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.

To register for one of the research topics, simply complete the registration form.

College Credit

We are excited to announce that Math Circle has now been given approval 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

A Mathematical Model to Mitigate the Spread of Environmentally Transmissible Diseases

Virtual Math Circle Research Proposal

Session

Session 1: Jun 9, 2025 – Jun 28, 2025

Mentor

Hemaho Beaugard Taboe
Ph.D. Candidate in Mathematical Biology
Department of Mathematics
University of Florida

Project Title

A Mathematical Model to Mitigate the Spread of Environmentally Transmissible Diseases

Topic Area

Mathematical Biology, Epidemiology

Background

High school calculus and first-order differential equations skills will be useful. These concepts will be taught or revised at the beginning of the project.

Abstract

Mathematical models provide guidance on controlling communicable diseases. In recent years, many scientific papers have focused on epidemic models, particularly due to COVID-19. However, one often overlooked aspect of these studies is the possible transmission of pathogens via surfaces, which we refer to as environmental transmission.

The objective of this study is to develop a basic mathematical model with logistic recruitment into the uninfected (susceptible) population that incorporates transmission from the environment to humans. We will apply this framework to the well-known environmentally transmissible disease, Lassa Fever virus, on which I have published research. Certain model parameters will be sourced from existing literature, while others will be estimated by calibrating the model to weekly confirmed data from my publications. Additionally, we will analyze various control strategies such as surface disinfection, mask-wearing, and treatment of infected persons to determine scenarios that may help mitigate disease spread.

Possible Extension

This project may be extended by incorporating an additional group of individuals who recover from the infection but can contract the disease again after an immunity period. Furthermore, the extended version will investigate the impact of vaccination in the presence of environmental contamination.

Outline/Timeline

Week 1: Introduction to the project, revision of calculus and ordinary differential equations, tutorial on Matlab for solving differential equations, and Overleaf for research documentation.

Week 2: Learning compartmental modeling in epidemiology, refining the proposed model, computing basic reproduction numbers, and estimating parameters from literature and data fitting.

Week 3: Model simulation, answering research questions, assisting students with manuscript writing, and preparing the final research report.

References

1. Taboe, Hemaho B., Sergei S. Pilyugin, and Calistus N. Ngonghala. "Resolve Lassa Fever Persistence: A Compartmental Model with Environmental Virus-Host-Vector Interaction." (2024). Click here
2. Nagle, R. Kent, et al. Fundamentals of differential equations and boundary value problems. New York: Addison-Wesley, 1996.
3. Ngonghala, C. N., Taboe, H. B., Safdar, S., & Gumel, A. B. (2023). "Unraveling the dynamics of the Omicron and Delta variants of the 2019 coronavirus in the presence of vaccination, mask usage, and antiviral treatment." Applied Mathematical Modelling, 114, 447-465. Read more

Exploring the Distance Between Prime Numbers

Virtual Math Circle Research Proposal

Session

Session 1: Jun 9, 2025 – Jun 28, 2025

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.

Graphs of Commutative Rings

Virtual Math Circle Research Proposal

Session

Session 1: Jun 9, 2025 – Jun 28, 2025

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. Some familiarity with ring theory and/or graph theory is helpful but not necessary, as these concepts will be introduced during the first week.

Abstract

Algebra is a rich field, and one area of significant interest is commutative rings. A commutative ring is a set equipped with addition, subtraction, and multiplication, but not necessarily division. The most common example of a commutative ring is the set of integers. Some rings contain elements that multiply to zero without being zero themselves; these are known as zero-divisors.

A recent application of commutative rings within graph theory is the study of zero-divisor graphs. In these graphs, the elements of a ring serve as vertices, with edges connecting the zero-divisors. This project will introduce students to existing research on zero-divisor graphs, apply computational techniques to analyze examples, and explore their structural properties.

Possible Extension

An extension of this research could involve adapting the concept of zero-divisor graphs to complemented rings. The primary goal would be to modify the zero-divisor graph structure to include complemented elements. Students interested in pursuing this direction can reach out for further discussion.

Outline/Timeline

Week 1: Introduction to rings and fundamental concepts in graph theory, including notation, terminology, and examples.

Week 2: Exploration of zero-divisor graphs and related graph structures, including a review of recent research findings.

Week 3: Computational analysis of graphs derived from small commutative rings, with potential algorithmic implementation for automated graph generation (if time permits).

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, Commutative Algebra. Springer.

Extensions of Colley's Matrix and Ranking Methods

Virtual Math Circle Research Proposal

Session

Session 1: Jun 9, 2025 – Jun 28, 2025

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.

Linear Optimization and the Simplex Method

Virtual Math Circle Research Proposal

Session

Session 1: Jun 9, 2025 – Jun 28, 2025

Mentor

Van Le
Ph.D. Student
Department of Operations Research
North Carolina State University

Project Title

Linear Optimization and the Simplex Method

Topic Area

Linear Optimization, Linear Programming, Applied Mathematics

Background

Required: High school algebra and basic problem-solving skills.
Recommended: Some familiarity with matrix operations, coordinate geometry, and systems of equations.
Optional: Coding experience in Python is helpful but not required (basic coding concepts will be introduced if necessary).

Abstract

Optimization plays a crucial role in various fields such as economics, logistics, artificial intelligence, and engineering. This project introduces students to the Simplex Method, a powerful algorithm used to solve linear programming problems. We will explore the fundamental concepts of linear programming, understand how constraints and objectives define feasible solutions, and implement the Simplex Method to optimize real-world problems.

During the three-week summer session, students will learn the mathematical foundations of optimization, explore the geometric interpretation of linear programming, and apply the Simplex Method to practical problems. Through hands-on problem-solving and algorithm implementation, students will gain an understanding of how mathematical optimization influences decision-making.

Students interested in continuing their research will have the opportunity to extend the project into the following academic year, culminating in a poster presentation at LSU Discover Day.

Possible Extensions

Duality in Linear Programming: Understanding how primal and dual problems relate and exploring economic interpretations.

Real-World Applications: Investigating optimization problems in transportation, finance, and supply chain management.

Outline/Timeline

Week 1: Foundations of Linear Programming
- Introduction to optimization and real-world applications.
- Understanding constraints, objective functions, and feasible regions.
- Graphical representation of two-variable linear programming problems.
- Introduction to matrices and systems of linear equations.

Week 2: The Simplex Method
- Formulating standard linear programs.
- Understanding the structure of the Simplex tableau.
- Implementing the Simplex algorithm step by step.
- Solving optimization problems using Python.

Week 3: Applications and Final Presentations
- Real-world optimization case studies.
- Sensitivity analysis: understanding changes in constraints.
- Final project presentations and discussions.

References

1. Bertsimas, Dimitris & Tsitsiklis, John (1997). Introduction to Linear Optimization. Athena Scientific, 1st Edition.

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.

A Combinatorial Interpretation of a Matroid

Virtual Math Circle Research Proposal

Session

Session 1: Jun 9, 2025 – Jun 28, 2025

Mentor

Emmanuel Asante
Ph.D. Student
Department of Mathematics
Louisiana State University

Project Title

A Combinatorial Interpretation of a Matroid

Topic Area

Combinatorics and Geometry

Background

An understanding of basic high school algebra and arithmetic is enough to approach this project. High school calculus is not a prerequisite. The instructor will teach necessary concepts such as matroids, partially ordered sets, and abstract simplicial complexes.

Abstract

Although many are familiar with concepts such as vector spaces and graphs, very few know of their common generalization, matroids. A most versatile concept, matroids appear in both combinatorics and topology as we’ll see in the course of this project. This makes it a great tool for investigating the relationship between combinatorial and topological objects.

Given a linear ordering on the elements of a matroid, we define a broken circuit complex whose geometric realization happens to be a polyhedral complex. We will then make use of Phillip Hall’s theorem to draw an interesting relationship between the characteristic polynomial of the matroid and the faces of the broken circuit complex.

Possible Extension

The lattice of flats of a matroid and the intersection poset of a hyperplane arrangement are both geometric lattices. The project can be extended by studying whether the results obtained in this research replicate themselves in the case of hyperplane arrangements.

Outline/Timeline

Week 1 (Background): Familiarize with concepts of matroids, abstract simplicial complexes, and partially ordered sets (POSETS).

Week 2 (Examples and Main Results): Apply tools to test the main results through examples and write a rigorous proof of the theorem.

Week 3 (Finalizing Results and Presentation): Finalize results and prepare the final presentation using Beamer.

References

1. Stanley, Richard P. “An introduction to hyperplane arrangements.” Geometric Combinatorics 13.389-496 (2004).
2. Kozlov, Dimitry. Combinatorial Algebraic Topology. Vol. 21. Springer Science Business Media, 2007.

Session 2: Mon, July 14, 2025 – Sat, Aug 2, 2025

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

Modeling COVID-19 with Comorbidities in the USA

Virtual Math Circle Research Proposal

Session

Session 2: Jul 14, 2025 – Aug 2, 2025

Mentor

Hemaho Beaugard Taboe
Ph.D. Student in Mathematical Biology
Department of Mathematics
University of Florida, Gainesville, FL 32611, USA

Project Title

Modeling COVID-19 with Comorbidities in the USA

Topic Area

Mathematical Biology, Epidemiology

Background

High school calculus and first-order differential equations will be useful. These concepts will be taught or revised at the beginning of the project.

Abstract

The unprecedented COVID-19 pandemic continues to unfold with unimaginable consequences, despite the implementation of stringent control measures since its initial outbreak in China in 2019. The burden of the disease, in terms of its associated death toll, is higher in the United States compared to other parts of the world. One of the factors cited in the literature to explain this phenomenon is the prevalence of preexisting health conditions among Americans.

The objective of this study is to develop a deterministic basic COVID-19 model that incorporates patients who concurrently have COVID-19 and either Cancer or Diabetes (Comorbidity), as well as those who are solely positive for COVID-19. We will utilize this framework to assess the magnitude of COVID-19-related deaths during the primary wave of the delta variant in the US. Certain parameters of the model will be sourced from existing literature, while the remainder will be estimated by calibrating the model to the daily COVID-19 death data. Additionally, various control strategies (such as vaccination and mask-wearing) will be analyzed to determine the scenarios in which the burden may be alleviated.

Possible Extension

This project may be extended to countries such as China, France, and Nigeria for comparison purposes. The model will remain the same, but parameters may vary for each country.

Outline/Timeline

Week 1: Introduction to the project, revision of calculus and ordinary differential equations, tutorial on Matlab for solving differential equations, and Overleaf for research documentation.

Week 2: Learning compartmental modeling in epidemiology, refining the proposed model, computing basic reproduction numbers, and estimating parameters from literature and data fitting.

Week 3: Model simulation, answering research questions, assisting students with manuscript writing, and preparing the final research report.

References

1. Chatterjee, Sayan, et al. "Association of COVID-19 with comorbidities: an update." ACS Pharmacology & Translational Science 6.3 (2023): 334-354. Read here
2. Nagle, R. Kent, et al. Fundamentals of differential equations and boundary value problems. New York: Addison-Wesley, 1996.
3. Eikenberry, Steffen E., et al. "To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic." Infectious Disease Modelling 5 (2020): 293-308.
4. Ngonghala, C. N., Taboe, H. B., Safdar, S., & Gumel, A. B. (2023). "Unraveling the dynamics of the Omicron and Delta variants of the 2019 coronavirus in the presence of vaccination, mask usage, and antiviral treatment." Applied Mathematical Modelling, 114, 447-465. Read more

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 Linear Regression: From Theory to Application

Virtual Math Circle Research Proposal

Session

Session 2: Jul 14, 2025 – Aug 1, 2025

Mentor

Van Le
Ph.D. Student
Department of Operations Research
North Carolina State University

Project Title

Exploring Linear Regression: From Theory to Application

Topic Area

Statistics, Probability, Machine Learning

Background

Recommended: Familiarity with high school algebra, basic probability concepts, and functions.
Optional: Some exposure to statistics will be helpful but not required. Basic coding experience (Python or R) is encouraged but will be introduced as needed.

Abstract

Linear regression is a fundamental statistical technique used to model relationships between variables. It has applications in various fields, including economics, science, and artificial intelligence. In this project, students will explore both simple and multiple linear regression models used in causal inference. They will learn how to formulate regression equations, interpret coefficients, evaluate model fit, and apply their understanding to real-world datasets.

The project will involve hands-on computational work, where students use Python or R to analyze data, visualize trends, and make predictions. By the end of the session, students will gain a solid foundation in regression analysis and be able to apply these concepts to independent research problems.

Additionally, students interested in further exploration will have the opportunity to extend their research into the academic year, culminating in a poster presentation at LSU Discover Day.

Possible Extension

Students can explore advanced topics such as polynomial regression and logistic regression. Another possible extension is applying regression models to a self-selected dataset, drawing insights, and validating findings with real-world applications.

Outline/Timeline

Week 1: Foundations of Linear Regression
- Understanding simple linear regression: assumptions, model interpretation, and visualization.
- Hands-on coding exercises using Python or R.

Week 2: Foundations of Multiple Linear Regression
- Evaluating model performance using statistical metrics and interpreting results.
- Feature selection and model tuning.
- Data-driven applications with real-world datasets.

Week 3: Applications and Final Presentations
- Regression analysis on a dataset of choice.
- Summarizing findings and preparing presentations.
- Final project presentations.

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:

• Developing new irreducibility criteria similar to Schönemann–Eisenstein–Dumas-Type Irreducibility.
• Applying numerical methods to solve polynomials in alternative number systems like p-adic fields (reference).
• Exploring algebraic geometry techniques for solving polynomial systems with multiple variables.
• Reviewing Finding Roots of Polynomials in p-adic Fields for a deeper insight into alternative root-finding techniques.

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.

Questions?

Contact Isaac Michael <imichael@lsu.edu>.