Virtual Math Research Circle — 2026 Summer Research Projects

Math Circle is a virtual math research camp for high school students.

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.

Registration Pending 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 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.

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

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
Louisiana State University
Department of Mathematics

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.

Questions

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