Virtual Math Circle: Summer 2025 Research Projects

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About

Math Circle is a virtual summer camp for high school students. You can read more about how the program works.

Cost & Registration

The cost per session for 2025 is \$1,200 with a deposit of \$300 due at the time of registration.

UPDATE: Registration is now closed for VMC 2025.

Financial Assistance

We are pleased to announce that Math Circle has been awarded the 2024 AMS Epsilon Grant to provide financial assistance to students in the form of full and partial fee reductions! Students who wish to be considered for financial assistance can do so at registration.

College Credit

We are excited to announce that Math Circle has now been approved to offer math credit - one hour of choice college credit* for Math 1999 - to high school students who want to continue their research project after the summer and present at LSU Discover Day in April 2026. Participants have the option to present virtually or in person.

*Choice College Credit: Students may opt to receive regular LSU credit if satisfied with their final grade.

Session 1: Mon, Jun 9, 2025 – Sat, Jun 28, 2025 (SESSION NOW CLOSED)

Multi-players Ballot and 3D Catalan Numbers

Virtual Math Circle Research Proposal

Session
Session 1: Jun 9, 2025 – Jun 28, 2025
Mentor
Dr. Zequn Zheng
Postdoctoral Researcher
Department of Mathematics
Louisiana State University

Project Title
Multi-players Ballot and 3D Catalan Numbers
Topic Area
Combinatorial Mathematics, Discrete Mathematics

High School Calculus
Not a prerequisite
Background
Basic combinatorics is recommended, but not required and will be introduced when needed. No particular background knowledge is required.

Abstract
Suppose three candidates, A, B, and C, are competing in an election. The votes are counted sequentially. If candidate A receives a votes while candidate B receives b votes and candidate C receives c votes with a > b > c, A is elected. What is the probability that A stays ahead of B and B stays ahead of C throughout the election? This is a generalization of the Ballot Problem dating back to the nineteenth century. The answer to this question is related to the generalization of a family of special integers named by the French-Belgian mathematician Eugène Charles Catalan. This number has numerous applications in computer science and other areas.

In this research project, we will learn how to calculate Catalan numbers and how to apply those numbers to solve some real-life problems. In addition, we will also write code for computing Catalan numbers and their variations.

Possible Extension
We can generalize to dimension m Catalan numbers, and proving its formula will be of interest.

Outline/Timeline
Week 1: Introduction to basic combinatorics and definition of Catalan numbers. Learn a proof for the formula of Catalan numbers.

Week 2: Continue learning more versions of proof for Catalan numbers. Try to compute 3D Catalan numbers.

Week 3: Visualizing and summarizing our results. Try to develop a formula for the 3D Catalan number. Preparation of the final presentation using either LaTeX or PowerPoint.

References
  1. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995.
  2. Hilton, P., & Pedersen, J. (1991). Catalan Numbers, Their Generalization, and Their Uses. The Mathematical Intelligencer, 13, 64–75. https://doi.org/10.1007/BF03024089.
  3. Selim, Aybeyan, & Saračević, Muzafer. (2019). Catalan Numbers and Applications, 4, 99-114.

Session 2: Mon, July 14, 2025 – Sat, Aug 2, 2025 (SESSION NOW CLOSED)

Extensions of Colley's Matrix and Ranking Methods

Virtual Math Circle Research Proposal

Session
Session 1: Jun 9, 2025 – Jun 28, 2025 (Rescheduled to Session 2)
Mentor
Jonathan Engle
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
Week 1: 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.

Week 2: 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.

Week 3: 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.

Exploring the Distance Between Prime Numbers

Virtual Math Circle Research Proposal

Session
Session 1: Jun 9, 2025 – Jun 28, 2025 (Rescheduled to Session 2)
Mentor
Miraj Samarakkody
Assistant Professor
Tougaloo College, MS

Project Title
Exploring the Distance Between Prime Numbers
Topic Area
Number Theory, Computational Mathematics

Background
High school algebra (functions, exponentials, and logarithms) is required. Some experience with coding (preferably Python) is beneficial, but not mandatory. High school calculus is not required.

Abstract
Prime numbers play a crucial role in number theory, yet their distribution remains an open problem in mathematics. This project will explore the spacing between consecutive prime numbers, analyzing patterns in their occurrence. We will examine famous conjectures such as the Twin Prime Conjecture and Cramér’s Conjecture, comparing theoretical predictions with computational data.

Using programming tools like Python, students will generate large sets of prime numbers, measure the gaps between them, and visualize these patterns through graphs and histograms. The project provides hands-on experience in mathematical research and computational data analysis while fostering an appreciation for unsolved problems in mathematics.

Students interested in continuing beyond the summer will have opportunities to extend the research into a formal project culminating in a poster presentation at LSU Discover Day.

Possible Extension
Prime Gaps and Probabilistic Models: Investigate whether prime gaps follow statistical distributions.

Computational Approaches to Twin Primes: Implement different algorithms to test the Twin Prime Conjecture for larger primes.

Graph Theory Connections: Explore prime numbers through graph structures, such as constructing prime distance graphs.

Outline/Timeline
Week 1: Introduction to prime numbers, famous conjectures, Python basics, and generating prime numbers.

Week 2: Computational analysis of prime gaps, statistical visualization, and comparison with conjectures.

Week 3: Investigating prime gap patterns, discussing open questions, presenting research findings, and brainstorming future research directions.

References
  1. Apostol, T. M. (1976). Introduction to Analytic Number Theory. Springer.
  2. Hardy, G. H., & Wright, E. M. (2008). An Introduction to the Theory of Numbers. Oxford University Press.
  3. Riemann, B. (1859). On the Number of Primes Less Than a Given Magnitude.
  4. Online resources such as the OEIS (Online Encyclopedia of Integer Sequences) and Prime Number Theorem resources.

Predicting Future Events Using Markov Chains and Machine Learning

Virtual Math Circle Research Proposal

Session
Session 2: Jul 14, 2025 – Aug 2, 2025
Mentor
Jacob Kapita
Ph.D. Student
Louisiana State University
Department of Mathematics

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

Background
Required: Basic arithmetic and algebra skills.
Optional: Coding experience is not required but can be helpful. All 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 for computations and LaTeX's Beamer in Overleaf for presentations.

Abstract
Modeling real-life events is vital in various disciplines. Sometimes, we aim to predict the likelihood of future random events. Markov Chains provide a good approximation based on the principle that future events depend only on the present state and not on past states—this property makes Markov Chains "memory-less."

This project will focus on discrete-time Markov Chains. Initially, we will predict the next word in a sentence using machine learning models built on the Markov property. By the end of the project, students will gain experience in probability modeling and its application to data-driven predictions.

Possible Extension
The project can be extended by exploring continuous-time Markov Chains in the field of life insurance (Actuarial Mathematics). Additionally, we may investigate inference using Markov Chains to determine the probability of reaching a particular state within a set number of steps.

Outline/Timeline
Week 1: Introduction to probability theory, probability laws, independence, conditional probability, and Markov Chains. We will also cover matrix operations, the Markov Transition Matrix, and properties of Markov Chains.

Week 2: Introduction to Python in Google Colab, fundamentals of machine learning, and applications of Markov Chains in predictive modeling. We will model real-life data, predict future events, and analyze results using Python.

Week 3: Introduction to Beamer in Overleaf for presentations. Students will finalize their findings and prepare final presentations.

References
  1. Fewster, Rachel. Chapter 8: Markov Chains, in Lecture Notes for Stats 325, Stochastic Processes. 2014. Read here
  2. Patel, Vatsal. Markov Chain Explained, Built In, 2022. Read here
  3. Verma, Yugesh. A Guide to Markov Chain and its Applications in Machine Learning, 2021. Read here

Hyperbolic Functions and Their Applications in Engineering and Architecture

Virtual Math Circle Research Proposal

Session
Session 2: Jul 14, 2025 – Aug 2, 2025
Mentor
Miraj Samarakkody
Assistant Professor
Tougaloo College, MS

Project Title
Hyperbolic Functions and Their Applications in Engineering and Architecture
Topic Area
Differential Geometry, Mathematical Modeling

Background
Students should be familiar with high school algebra, geometry, and basic trigonometry. No prior knowledge of calculus or coding is required but will be introduced as needed.

Abstract
Hyperbolic functions play a fundamental role in engineering and architecture, particularly in the study of catenary curves and structural stability. The catenary, described by the hyperbolic cosine function, appears in suspension bridges, arches, and hanging cables, making it essential for engineers and architects.

This project will explore the mathematical properties of hyperbolic functions and their real-world applications, with a primary focus on structural design. Students will develop an understanding of hyperbolic functions through theoretical study, computational modeling, and hands-on physical experiments. We will analyze famous structures such as the Gateway Arch and the Golden Gate Bridge to understand how catenary curves optimize structural efficiency.

Additionally, participants will learn to use graphing tools and simple Python simulations to model these curves. The project will culminate in a final report and presentation, showcasing mathematical and computational models of real-world structures.

For interested students, this project may be extended into the following academic year, culminating in a research poster presentation at LSU Discover Day. Possible extensions include advanced analysis of stability in different structural designs or computational modeling of load distributions in suspension bridges.

Possible Extension
Advanced Stability Analysis: Investigate the stability of different catenary-based structures.

Computational Modeling: Explore engineering applications of hyperbolic functions using simulations.

Historical and Modern Uses: Research how catenary curves have been used in past and present architecture.

Outline/Timeline
Week 1: Introduction to hyperbolic functions, their properties, and graphing tools.

Week 2: Derivation of catenary equations, structural applications, and basic calculus introduction.

Week 3: Python simulations, physical modeling of catenary curves, and research report preparation.

References
  1. Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
  2. Do Carmo, M. P. (2016). Differential Geometry of Curves and Surfaces. Prentice Hall.
  3. Billington, D. P. (1985). The Tower and the Bridge: The New Art of Structural Engineering. Princeton University Press.
  4. Gordon, J. E. (1978). Structures: Or Why Things Don't Fall Down. Da Capo Press.

Modelling Biological Systems

Virtual Math Circle Research Proposal

Session
Session 2: Jul 14, 2025 – Aug 2, 2025
Mentor
James Branca
Ph.D. Student in Biomathematics
Department of Mathematics
Florida State University

Project Title
Modelling Biological Systems
Topic Area
Mathematical Biology

Background
Knowledge of calculus and differential equations is preferred but not necessary. All required concepts will be introduced during the first week.

Abstract
Many biological phenomena, such as the spread of disease, the human nervous system, and predator-prey interactions, are modeled using systems of first-order differential equations. Students will learn the fundamentals of bifurcation theory, including equilibrium stability analysis and classification of bifurcations, as well as numerical solving techniques.

The goal of this project is to explore key models in mathematical biology, analyzing their stability, potential bifurcations, and verifying results numerically. The biological significance of these models will be discussed throughout the project.

Possible Extension
A potential extension of this project could involve adding a delay or stochastic component to an existing model, significantly increasing its complexity. Another possible extension is the creation or modification of a mathematical model to better represent a biological phenomenon.

Outline/Timeline
Week 1: Review of calculus, ordinary differential equations (ODEs), and linear algebra. Introduction to analyzing linear and nonlinear systems and bifurcation theory.

Week 2: Analysis of key models in mathematical biology, including Population Dynamics, Lotka-Volterra, SIR/SIS epidemiological models, and Fitzhugh-Nagumo equations.

Week 3: Numerical simulations using MATLAB (or a free alternative if necessary). Simulating solutions with realistic parameter values and different initial conditions.

References
  1. Chou, C.S. & Friedman, A. (2016). Introduction to Mathematical Biology. Springer.
  2. Strogatz, S. (2015). Nonlinear Dynamics and Chaos. CRC Press.

Exploring Roots of Polynomials: A Journey Through Different Branches of Mathematics

Virtual Math Circle Research Proposal

Session
Session 2: Jul 14, 2025 – Aug 2, 2025
Mentor
Saayan Mukherjee
Graduate Student
Department of Mathematics
Oklahoma State University

Project Title
Exploring Roots of Polynomials: A Journey Through Different Branches of Mathematics
Topic Area
Analysis and Algebra

Background
Knowledge of basic high school mathematics is sufficient. Some familiarity with polynomials is ideal but not required. Lectures will be self-contained, and no advanced mathematics will be assumed.

Abstract
Solving equations is one of the most exciting parts of mathematics. While solving quadratic equations is straightforward with the quadratic formula, solving higher-degree polynomials (such as fifth-degree equations) presents unique challenges. Evariste Galois proved that no general formula exists for polynomials of degree five or higher using basic operations.

Mathematicians have developed various techniques for finding polynomial roots. Algebraic methods like the Rational Root Theorem and Eisenstein’s Criterion help determine whether a polynomial has rational roots. Sir Isaac Newton's iterative process provides an alternative numerical approach. When considering complex numbers, theorems such as Rouche’s Theorem and the Argument Principle allow us to locate roots without solving equations directly.

Additionally, matrices offer powerful tools for solving systems of equations, which are widely applied in fields like Artificial Intelligence, Machine Learning, and Quantum Computing. Exploring equations with multiple variables leads us to algebraic geometry, which studies geometric properties of polynomial solutions.

This project provides an interactive journey through algebra, analysis, geometry, and computer science, offering students hands-on experience in exploring different techniques for finding polynomial roots.

Possible Extension
Students may explore advanced topics such as:

Outline/Timeline
Week 1: Newton’s geometric approach, introduction to Galois’ theorem, algebraic methods for determining rational roots, and an introduction to matrices.

Week 2: Exploring functions on the complex plane, using analysis techniques to locate polynomial roots, and comparing results with real-number polynomials.

Week 3: A deep dive into a selected area of interest, working on research papers and problem-solving, followed by final presentation preparation.

References
  1. V. V. Prasolov. Polynomials. Springer Berlin, Heidelberg, 2004.
  2. "Finding Roots of Polynomials in p-adic Fields" (Read here).

How to Lie? Game Theory behind the Computer Game Liar's Bar

Virtual Math Circle Research Proposal

Session
Session 2: July 14, 2025 – Aug 2, 2025
Mentor
Dr. Zequn Zheng
Postdoctoral Researcher
Department of Mathematics
Louisiana State University

Project Title
How to Lie? Game Theory behind the Computer Game Liar's Bar
Topic Area
Game Theory, Computer Science, Discrete Mathematics

High School Calculus
Not a prerequisite
Background
Basic linear algebra is recommended but not required and will be introduced when needed. No particular background knowledge is required. Python coding and implementation will be taught in the class.

Abstract
Liar's Bar is a famous computer game with more than 46,000 likes on Steam. Players must make statements—some true, some false—while determining which statements are lies and which are truths. Game theory, an interdisciplinary field studying strategic interactions among rational decision-makers, provides a framework for analyzing behavior in situations where outcomes depend not only on individual choices but also on the choices of others.

In this research project, we will use game theory to analyze the best strategy for winning in Liar's Bar. We will explore optimal strategies, Nash equilibrium, and AI-based decision-making. Ultimately, we will design an AI based on our strategy and test it against real players or large language models like ChatGPT or Llama.

Possible Extension
Liar's Bar has multiple game modes. We will start with the easiest one and analyze other variations if time permits.

Outline/Timeline
Week 1: Introduction to game theory and the rules of the easiest game. Setup computer environment for Python coding.

Week 2: Develop an AI strategy to maximize success in the game. Implement and test the strategy through coding experiments.

Week 3: Visualize and analyze results. Test AI performance against real players or large language models like ChatGPT or Llama. Prepare the final presentation using either LaTeX or PowerPoint.

References
  1. Osborne, M. J. (2004). An Introduction to Game Theory. Oxford University Press.
  2. Guardiola, Emmanuel & Natkin, Stéphane. (2005). Game Theory and Video Games: A New Approach to Analyze and Design Game Systems. CGAMES'05, Int. Conf. on Computer Games, Angoulême, France, pp.166-170.
  3. Mark Taylor, Mike Baskett, Denis Reilly, & Somasundaram Ravindran. (2019). Game Theory for Computer Games Design. Games and Culture, 14(7-8), 843-855.

Choose a topic that excites you, and get ready to explore, discover, and present like a real mathematical researcher.

Questions?

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