Current Circle projects.

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.

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.

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.

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.

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.

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.

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.

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.

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:



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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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



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.

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.

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.

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.

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.

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.

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.

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

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,



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.

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.

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.

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.

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.

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.

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.

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.