Virtual Math Circle — 2023 Summer Research Projects

Virtual Math Circle — 2023 Summer Research Projects

Overview

Our Virtual Math Circle research sessions bring motivated high-school students together with university mentors to explore 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 · Jun 12, 2023 – Jul 1, 2023

Lagrange Multipliers in Portfolio Optimization with Mixed Constraints

Virtual Math Circle Research Proposal

Session
Session 1: Jun 12, 2023 – Jul 1, 2023
Mentor
Minh Vu
PhD Student
Department of Mathematics
Louisiana State University

Project Title
Lagrange Multipliers in Portfolio Optimization with Mixed Constraints
Topic Area
Optimization

Background
This project requires a basic understanding of calculus (derivatives of polynomials to determine extrema) and solid high school algebra. Additional calculus concepts will be introduced as needed.

Abstract
Portfolio optimization seeks the best mix of assets subject to goals and constraints. We will use the method of Lagrange multipliers to build an optimal two-stock portfolio with known expected returns and standard deviations (uncorrelated assets, nonzero variance). We formulate a nonlinear constrained optimization problem to maximize expected return while controlling portfolio risk under budget and risk constraints, and we study how modifying the risk limit affects the optimal allocation. The project develops fluency with Lagrange multipliers and their practical application to portfolio design.

Possible Extension
Extend to multi-asset portfolios to study diversification, or add practical constraints (e.g., sector constraints) and compare solutions.

Outline/Timeline
Week 1: Review derivatives, extrema, and Lagrange multipliers; introduce mixed equality/inequality constraints via slack variables; guided exercises.

Week 2: Introduce portfolio optimization concepts; define expected return and risk for two-asset portfolios; set parameters and solve constrained problems under alternative risk limits; interpret solutions.

Week 3: Finalize results, write the report, and prepare LaTeX slides.

References
  1. Z. Bodie, A. Kane, A. J. Marcus, Investments, 10th ed., McGraw-Hill, 2013 (Ch. 7).
  2. F. Fischer, The Method of Lagrange Multipliers, QMUL lecture notes.
  3. J. Stewart, Calculus: Early Transcendentals, 8th ed., Cengage, 2015 (Ch. 3.1, 4.1, 14.8).
  4. R. J. Vanderbei, Linear Programming: Foundations and Extensions, 3rd ed., Springer, 2008 (Ch. 2).

Mathematical Modeling of the Transmission of Multi-strain Infectious Diseases

Virtual Math Circle Research Proposal

Session
Session 1: Jun 12, 2023 – Jul 1, 2023
Mentor
Dr. Olusegun Michael Otunuga
Assistant Professor of Mathematics
Department of Mathematics
Augusta University

Project Title
Mathematical Modeling of the Transmission of Multi-strain Infectious Diseases
Topic Area
Mathematical Biology; Epidemiology

Background
High school calculus and introductory differential equations are helpful; key concepts will be taught or reviewed as needed.

Abstract
We develop a multi-strain susceptible–vaccinated–exposed asymptomatic–symptomatic–recovered (S-V-Ea-Es-R) model to analyze breakthrough and rebound infections. The population is initially susceptible to n variants; vaccination or recovery confers immunity to a strain and its predecessors but not necessarily to newer variants. We visualize transmission in Simulink (MATLAB), estimate epidemiological parameters for COVID-19 variants in the U.S., and compute the basic reproduction number to assess endemicity and variant coexistence.

Possible Extension
Incorporate stochastic or seasonally varying transmission rates to reflect environmental perturbations.

Outline/Timeline
Week 1: Meet, curate readings, review calculus/ODEs, introduce numerical methods and MATLAB/ODE45; short reports and exercises.

Week 2: Collect/clean data; design and analyze the compartmental model; parameter estimation workflow.

Week 3: Complete analysis and estimation; write manuscript in LaTeX (Overleaf).

References
  1. S. Nagle, E. Saff, A. Snider, Fundamentals of Differential Equations, 9th ed., Pearson, 2018.
  2. O. M. Otunuga, “Analysis of multi-strain infection of vaccinated and recovered population…,” PLOS ONE 17(7): e0271446, 2022.
  3. O. M. Otunuga, “Stochastic modeling and forecasting of COVID-19 deaths…,” Acta Biotheoretica 70:25, 2022.
  4. P. van den Driessche, J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria…,” Math. Biosci. 180:29-48, 2002.

Walking to Infinity Along Primes

Virtual Math Circle Research Proposal

Session
Session 1: Jun 12, 2023 – Jul 1, 2023
Mentor
Bencheng Li
Graduate Student
Department of Mathematics
Georgia Institute of Technology

Project Title
Walking to Infinity Along Primes
Topic Area
Elementary Number Theory and Probability Theory

Background
Elementary algebra and sequences; number theory/probability will be introduced as needed. We will use Python or R for simulations and Beamer/PowerPoint for presentation.

Abstract
Starting from a prime (e.g., 3), append a single digit to the right to obtain another prime (31, 311, …). How long can such a “prime walk” continue? We design algorithms to compute longest walks from a given starting prime and compare across starts. Extensions consider appending multiple digits at a time and whether this enables walks to infinity. The project is inspired by results from the 2020 Polymath REU.

Possible Extension
Investigate analogous walks along other sequences (e.g., squares), and adapt code to simulate and compare their behavior.

Outline/Timeline
Week 1: Primes, sequences, Dirichlet’s theorem (overview); set up coding environment and basics in Python/R.

Week 2: Implement algorithm to compute longest one-digit prime walks from a given start; compare across starts.

Week 3: Explore bounded multi-digit appends; finalize results and present in Beamer/PowerPoint.

References
  1. S. J. Miller et al., “Walking to Infinity Along Some Number Theory Sequences,” arXiv:2010.14932.

Session 2 · Jul 17, 2023 – Aug 5, 2023

An Introduction to Stochastics in Physics

Virtual Math Circle Research Proposal

Session
Session 2: Jul 17, 2023 – Aug 5, 2023
Mentor
Moises Gomez Solis
PhD Student
Department of Mathematics
Louisiana State University

Project Title
An Introduction to Stochastics in Physics
Topic Area
Applied Mathematics, Stochastic Processes, Mathematical Physics

Background
High school algebra and basic coding; we introduce elementary probability, linear algebra, and Python for Markov Chain Monte Carlo (MCMC). Presentation tools: PowerPoint/LaTeX.

Abstract
We study discrete-time stochastic dynamics via recursions of the form $X_{n+1}=f(X_n, Z_{n+1})$. As a model problem from statistical mechanics, consider $N$ particles distributed between compartments A and B, with random single-particle moves between compartments: $X_{n+1}=X_n+Z_{n+1}$, where $Z_{n+1}\in\{-1,1\}$. We explore hitting/cover times, equilibrium behavior, and connections to thermodynamic intuition, supported by simulation.

Possible Extension
Simulate diffusion processes and compare with limiting continuous models.

Outline/Timeline
Week 1: Probability and linear-algebra refreshers; set up tools; minimal proof technique as needed.

Week 2: Numerical experiments and visualization of stochastic models.

Week 3: Validate models and code; prepare final presentation (PowerPoint/LaTeX).

References
  1. P. Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, Springer, 1999.
  2. F. Mendivil, R. W. Shonkwiler, Explorations in Monte Carlo Methods, Springer, 2009.

Walking to Infinity Along Squares

Virtual Math Circle Research Proposal

Session
Session 2: Jul 17, 2023 – Aug 5, 2023
Mentor
Bencheng Li
Graduate Student
Department of Mathematics
Georgia Institute of Technology

Project Title
Walking to Infinity Along Squares
Topic Area
Elementary Number Theory and Probability Theory

Background
Elementary algebra and sequences; we will introduce needed number theory/probability. Python or R used for simulations; final slides in Beamer/PowerPoint.

Abstract
Starting from a square $x^2$, append digits to the right and ask whether the resulting number is again a perfect square (e.g., $1 \to 16 \to 169$). How long can such a “square walk” last? We devise algorithms to compute longest walks from a given $x^2$ and compare across starts. We also explore appending multiple digits at a time and whether infinite walks are possible, adapting ideas from the Polymath REU study of prime walks.

Possible Extension
Compare behavior with prime-walk algorithms; investigate other sequences and constraints; study density heuristics empirically.

Outline/Timeline
Week 1: Squares, sequences, key background; set up coding environment and basics in Python/R.

Week 2: Implement longest one-digit square-walk search; benchmark starts; analyze obstacles and pruning rules.

Week 3: Explore bounded multi-digit appends; finalize results and prepare Beamer/PowerPoint presentation.

References
  1. S. J. Miller et al., “Walking to Infinity Along Some Number Theory Sequences,” arXiv:2010.14932.

Questions

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