Program Overview

Virtual Math Research Circle

The Virtual Math Research Circle is a program that brings motivated high-school students together with university mentors to pursue authentic, publishable mathematics.

Overview

The program is designed to provide students with the opportunity to work on real mathematical research projects, with the guidance of experienced mentors. The program consists of two sessions, each lasting three weeks.

Session 1

Session 1 will take place from June 10, 2024, to June 29, 2024. The following research projects will be offered:

• Random Walks on Graphs
  • Mentor: Nicolae Sapoval, Ph.D. Student, Department of Computer Science, Rice University
  • Project Title: Random Walks on Graphs
  • Topic Area: Graph Theory, Combinatorics, Probability Theory
  • Background: Familiarity with basic algebra and arithmetic, and comfort with mathematical notation
  • Abstract: This project explores the concept of random walks on graphs, which is central to mathematics, computer science, and the natural sciences
  • Possible Extension: Formalizing "how many shuffles randomize a deck?", extending from finite to infinite graphs and studying return probabilities
  • Outline/Timeline:
    • Week 1: Graph theory fundamentals, core concepts in probability
    • Week 2: Continue graph theory and probability, build simulations
    • Week 3: Visualize simulation results, compare against theory, prepare final presentation
• Nodal Sets on a Square Membrane
  • Mentor: Andrew Lyons, Ph.D. Student, Department of Mathematics, University of North Carolina, Chapel Hill
  • Project Title: Nodal Sets on a Square Membrane
  • Topic Area: Analysis and Geometry
  • Background: Strong algebra and arithmetic foundations, knowledge of derivatives of sinx and cosx
  • Abstract: This project explores the concept of nodal sets on a square membrane, which is related to Dirichlet-Laplacian eigenfunctions
  • Possible Extension: Explore Neumann-Laplacian eigenfunctions, replace the square with a flat torus
  • Outline/Timeline:
    • Week 1: Review derivative properties, study eigenfunction examples
    • Week 2: Construct eigenfunctions with four nodal domains, analyze superposition effects
    • Week 3: Write up results in LaTeX, prepare slides in PowerPoint or Beamer
• From the Ballot Problem to Catalan Numbers and Their Variations
  • Mentor: Dr. Zequn Zheng, Postdoctoral Researcher, Department of Mathematics, Louisiana State University
  • Project Title: From the Ballot Problem to Catalan Numbers and Their Variations
  • Topic Area: Combinatorial Mathematics, Discrete Mathematics
  • Background: Basic combinatorics, coding and implementation in Matlab or Python
  • Abstract: This project explores the connection between the Ballot Problem and Catalan numbers, which have numerous applications in computer science
  • Possible Extension: Explore variations and generalizations of Catalan numbers, derive formulas and write programs to compute them
  • Outline/Timeline:
    • Week 1: Introduction to combinatorics, definition and proofs for Catalan numbers
    • Week 2: Additional proofs and variations, implement a program to compute a Catalan-number variant
    • Week 3: Visualize and summarize results, attempt a formula for a selected variation, prepare final presentation

Session 2

Session 2 will take place from July 15, 2024, to August 3, 2024. The following research projects will be offered:

• Game Theory and the Best Strategy for a Game
  • Mentor: Dr. Zequn Zheng, Postdoctoral Researcher, Department of Mathematics, Louisiana State University
  • Project Title: Game Theory and the Best Strategy for a Game
  • Topic Area: Game Theory, Computer Science, Discrete Mathematics
  • Background: Basic linear algebra, no prior background assumed, Python coding and implementation
  • Abstract: This project explores the concept of game theory and the best strategy for a game, with applications in social science, biology, and computer science
  • Possible Extension: Analyze more complex games, build models to study game balance
  • Outline/Timeline:
    • Week 1: Introduction to game theory, study the rules of the chosen game
    • Week 2: Develop and implement an AI strategy, run tests and iterate
    • Week 3: Visualize and analyze results, prepare final presentation
• Study of Infinity
  • Mentor: Saayan Mukherjee, Ph.D. Student, Department of Mathematics, Oklahoma State University
  • Project Title: Study of Infinity
  • Topic Area: Set Theory
  • Background: Knowledge of basic high school mathematics, lectures are self-contained
  • Abstract: This project explores the concept of infinity, including countable vs. uncountable sets, Cantor's diagonal argument, and the Continuum Hypothesis
  • Possible Extension: Investigate foundational questions related to the Continuum Hypothesis, the Axiom of Choice, and alternative set-theoretic frameworks
  • Outline/Timeline:
    • Week 1: Sets, power sets, cardinality, types of infinity, Cantor's diagonal, CH
    • Week 2: Approaches to CH, variants of the Cantor construction and their properties
    • Week 3: Further exploration of Cantor set/function, prepare final presentation