Virtual Math Circle: Summer 2024 Research Projects
About
Math Circle is a virtual summer camp for high school students. You can read more about how the program works.
Project posters, papers, and presentations appear in the public archive of Math Circle projects.
Session 1: Mon, Jun 10, 2024 – Sat, Jun 29, 2024
Random Walks on Graphs [5/4 CLOSED]
Virtual Math Circle Research Proposal
 Session
 Session 1 (Jun 10 – Jun 29, 2024)
 Mentor
 Nicolae Sapoval
PhD Student
Department of Computer Science
Rice University  Project title
 Random Walks on Graphs
 Topic area
 Graph theory, Combinatorics, Probability Theory
 High school calculus
 Not required, some basic notions of limits will be introduced during the project.
 Skills and background
 This research project requires familiarity with basic algebra and arithmetic including notions of functions and polynomials, and ability to understand and operate basic mathematical formalism (set notation, understanding proof structure). Some induction might be required, but will be explained during the project. Familiarity with basic notions from combinatorics and graph theory is highly encouraged, but is not required and will be introduced by the mentor as needed. Basic understanding of discrete probability theory will be helpful, but key concepts will be explained. We will also use Python or R (mentees preference) to conduct some empirical experiments and algorithm testing. Results for the final presentation will be formalized using PowerPoint or Beamer via Overleaf (mentees preference).
 Abstract

If you randomly wander around Manhattan, what are the chances that you'll come back to the starting point? What if a bird decides to do the same, will the outcome be identical? How many shuffles does it take to get a random order in a deck of cards? All these questions connect to the fundamental concept of a random walk on a graph. Graphs have come to play a crucial role in many areas of mathematics, computer science, and natural sciences. The idea of a random walk on a graph can seem trivial at first, we simply keep wandering from one city to another along one of the possible roads. However, as we will see throughout this project this seemingly simple idea can help us give rigorous answers to these questions.
In this research project, we will learn the basic tools that allow us to talk about random walks on graphs and draw connections between the graph and probability theory. We will also empirically test some of the theoretical results and use code to generate hypotheses about the behavior of different systems. By the end of this project, students will gain familiarity with some key concepts in probability and graph theory, as well as some practice for testing algorithmic and mathematical ideas via coding. Finally, don't worry since with probability 1 we will make it back to our starting point!
 Possible extension
 There are several potential extensions to this project and the exact path will be determined by the student interest. A first potential extension can explore the formalization of the question ``how many shuffles will make a deck of cards random?'' [1]. Another possible extension consists of expanding our framework from finite to infinite graphs and focusing on the question of whether a random walk returns to its starting point [3].
 Outline/timeline

Week 1: Introduction to basic graph theory definitions and results. Introduction to basic concepts in probability theory. The aim is to be comfortable with Sections 01, and parts of Sections 2 and 6 of the review by Lovász [2]. Additionally setting up the Python/R environment and getting familiarity with basic coding skills relevant for the project.
Week 2: Continuing developing a working knowledge of parts of Sections 2 and 6 of Lovász. In particular, focusing on the proof of Theorem 6.6 and setting up a set of numerical simulations to computationally explore parameters in Theorem 6.6.
Week 3: Visualizing results of the simulations and comparing them with theoretical guarantees. Synthesizing an understanding of limitations of simulationbased validation and basic statistical analysis of simulated results. Preparation of the final presentation.  References References are given below.
 David Aldous and Persi Diaconis, Shuffling cards and stopping times , The American Mathematical Monthly, 93(5):333–348, 1986.
 László Lovász, Random walks on graphs , Combinatorics, Paul Erd˝os is eighty, 2(146):4, 1993.
 Georg Pólya, Über eine aufgabe der wahrscheinlichkeitsrechnung betreffend die irrfahrt im straßennetz , Mathematische Annalen, 84(12):149–160, 1921.
Nodal Sets on a Square Membrane [5/4 CLOSED]
Virtual Math Circle Research Proposal
 Session
 Session 1 (Jun 10 – Jun 29, 2024)
 Mentor
 Andrew Lyons
PhD Student
Department of Mathematics
University of North Carolina, Chapel Hill  Project title
 Nodal Sets on a Square Membrane
 Topic area
 Analysis and Geometry
 High school calculus
 Familiarity with derivatives is necessary.
 Skills and background
 This research project requires knowledge of derivatives. Namely, students should be familiar with derivatives of $\sin(x)$ and $\cos(x)$; otherwise, a healthy understanding of algebra and arithmetic is all that is necessary. The techniques to approach this problem will be introduced by the instructor. This project connects ideas in analysis, geometry, and number theory.
 Abstract

A DirichletLaplacian eigenfunction is a function that is proportional to its second derivative and equal to zero at the endpoints of its domain. On an interval $[0,\pi]$, all DirichletLaplacian eigenfunctions are known; they take the form:
\begin{equation}
f_n(x)=\sin(n x) \nonumber
\end{equation}
where $n=1,2,3,\dots$ These functions are known to model several phenomena in acoustics, data science, and even quantum mechanics! Of particular interest are the points where eigenfunctions are equal to zero. The collection of such points separates the domain into disjoint regions called nodal domains.
In $1924$, Antonie Stern constructed a sequence of DirichletLaplacian eigenfunctions on the square which have exactly $2$ nodal domains. Is this still possible if we demand more nodal domains? For this project, we strengthen Stern's result by constructing a sequence of DirichletLaplacian eigenfunctions on the square which have exactly $m$ nodal domains, for any choice of $m=4,6,8,\dots$ Students will also form a conjecture on whether this statement holds when $m$ is odd.
Upon student interest, the instructor is available for guidance in extending this research project through the $2024$$2025$ school year, culminating in a poster presentation at LSU Discovery Day.
Students will be given exercises to further preliminary understanding. They will also read about why eigenfunctions and nodal domains are worth studying from a physical perspective. By the end of this week, students should understand Stern's construction.
 Possible extension

A NeumannLaplacian eigenfuncton is a function that is proportional to its second derivative and whose first derivative is equal to zero at the endpoints of its domain. It does not seem possible to construct a sequence of NeumannLaplacian eigenfunctions on the square membrane with an equal number of nodal domains, although this has not yet been proven. Additionally, if we replace the square membrane with a flat torus, even less is known.
Remark. All of the functions discussed above, both on the square membrane and the flat torus, are explicit and can be written in terms of sine and cosine functions; this project requires no knowledge of differential equations.  Outline/timeline

A general timeline is below; suitable adjustments will be made as we progress.
Week 1 (Background): In the first week, students will review derivative properties, tackle eigenfunction examples, and be introduced to techniques in analyzing eigenfunctions [2]. Namely, students will learn about Stern’s checkerboard method, separation lemma, and critical zero analysis as in [1].
Students will be given exercises to further preliminary understanding. They will also read about why eigenfunctions and nodal domains are worth studying from a physical perspective. By the end of this week, students should understand Stern’s construction.
Week 2 (Construct Sequences): In the second week, students will use Stern’s techniques to construct a sequence of eigenfunctions with exactly 4 nodal domains. They will explore the impacts of eigenfunction multiplicity (superposition) on the number of nodal domains and determine a method for extending this result to all other even (positive) numbers.
Students will use accessible tools like Desmos to visualize the nodal domains and understand the role of domain symmetry. In particular, they will explore the limitations of Stern’s method when seeking an odd number of nodal domains.
Week 3 (Compile Results): In the final week, students will finalize their findings in a paper written in Latex (with typesetting assistance from the instructor). If time permits, figures will be generated in Matlab.
In the final days, students will construct a presentation (either PowerPoint or Beamer) and practice communicating their ideas. The final result should include a formal statement regarding the existence of eigenfunctions with an even number of nodal domains, a rigorous proof, and an informed conjecture in the odd case.  References All necessary resources/readings will be provided by the instructor. General references are given below.
 P. Bérard and B. Helffer, Nodal sets of eigenfunctions, Antonie Stern’s results revisited, Séminaire de th´eorie spectrale et géométrie, Volume 32 (20142015), pp. 122.
 J. Stewart, Calculus: Early Transcendentals, 6th ed., Thomson Learning, 2008.
From the Ballot Problem to Catalan Numbers and Their Variations [2/4]
Virtual Math Circle Research Proposal
 Session
 Session 1 (Jun 10 – 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.
 High school calculus
 Not a prerequisite.
 Skills and background
 Basic combinatorics is recommended, but is not required and will be introduced when needed. No particular background knowledge is required. Matlab Coding and Implementation will be taught in the class.
 Abstract

Suppose two candidates, A and B, are competing in an election. The votes are counted sequentially. If candidate A receives $a$ votes while candidate B receives $b$ votes with $a>b$. A is elected. What is the probability that A stay ahead of B throughout the whole election? This is called the Ballot Problem back to nineteenth century. The answer to this question is related to a family of special integers named by a FrenchBelgian 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 codes for computing Catalan numbers and their variations.
 Possible extension
 There are different variations and generalization to Catalan numbers. Formulas and programs to compute them will be of our interest.
 Outline/timeline

A general timeline is below; suitable adjustments will be made as we progress.
Week 1: Introduction to basic combinatorics and definition of Catalan number. Learn a proof for the Formula of Catalan number. Additionally setting up Matlab and run some basic examples.
Week 2: Continuing learning more versions of proof for Catalan numbers. Learn a variation of a Catalan number and try programming a code to calculate it.
Week 3: Visualizing and summarize our results. Try to develop a formula for a variation of Catalan number. Preparation of the final presentation with either Latex or PowerPoint.  References References are given below.
 Hilton, P. Pedersen, J, Catalan Numbers, Their Generalization, and Their Uses. <\cite> The Mathematical Intelligencer 13, 64–75 (1991). https://doi.org/10.1007/BF03024089.
 Selim, Aybeyan. Saračević, Muzafer, Catalan Numbers and Applications , 4. 99114(2019).
Session 2: Mon, Jul 15, 2024 – Sat, Aug 3, 2024
Game Theory and the best strategy for a game [6/4 CLOSED]
Virtual Math Circle Research Proposal
 Session
 Session 2 (Jul 15 – 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
 High school calculus
 Not a prerequisite.
 Skills and background
 Basic linear algebra is recommended, but is 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

Game theory is an interdisciplinary field studying strategic interactions among rational decisionmakers. Originating from mathematics and economics, game theory has many applications in diverse fields such as social science, biology, computer science, and more. Game theory provides a framework for analyzing the behavior of individuals in situations where the outcome of their actions depends not only on their own choices but also on the choices of others.
In this research project, we use game theory to analyze the best strategy for a computer game called. We will also analyze the Strategies and Nash equilibrium in this game. Then we design an AI based on our best strategy and test our strategy.
 Possible extension

There are many successful computer games. We could analyze some more complicated games like
and build a model to study its game balance.  Outline/timeline

A general timeline is below; suitable adjustments will be made as we progress.
Week 1: Introduction to Game theory and get familiar with the game that we analyze. Setup computer environment for writing python code
Week 2: Find out the best AI to win the game. Write a code to test our strategy.
Week 3: Visualizing and analyzing our results. Preparation of the final presentation with either Latex or PowerPoint.
 References References are given below.
 Guardiola, Emmanuel \& Natkin, Stéphane, Game Theory and video game, a new approach of game theory to analyze and conceive game systems, CGAMES'05, Int. Conf. on Computer Games, AngoulÃ¨me, France, pp.166170 (2005).
 Mark Taylor, Mike Baskett, Denis Reilly, and Somasundaram Ravindran, Game Theory for Computer Games Design, Games and Culture. Vol. 14(78) 843855 (2019).
Study of Infinity [4/4 CLOSED]
Virtual Math Circle Research Proposal
 Session
 Session 2 (Jul 15 – Aug 3, 2024)
 Mentor
 Saayan Mukherjee
PhD Student
Department of Mathematics
Oklahoma State University  Project title
 Study of Infinity
 Topic area
 Set Theory
 High school calculus
 Not a prerequisite.
 Skills and background
 Knowledge of basic high school mathematics is sufficient, some familiarity with Set Theory is ideal but not required. Lectures will be selfcontained, and no amount of advanced mathematics will be assumed.
 Abstract

Consider two sets, $A=\{1,2,3,4,5,6\}$ and $B=\{1,2,3,4,5\}$. A natural question to ask then is which set is bigger? Counting the number of elements, the answer is easy, the set $A$. The same question when asked for the sets $\bN=\{1,2,3,4,,5,6,7,\cdots \}$ and $K= \{2,3,4,,5,6,7,\cdots \}$ would have a different answer. It turns out both sets have the same number of elements although the set $\bN$ has the element $1$ which is missing in the set $K$. \textbf{WHY!!!} Notice that both $\bN$ and $K$ have infinitely many elements, and we will learn later that these types of infinite sets are called $\textbf{countably}$ infinite. An example of a set that is bigger than both the sets above is the set of all real numbers, usually denoted by $\bR$ which is an $\textbf{uncountably}$ infinite set. An intriguing question is to investigate whether there are sets that are smaller than $\bR$ but larger than $\bN$. This is famously known as the $\textbf{continuum}$ $\textbf{hypothesis}$.
In this fun course, we will learn about some of the rich research on the continuum hypothesis. Along the way, we will browse through the $\textbf{Cantor}$ $\textbf{set}$, $\textbf{Cantor}$ $\textbf{function}$, the \href{https://en.wikipedia.org/wiki/Sierpi%C5%84ski_carpet}{\textcolor{blue}{Sierpi{\' n}ski carpet}}, and the \href{https://en.wikipedia.org/wiki/Menger_sponge}{\textcolor{blue}{Menger sponge}}, see some of their exquisite properties, and some cool applications in different areas of mathematics.
After constructing the Cantor set we will also investigate all the possible changes in the construction of the Cantor set and examine all properties of the resulting set and their characteristics.
 Possible extension

We can investigate foundational questions in mathematics related to set theory, such as the continuum hypothesis, the axiom of choice, or the nature of mathematical truth. Explore alternative settheoretic axioms or foundational frameworks and analyze their consequences for the rest of mathematics. Consider philosophical implications and debates surrounding foundational issues in mathematics.
 Outline/timeline

A general timeline is below; suitable adjustments will be made as we progress.
Week 1: We will start with a historical development of set and function. Then we will start with the definition of a set, Power set, Cardinal number, various types of Infinity, Cantor's diagonal argument, the Continuum hypothesis, the Construction of the Cantor set, the Cantor function, and their nice applications.
Week 2: We will continue with week 1 material initially. Then we will study various nice and simple approaches to solving the continuum hypothesis and also investigate all the possible changes in the construction of the Cantor set and examine all properties of the resulting sets.
Week 3: We'll keep investigating the Cantor set and Cantor function. Then, in the last week, we'll begin preparing our final presentation.  References All necessary resources/readings will be provided by the instructor. General references are given below.
 A. Shen, N. K. Vereshchagin, Basic Set Theory , American Mathematical Society 2002
Counting Connected Regions of a Hyperplane Arrangement [6/4 CLOSED]
Virtual Math Circle Research Proposal
 Session
 Session 2 (Jul 15 – Aug 3, 2024)
 Mentor
 Emmanuel Asante
PhD Student
Department of Mathematics
Louisiana State University  Project title
 Counting Connected Regions of a Hyperplane Arrangement
 Topic area
 Combinatorics
 High school calculus
 Not a prerequisite.
 Skills and background
 An understanding of basic high school algebra and arithmetic is enough to approach this project. However, the instructor will introduce and teach some concepts such as hyperplanes and partially ordered sets to the students.
 Abstract

If you cut across a sheet of paper a finite number of times, you will be
left with two or more pieces of paper depending on the number of cuts you
you made and how the cuts you made meet or intersect with another.
How many pieces of paper do you have? Considering the real plane as a
sheet of paper of infinite area, and the cuts as straight lines on the plane,
how many regions does the lines divide the plane into and how many of
these regions are of finite area or bounded. What if weclosed divided the 3dimensional
space into two or more regions using planes?
In the case of the plane, the lines are the hyperplanes and in the case of the threedimensional space, the planes are the hyperplanes. By studying the intersections of the hyperplanes that divide the space, we are able to obtain a polynomial known as the characteristic polynomial which will will tell us how many regions the space is divided into and how many of them are of finite area.
 Possible extension

This project can be extended to provide a rigorous mathematical proof for the hyperplane arrangement of a general real ndimensional space.
 Outline/timeline

This is a weekly schedule of our activities during the period of our research.
Week 1 (Background): The first week will be allotted to introducing concepts such as hyperplanes. We will focus on hyperplanes of the spaces $\mathbb{R}, \mathbb{R}^{2},$ and $\mathbb{R}^{3}$ since they are easier to conceive geometrically. We will also introduce partial relations and define a relation on the set of hyperplanes and their intersections which will make it a partially ordered set (POSET).
Finally, we will learn how to draw the lattice that corresponds to the POSET of the hyperplanes and their intersections.
Week 2 (Examples and Main Results): Beginning with examples of hyperplane arrangements of the real number line $\mathbb{R}$ and $\mathbb{R}^{2}$ we will see how we can obtain the characteristic polynomial from the lattice of hyperplanes and their intersections. The characteristic polynomial is a single variable polynomial that will give us, upon substituting the single variable for a particular number, the total number of connected regions as well as how many of them are bounded.
We will then begin to look at some popular arrangements in $\mathbb{R}^{2}$ such as the Type A coordinate hyperplane arrangement and arrangement of lines in general position( but not with more than 5 lines). We conclude the week by obtaining a formula for counting the number of connected and bounded regions in these special hyperplane arrangements given an arbitrary number of hyperplanes.
Week 3 (Finalizing Results and Presentation): The final week will be spent finalizing the results from the last week, writing the report, and preparing for the presentation. The final presentation will be prepared using Beamer.
In the final days, students will construct a presentation (either PowerPoint or Beamer) and practice communicating their ideas. The final result should include a formal statement regarding the existence of eigenfunctions with an even number of nodal domains, a rigorous proof, and an informed conjecture in the odd case.  References References are given below.
 Stanley, Richard P., An introduction to hyperplane arrangements, Geometric combinatorics 13.389496 (2004).
 Zaslavsky, Thomas, Facing up to arrangements: Facecount formulas for partitions of space by hyperplanes, Vol. 154. American Mathematical Soc., 1975.
Questions?
Contact Isaac Michael <imichael@lsu.edu>.