Virtual Math Research Circle — 2026 Spring Research Projects

Virtual Math Research Circle — 2026 Spring 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

Session 1 · Jan 12, 2026 – Feb 21, 2026

Extensions of Colley's Matrix and Ranking Methods

Research Proposal — Virtual Math Research Circle

Session
Session 1: Jan 12, 2026 – Feb 21, 2026
Mentor
Jonathan Engle
PhD Student
Department of Mathematics
Florida State University

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

Background
This research project requires only basic algebra and arithmetic. All necessary linear algebra concepts will be taught by the instructor. This project serves as an excellent introduction to linear algebra, introductory computer science, statistics, or modeling courses. During the computational phase, we will use tools such as Microsoft Excel, Matlab, and Julia. For the final presentation, students will use PowerPoint and/or Beamer.

Abstract
Colley's matrix method for ranking college football teams is an exciting application of linear algebra. Traditional ranking methods rely on team records, margin of victory, or subjective committee decisions, which may not always produce accurate rankings. Colley's method improves upon this by adjusting team rankings based on strength of schedule without considering conference bias or margin of victory.

Over three weeks, we will replicate Colley's Matrix for a small example using recent college football data. We will analyze these rankings to determine whether Colley's method predicted upsets that the College Football Committee overlooked. This implementation requires solving a linear system:

C * r = b, where C is Colley's matrix, r is the ranking vector, and b is one plus the average win rate for a team. This leads to the research question: how can Colley's Matrix be improved with additional information?

Possible Extension
Beyond sports, students will explore applications of ranking methods in other fields, such as resource distribution, natural disaster response, and stock rankings.

Outline/Timeline
Weeks 1-2: Introduction to traditional ranking methods and how linearization can improve rankings. Basic concepts of matrices, vectors, matrix operations, and inverses. Literature review on Colley's method.

Weeks 3-4: Computational experiments: implementing Colley's matrix in Matlab, Julia, or another preferred coding language. Data mining and extracting elements for Colley's matrix. Comparing Colley rankings with College Football Committee rankings and actual results. Students will modify Colley's method to include additional parameters and compare their results.

Weeks 5-6: Refining the modified ranking methods, analyzing differences, and preparing final results. Constructing and practicing presentations using PowerPoint, Google Slides, or Beamer.

References
  1. Boginski, V., Butenko, S., & Pardalos, P. M. (2004). Matrix-based methods for sports rankings: A survey.
  2. Colley, W. N. (2002). Colley's Bias-Free College Football Ranking Method.

Predicting Future Events Using Markov Chains and Machine Learning

Research Proposal — Virtual Math Research Circle

Session
Session 1: Jan 12, 2026 – Feb 21, 2026
Mentor
Jacob Kapita
PhD Student
Department of Mathematics
Louisiana State University

Project Title
Predicting Future Events Using Markov Chains and Machine Learning
Topic Area
Stochastic Probability Theory, Machine Learning, Data Science

Background
Basic arithmetic and algebra skills are sufficient. Coding is not required, but it may be helpful. Necessary skills will be taught as the project progresses. This project serves as an excellent introduction to probability theory, stochastic analysis, data analysis, and machine learning. We will use Python in Google Colab and create the final presentation using LaTeX Beamer in Overleaf.

Abstract
Modeling real-life events is essential across many disciplines, especially when predicting the likelihood of future random outcomes. Markov Chains offer a powerful approximation tool because future states depend only on the present state, not on past events—making the process memory-less. In this project, we focus on discrete-time Markov Chains, using next-word prediction as our primary example. We will develop machine learning models grounded in the Markov property, where transition probabilities between words estimate the most likely next word. This demonstrates how Markovian behavior applies to real-world predictive tasks. For the final presentation, students will choose an application area in which to apply the concepts learned.

Possible Extension
The project may be extended to continuous-time Markov Chains, particularly in life insurance (actuarial mathematics). We may also explore Markov Chain inference, such as determining the probability of being in a particular state after a number of steps.

Outline/Timeline
A bi-weekly outline is provided below; adjustments may be made as necessary.

Weeks 1–2 (Background):
We will begin with the fundamentals of probability theory, focusing on probability laws, independence, and conditional probability. After building this foundation, we will introduce Markov Chains and the Markov property. These weeks will conclude with an exploration of matrix operations—particularly matrix multiplication—followed by the construction and analysis of transition matrices and introductory examples of Markov Chains.

Weeks 3–4 (Markov Chains in Machine Learning):
Students will be introduced to Python using Google Colab. We will then discuss key machine learning concepts and highlight how Markov Chain principles are applied in predictive modeling. Using real-world text data, students will build a next-word prediction model based on transition probabilities.

Weeks 5–6 (Finalizing Results and Presentation):
Students will learn to create professional slides using Beamer in Overleaf. During this period, we will finalize the results, refine visualizations, and assemble the final presentation for the research showcase.

References
  1. Fewster, Rachel. Chapter 8: Markov Chains, Lecture Notes for Stats 325, Stochastic Processes, 2014. Available at https://www.stat.auckland.ac.nz/~fewster/325/notes/ch8.pdf
  2. Patel, Vatsal. Markov Chain Explained, Built In, 2022. Available at https://builtin.com/machine-learning/markov-chain
  3. Verma, Yugesh. A Guide to Markov Chain and its Applications in Machine Learning, 2021. Available at https://analyticsindiamag.com/deep-tech/a-guide-to-markov-chain-and-its-applications-in-machine-learning/

Session 2 · Feb 24, 2026 – Apr 4, 2026

Modeling Cooperation

Research Proposal — Virtual Math Research Circle

Session
Session 2: Feb 24, 2026 – Apr 4, 2026
Mentor
James Branca
Ph.D. Student in Biomathematics
Department of Mathematics
Florida State University

Project Title
Modeling Cooperation
Topic Area
Game Theory

Background
No calculus needed. Knowledge of set theory will be helpful. Necessary concepts will be taught along the way.

Abstract
In 1951, John Nash published one of his seminal papers on game theory, helping to formalize how rational players make decisions in competitive situations. A parallel line of work, cooperative game theory, focuses on how groups (coalitions) form and how the benefits of cooperation are divided among participants.

In this project we will study simple cooperative games, where a coalition either “wins” or “loses” depending on which players are included. A classical example is a voting game with 101 participants in which coalitions of 51 or more people win and coalitions with 50 or fewer people lose. Such models raise natural questions: Which players are most “powerful”? How should we measure influence or fairness in a voting system?

Goals for this project are to learn the basic ideas of cooperative game theory, work through concrete examples (especially simple games and voting games), and see how these tools are used to model real-world situations involving alliances, negotiations, and shared decision-making.

Possible Extension
There are many possible extensions of this research. An interested student may, for example, investigate more sophisticated power indices, explore applications to real voting bodies (such as councils or committees), or examine how changing the rules (such as the winning threshold) affects the distribution of power. Students who wish to pursue an extension should reach out to discuss options.

Outline/Timeline
The schedule below is designed for a 6-week Session 2 program and may be adjusted as needed based on student progress and interests.

Week 1: Introduction to game theory.
• Overview of game theory and the distinction between cooperative and noncooperative games.
• Work through small examples of strategic and cooperative situations and discuss basic terminology (players, coalitions, payoffs).

Week 2: Simple games and voting models.
• Introduce simple games formally and study majority voting games (e.g., 101 voters where coalitions of 51 or more win).
• Identify winning and losing coalitions in specific examples and discuss informal notions of power and fairness.

Week 3: Power and influence in cooperative games.
• Explore ideas of player “desirability” and relative influence using examples from the literature.
• Compute basic power measures for small voting games by hand and interpret the results.

Week 4: Applications and literature exploration.
• Read selected sections from introductory texts on cooperative game theory and simple games.
• Connect the theory to real-world voting systems or decision-making bodies chosen by the group.

Week 5: Computation and analysis.
• For selected games, systematically enumerate coalitions and compute power measures or related quantities (by hand and/or with light computational assistance).
• Begin drafting preliminary results, tables, and figures for the final presentation.

Week 6: Synthesis and presentation.
• Finalize calculations and interpretations; organize examples into a coherent narrative.
• Prepare and practice a Beamer or PowerPoint presentation explaining the model, examples, and what the group learned about cooperation and fairness.

References
  1. Curiel, I. Cooperative Game Theory and Applications, 1997.
  2. Taylor, A. and Zwicker, W. Simple Games: Desirability Relations, Trading, Pseudoweightings, 1999.

How Random is a Shuffle?

Research Proposal — Virtual Math Research Circle

Session
Session 2: Feb 24, 2026 – Apr 4, 2026
Mentor
Mae Lee
Graduate Teaching Assistant
Bridge-to-Doctorate Affiliate graduate student
Department of Mathematics
University of Texas at Arlington

Project Title
How Random is a Shuffle?
Topic Area
Combinatorics

Background
Students should be comfortable with basic algebra and arithmetic-style reasoning. All needed combinatorics concepts (permutations, simple counting arguments, etc.) will be introduced during the program. No prior exposure to combinatorics is assumed.

Abstract
It is your turn to shuffle and deal a deck of cards for a game. Everyone wants the deck to be “well shuffled,” but what does that actually mean? Is every possible order really “equally likely,” or are some orders still much more likely than others?

In this project, students will explore how to model a shuffle mathematically and how to measure “how random” the resulting deck order is. We will treat a deck as an ordered list (a permutation) and choose a specific shuffle rule—such as a simple riffle shuffle or a “top-in-at-random” shuffle—to study in detail.

Students will construct small examples, compute how a single shuffle moves the deck from one order to another, and then investigate what happens after many shuffles. They will design and analyze measures of “closeness to random” and use these to ask questions like: How many shuffles are needed before the deck looks “close to random”? The project will culminate in a research-style presentation explaining the models, experiments, and findings.

Possible Extension
As a possible extension, students may compare multiple shuffle rules (for example, different riffle-style shuffles or a “cut and interleave” rule), studying how fast each rule appears to mix the deck. They may also investigate how mixing behavior changes when the deck size changes (e.g., 10 cards vs. 52 cards), using a combination of exact calculations for very small decks and larger-scale computer simulations for bigger decks.

Outline/Timeline
This is a tentative schedule for the 6-week Session 2 program and may be adjusted as needed as the session progresses.

Week 1 (Cards and permutations).
• Introduce decks as ordered lists and review permutations and basic counting ideas.
• Discuss what “random” should mean in this context and brainstorm simple tests for randomness (for example, looking at positions of certain cards or simple statistics on the order).

Week 2 (Modeling a shuffle).
• Choose a specific shuffle rule and describe precisely how it acts on the deck.
• For a very small deck (e.g., 3–4 cards), build a transition table or diagram that shows how one shuffle moves between different deck orders.

Week 3 (Repeated shuffles and small-deck analysis).
• Use the transition table to study what happens after several shuffles of a small deck.
• Track how the distribution over deck orders changes with the number of shuffles and begin to define simple quantitative measures of “closeness to random.”

Week 4 (Scaling up: larger decks and simulations).
• Move from hand calculations to computer simulation for larger decks.
• Students write simple code (or use prepared notebooks) to simulate repeated shuffles, collect data, and visualize how quickly the deck appears to mix under their chosen rule.

Week 5 (Analyzing and comparing shuffle rules).
• Analyze simulation results, looking for patterns in how measures of randomness evolve with the number of shuffles.
• If time permits, compare two different shuffle rules and discuss which one appears to mix faster and why.

Week 6 (Prepare presentation).
• Students organize their definitions, models, and experimental results into a coherent story about “how random” their shuffle is.
• Prepare a final Beamer or PowerPoint presentation (and/or short write-up) explaining how many shuffles are needed before the deck looks “close to random” under their tests.

References
  1. Joy Morris, Combinatorics: An Upper-Level Introduction to Combinatorics and Design Theory, Version 2.1.1, University of Lethbridge, 2023.

Graphs of Commutative Rings

Research Proposal — Virtual Math Research Circle

Session
Session 2: Feb 24, 2026 – Apr 4, 2026
Mentor
James Branca
Ph.D. Student in Biomathematics
Department of Mathematics
Florida State University

Project Title
Graphs of Commutative Rings
Topic Area
Algebra, Graph Theory

Background
Knowledge of calculus is not required for this project. Some familiarity with algebra and mathematical notation is helpful, but all necessary concepts will be taught during the first week.

Abstract
Algebra is a very rich subject, and one large area of interest is the study of commutative rings. In such a ring, it can happen that the product of two nonzero elements is zero; elements that do this are called zero-divisors.

A recent application of commutative rings within graph theory is the construction of graphs whose vertices come from elements (or special subsets) of the ring and where adjacency is determined by a zero-divisor relationship. These constructions link algebraic properties of the ring to structural properties of the resulting graph.

In this project, students will study graphs associated with commutative rings, explore examples, prove basic results from the literature, and apply computation to build and analyze specific zero-divisor graphs.

Possible Extension
A possible extension of this research is to apply these ideas to other families of rings, or to related constructions such as graphs built from ideals or other special elements. Depending on interest and time, students may explore variations that appear in current research papers and compare how different graph constructions capture the algebraic structure of a ring.

Outline/Timeline
Weeks 1-2: Introduction to rings and basic ring theory. We will review operations in rings, standard examples, and key terminology. All new terms will be thoroughly explained and illustrated with many examples.

Weeks 3-4: The concept of the zero-divisor and the zero-divisor graph. We will introduce standard definitions, look at important examples, and read portions of the research literature, discussing how algebraic conditions show up in the associated graphs.

Weeks 5-6: Computing graphs by hand for small or familiar rings (such as \(\mathbb{Z}_n\)) and, if time permits, beginning to create or adapt an algorithm to generate zero-divisor graphs more systematically. Students will analyze the resulting graphs and relate their observations back to the ring structure.

References
  1. Beck, I. Coloring of commutative rings, J. Algebra 116 (1988), 208–226.
  2. Anderson, D., Axtell, M., & Stickles, J. Zero-divisor graphs in commutative rings, in Commutative Algebra, Springer.

Questions

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