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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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.