Virtual Math Circle: Summer 2024 Research Projects

Math Circle Logo

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 2024 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.

Session 1: Mon, Jun 10, 2024 – Sat, Jun 29, 2024

Random Walks on Graphs [0/4]

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 0-1, 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 simulation-based validation and basic statistical analysis of simulated results. Preparation of the final presentation.


References
References are given below.
  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 , Combinatorics, Paul Erd˝os is eighty, 2(1-46):4, 1993.
  3. Georg Pólya, Über eine aufgabe der wahrscheinlichkeitsrechnung betreffend die irrfahrt im straßennetz , Mathematische Annalen, 84(1-2):149–160, 1921.

The Möbius function of a Kohnert Poset [1/4]

Virtual Math Circle Research Proposal

Session
Session 1 (Jun 10 – Jun 29, 2024)
Mentor
Dr. Nicholas Mayers
Postdoctoral Research Scholar
Department of Mathematics
North Carolina State University

Project title
The Möbius function of a Kohnert Poset
Topic area
Combinatorics

High school calculus
Not a prerequisite.
Skills and background
No particular background knowledge is required. Code, written in Sage, to do experiments will be provided together with instructions on how to download the software and run the code. Students will be able to run examples from the first day.

Abstract
Partially ordered sets (or posets for short) are collections of objects with an ordering. A familiar example is the collection of positive integers where we have $1<2<3<\cdots$. In this project, we will focus on a family of posets whose objects are diagrams consisting of boxes arranged into rows and columns in the first quadrant and whose order is described in terms of sequences of valid moves. Such posets are called “Kohnert posets.” Unlike in the case of the positive integers with the ordering discussed above, Kohnert posets often have many pairs of elements for which there is no relation (i.e., neither is bigger or smaller than the other); consequently, Kohnert posets are more complicated posets.

When studying new and interesting posets, mathematicians often consider the posets’ “Möbius functions.” These are functions defined on pairs of comparable elements (i.e., one is bigger than the other) in the post. To compute the value of the Möbius function for such a pair, there is a simple (but cumbersome) recipe, starting from the smaller element and working up to the larger one. The goal of this project is to study whether one can compute the values of Möbius functions on Kohnert posets by just looking at the pairs of diagrams, bypassing the aforementioned time-consuming recipe.


Possible extension
Related to the Möbius function, we could consider computing the $f$-vectors associated with Kohnert posets which enumerate the totally ordered subsets, called chains, of varying sizes.

Outline/timeline
Week 1: The first week will focus on introducing poset basics, Möbius functions, and Kohnert posets. In addition, we will learn how to “play around” with the code, looking at several examples and exploring what happens, aiming to state some conjectures.

Week 2: During the second week we will focus on studying promising families of Kohnert posets noted during the previous week. In particular, we will gather further data concerning the M¨obius functions of special families of posets for which it appears that a combinatorial method of computation is possible. Moreover, we will discuss proof approaches and attempt to prove any conjectures.

Week 3: For the third week, we will continue to pursue proofs and create the final presentation using Beamer.


References
References are given below.
  1. S. Assaf and D. Searles, Kohnert Polynomials , Experiment. Math., 31(1): 93–119, 2019.
  2. S. Assaf, Demazure crystals for Kohnert polynomials , Trans. Amer. Math. Soc., 375(3): 2147–2186, 2022.
  3. L. Colmenarejo, F. Hutchins, N. Nayers, and E. Phillips, On ranked and bounded Kohnert posets , arXiv preprint arXiv: 2309.07747 (2023).

Nodal Sets on a Square Membrane [0/4]

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 Dirichlet-Laplacian 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 Dirichlet-Laplacian 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 Dirichlet-Laplacian 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 Dirichlet-Laplacian 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 Neumann-Laplacian 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 Neumann-Laplacian 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.
  1. P. Bérard and B. Helffer, Nodal sets of eigenfunctions, Antonie Stern’s results revisited, Séminaire de th´eorie spectrale et géométrie, Volume 32 (2014-2015), pp. 1-22.
  2. J. Stewart, Calculus: Early Transcendentals, 6th ed., Thomson Learning, 2008.
Remark. Note that the instructor is available during either summer session, but not both.

Counting connected regions of a hyperplane arrangement [0/4]

Virtual Math Circle Research Proposal

Session
Session 1 (Jun 10 – Jun 29, 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 we divided the 3-dimensional 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 three-dimensional 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 n-dimensional 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.
  1. Stanley, Richard P., An introduction to hyperplane arrangements, Geometric combinatorics 13.389-496 (2004).
  2. Zaslavsky, Thomas, Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes, Vol. 154. American Mathematical Soc., 1975.

Session 2: Mon, Jul 15, 2024 – Sat, Aug 3, 2024

Questions?

Contact Isaac Michael <imichael@lsu.edu>.