Virtual Math Circle — 2024 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 10, 2024 – Jun 29, 2024

Random Walks on Graphs

Virtual Math Circle Research Proposal

Session
Session 1: Jun 10, 2024 – Jun 29, 2024
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 (functions, polynomials) and comfort with mathematical notation (set notation, proof structure) are helpful. Some induction may be used and will be explained during the project. Prior exposure to combinatorics, graph theory, and discrete probability is encouraged but not required. We will also use Python or R (mentee’s preference) for empirical experiments and algorithm testing. Final results will be prepared using PowerPoint or Beamer (via Overleaf).

Abstract
If you randomly wander around Manhattan, what are the chances you return to the starting point? How many shuffles does it take to randomize a deck of cards? These questions connect to random walks on graphs—a concept central to mathematics, computer science, and the natural sciences. Although the idea is simple (move along edges from vertex to vertex), it yields rigorous answers to deep questions.

In this project, we will learn the basic tools that connect graph structure and probability theory, test theoretical results empirically, and use code to generate hypotheses about system behavior. By the end, students will understand key concepts in probability and graph theory and practice testing mathematical ideas via coding.

Possible Extension
Potential directions include: formalizing “how many shuffles randomize a deck?” [1]; extending from finite to infinite graphs and studying return probabilities [3].

Outline/Timeline
Week 1: Graph theory fundamentals; core concepts in probability. Read Sections 0–1 and parts of Sections 2 and 6 of Lovász [2]. Set up Python/R and practice basic coding skills.

Week 2: Continue Sections 2 and 6 of Lovász, including Theorem 6.6. Build simulations to explore its parameters.

Week 3: Visualize simulation results and compare against theory. Discuss limits of simulation-based validation. Prepare final presentation.

References
  1. David Aldous and Persi Diaconis, Shuffling Cards and Stopping Times, The American Mathematical Monthly, 93(5):333–348, 1986.
  2. László Lovász, Random Walks on Graphs, in Combinatorics, Paul Erdős is Eighty, 2:1–46, 1993.
  3. Georg Pólya, Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz, Mathematische Annalen, 84(1–2):149–160, 1921.

Nodal Sets on a Square Membrane

Virtual Math Circle Research Proposal

Session
Session 1: Jun 10, 2024 – Jun 29, 2024
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
Students should know derivatives of \(\sin x\) and \(\cos x\); otherwise, strong algebra and arithmetic foundations suffice. We will introduce all techniques needed. The project connects analysis, geometry, and number theory.

Abstract
A Dirichlet–Laplacian eigenfunction is proportional to its second derivative and vanishes on the boundary. On \([0,\pi]\), these eigenfunctions are \(f_n(x)=\sin(nx)\) for \(n=1,2,3,\dots\). They model phenomena in acoustics, data science, and quantum mechanics. Points where eigenfunctions vanish partition the domain into nodal domains.

In 1924, Antonie Stern constructed eigenfunctions on the square with exactly two nodal domains. We will strengthen this by constructing a sequence with exactly \(m\) nodal domains for any even \(m=4,6,8,\dots\), and we’ll conjecture about the odd case.

Possible Extension
Explore Neumann–Laplacian eigenfunctions (derivative zero on the boundary) and whether analogous sequences exist. Consider replacing the square by a flat torus, where less is known.

Outline/Timeline
Week 1 (Background): Review derivative properties, study eigenfunction examples, and learn techniques such as Stern’s checkerboard method, separation lemma, and critical-zero analysis [1,2].

Week 2 (Construct Sequences): Use Stern’s techniques to construct eigenfunctions with four nodal domains; analyze superposition effects and extend to all even counts; visualize with Desmos; examine limits when seeking an odd count.

Week 3 (Compile Results): Write up in LaTeX (with figures in MATLAB if time permits). Prepare slides in PowerPoint or Beamer.

References
  1. P. Bérard and B. Helffer, Nodal Sets of Eigenfunctions, Antonie Stern’s Results Revisited, Séminaire de théorie spectrale et géométrie, 32 (2014–2015), 1–22.
  2. J. Stewart, Calculus: Early Transcendentals, 6th ed., Thomson Learning, 2008.
Remark. The instructor is available during either summer session, but not both.

From the Ballot Problem to Catalan Numbers and Their Variations

Virtual Math Circle Research Proposal

Session
Session 1: Jun 10, 2024 – Jun 29, 2024
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 is recommended but not required and will be introduced as needed. Coding and implementation (Matlab or Python) will be taught in class.

Abstract
Suppose candidates \(A\) and \(B\) are in an election and votes are counted sequentially. If \(a>b\), \(A\) is elected. What is the probability that \(A\) stays ahead of \(B\) throughout? This classical Ballot Problem connects to the Catalan numbers, which have numerous applications in computer science and beyond.

We will learn to compute Catalan numbers and apply them to real problems, including programming to compute variations.

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 the Catalan numbers; set up Matlab and run examples.

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 a final presentation in LaTeX or PowerPoint.

References
  1. Hilton, P., & Pedersen, J., Catalan Numbers, Their Generalization, and Their Uses, The Mathematical Intelligencer 13 (1991), 64–75. https://doi.org/10.1007/BF03024089
  2. Selim, A., & Saračević, M., Catalan Numbers and Applications, 2019, 4, 99–114.

Session 2 · Jul 15, 2024 – Aug 3, 2024

Game Theory and the Best Strategy for a Game

Virtual Math Circle Research Proposal

Session
Session 2: Jul 15, 2024 – Aug 3, 2024
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 is recommended but not required. No prior background is assumed. Python coding and implementation will be taught.

Abstract
Game theory studies strategic interactions among rational decision-makers and has applications across social science, biology, and computer science. In this project, we will analyze optimal strategies and Nash equilibria for the computer game Buckshot Roulette. We will design an AI based on our best strategy and test it empirically.

Possible Extension
Analyze more complex games (e.g., Balatro) and build models to study game balance.

Outline/Timeline
Week 1: Introduction to game theory; study the rules of the chosen game; set up the Python environment.

Week 2: Develop and implement an AI strategy; run tests and iterate.

Week 3: Visualize and analyze results; prepare the final presentation in LaTeX or PowerPoint.

References
  1. Guardiola, E., & Natkin, S., Game Theory and Video Games: A New Approach to Analyze and Design Game Systems, CGAMES’05, 2005, 166–170.
  2. Mark Taylor, Mike Baskett, Denis Reilly, & Somasundaram Ravindran, Game Theory for Computer Games Design, Games and Culture, 14(7–8), 843–855 (2019).

Study of Infinity

Virtual Math Circle Research Proposal

Session
Session 2: Jul 15, 2024 – Aug 3, 2024
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 is sufficient; lectures are self-contained, with no advanced prerequisites assumed.

Abstract
We will explore countable vs. uncountable sets, Cantor’s diagonal argument, and the Continuum Hypothesis. Along the way, we will study the Cantor set and function, the Sierpiński carpet, and the Menger sponge, examining properties and applications.

Possible Extension
Investigate foundational questions related to the Continuum Hypothesis, the Axiom of Choice, and alternative set-theoretic frameworks; consider philosophical implications.

Outline/Timeline
Week 1: Sets, power sets, cardinality, types of infinity, Cantor’s diagonal, CH, construction of the Cantor set and function, and applications.

Week 2: Approaches to CH; variants of the Cantor construction and their properties.

Week 3: Further exploration of Cantor set/function; prepare the final presentation.

References
  1. A. Shen & N. K. Vereshchagin, Basic Set Theory, American Mathematical Society, 2002.

Questions

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