introduces fundamental concepts of BIBDs, the second explores a special class known as symmetric designs, and the third examines structural relationships between designs, including isomorphisms and connections to other mathematical structures.

Possible Extension
Students may explore applications of this topic in coding theory, where combinatorial designs, including BIBDs, play a crucial role in constructing error-correcting codes. These codes are essential for ensuring reliable data transmission in communication systems by detecting and correcting errors. Additionally, combinatorial designs are useful in cryptography, particularly in the construction of secure encryption schemes. Further exploration could involve studying connections between BIBDs and graph theory, leading to advancements in network reliability and data security.

Outline/Timeline
This is a tentative timeline that we will adjust as needed throughout the duration of the project.

Week 1: We will begin with an overview of the fundamental tools required, such as combinatorial counting, matrices, and related concepts. Next, we will introduce the basic definitions, aiming to derive key properties of these fascinating structures from scratch. This approach will help students become familiar with rigorous mathematical proofs.

Week 2: We will explore symmetric designs and their properties, along with Hadamard matrices, projective and affine planes, and residual and derived BIBDs.

Week 3: We will study BIBD automorphisms, difference sets, and the Multiplier Theorem. The week will conclude with final project presentations.

References
  1. Stinson, D. R. Combinatorial Designs: Constructions and Analysis. Springer, 2004.
  2. Wan, Z. X. Design Theory. Higher Education Press and World Scientific, 2009.

Questions

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