# Summer 2022 Math Research Circle Projects

Math Research Circle is a virtual summer camp for high school students. You can read more about how the program works. The cost per session for 2022 is \$895, with a deposit of \$300 due at time of registration. To register for one of the research topics, simply complete the registration form.

## Session 1: Mon, Jun 13, 2022 – Sat, Jul 2, 2022

### Unavoidable Induced Subgraphs of Large Rooted Trees [Participants: 4/4 (CLOSED)]

- Topic area
- Combinatorics (Calculus not required)
- Instructor
- Samuel Weiner - Louisiana State University

In the field of graph theory, a graph is a collection of vertices and edges. Graphs are powerful tools that can be used to model any real-world system. For instance, consider air traffic: each airport can be viewed as a vertex, and two vertices are connected by an edge if there is a flight that travels between the corresponding airports. Since there are so many flights from so many airlines at so many different times of day, it can be difficult to find the cheapest or quickest travel option. This is an issue across any large or complex system: how do we sift through the noise to find what we need? This project seeks to answer this question by using an active area of research within graph theory known as Ramsey Theory, which helps us find the basic structural properties common to even the most complex systems in existence.

In the aforementioned graph modeling air traffic, let us imagine that we take some large collection of edges and color them red. If we can find all subgraphs which contain many of these red edges, then we would know exactly how some cities are connected through air travel. We could then use this discovery to know with certainty the cheapest or quickest ways to travel between any two given cities. As such, the crux of the issue is to know how to find the subgraphs that contain these red edges, and that is what this research project seeks to do.

References:

- R. Diestel, Graph Theory, 5th ed., Graduate Texts in Mathematics, 2016.
- Bogdan Oporowski, James Oxley, and Robin Thomas, Typical Subgraphs of 3- and 4 connected graphs, Journal of Combinatorial Theory, Series B57 (1993) 239-257.

### Points with Rational Distances [Participants: 0/4]

- Topic area
- Geometry (Calculus not required)
- Instructor
- Camille Schuetz - University of Kentucky

A well-known open problem in mathematics is whether you can find a point in the unit square such that the distance from that point to each corner of the square is a rational number. While this remains unsolved, a similar result is known about the triangle.

For a triangle that has one side of rational length and the remaining side length whose squares are rational, there exist infinitely many points such that that distance from that point to each of the vertices is rational, according to [1].

I am proposing we study a similar problem on general quadrilaterals (kites, rhombuses, and/or isosceles trapezoids). We will answer what restrictions on the side lengths and/or angles of a quadrilateral guarantee there will be a point in the quadrilateral that is a rational distance away from all four corners?

Possible Extension: If we are able to find a good result for quadrilaterals, we may look at a similar problem on the pentagon.

This research proposal has the potential for continued research after the program.

References:

- T. Berry, Points at rational distance from the vertices of a triangle, Acta Arithmetica 62 (1992), no. 4

### Modeling the COVID-19 Pandemic [Participants: 2/4]

- Topic area
- Epidemiology (Calculus required)
- Instructor
- Jacob Davis - Southern Methodist University

The COVID-19 Pandemic has caused a boom in the fields of epidemiology and disease dynamics. Mathematicians have used different methods to track and model past pandemics for years. Researchers finally have new relevant data to understand, and now high schoolers get their chance.

The *SIR Model* is a basic model, given as
\[
\frac{dS}{dt} = -rSI,
\qquad
\frac{dI}{dt} = rSI - aI,
\qquad
\frac{dR}{dt} = aI,
\]
where $r > 0$ is the infection rate and $a > 0$ is the removal rate of infectives. In this model, we
have three distinct classes: Susceptible ($S$), who can catch the disease; Infectives ($I$), who have the
disease and can transmit it; and the Removed ($R$), who have recovered, are immune, or isolated
until recovered.

Using this model, we will ask the following questions:

- Can this model accurately predict the first 100 days of the pandemic?
- Can we then use this model to predict the remainder of the pandemic?
- How can quarantining (decrease in $r$) or a more susceptible population (decrease in $a$) affect the outcome of the model?

References:

- J.D. Murray, Mathematical Biology I: An Introduction, 3rd ed., Springer, 2002.

### Generalizing the Derivative and Integral Operators to Fractional Order [Participants: 0/4]

- Topic area
- Fractional Calculus (Calculus recommended)
- Instructor
- Zachary Bradshaw - Tulane University

In calculus, one learns about the derivative and integral operators and how to apply them successively to achieve an order n operator for any whole number n. A natural question is whether these operators can be extended to allow fractional orders. In fact, this was discussed at least as early as 1695 in a correspondence between Leibniz and l'Hôpital, where they wrote of the possibility of extending the derivative operator to a fractional order. Thus, the field of fractional calculus was born. Though, it failed to develop until recently, with the discovery of many physical applications. One of the earliest applications is due to Abel in his 1823 treatment of a generalized version of the tautochrone problem. In this work, Abel laid the foundation for what would become the Riemann-Liouville fractional integral and the Caputo fractional derivative.

What other extensions of the derivative and integral operators exist? This is the question we set out to study here. We will construct our own notion of a fractional derivative and/or integral, prove that it generalizes the classical definition, and study its behavior. Does our derivative satisfy the classical derivative rules? Does a generalized integration by parts hold? Is our derivative definition local? What kinds of solutions do we get from fractional differential equations?

References:

- K. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, 1974.
- R. Khalil et al. A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics, 2014.
- J. Stewart Calculus: Early Transcendentals, 8th ed., Cengage Learning, 2016.

### Perfect Covers of Chessboards [Participants: 0/4]

- Topic area
- Combinatorics (Calculus not required)
- Instructor
- Runze Wang - University of Memphis

In this research, we will use the coloring method to investigate the problem of perfectly covering chessboards with different shapes (You can rotate the selected shape by $90$, $180$, or $270$ degrees, but not cut or fold it). Perfectly covering a chessboard means every square is covered by exactly one piece of the selected shape.

Let’s see an example: Can you use some dominoes to perfectly cover an 8*8 chessboard with the top left and the bottom right squares removed? To solve this problem, we color the squares in the way that a black one is next to a white one. Then every domino will cover one black square and one white square. (Then?)

In this research, we will start with a relatively easy problem: Can we use some “T” tetrominoes to perfectly cover a $10\times10$ chessboard? And then we will try to use some “L” trominoes to perfectly cover a $3 \times (2n+1)$ chessboard. On this basis, we will investigate if we can use “L” trominoes to perfectly cover an $m \times n$ chessboard, then what conditions $m$, $n$ should satisfy.

Possible Extension: If we can use “L” trominoes to perfectly cover an $m \times n$ chessboard by several layers (which means every square is covered by the same number of trominoes), then what conditions should $m$, $n$ satisfy?

References:

- R. A. Brualdi, Introductory Combinatorics, 5th ed., Pearson, 2010.
- Y. Zhao, Tiling: Coloring and Weights, 2007.

### Introduction to Algorithmic Combinatorics [Participants: 4/4 (CLOSED)]

- Topic area
- Combinatorics (Calculus not required)
- Instructor
- Frederic Marazzato - Louisiana State University

Imagine that $n$ different students apply to $p$ different universities. Each student ranks the universities by preference (he might not rank some of them) and each university ranks the students that apply to it (based on grades and extracurricular activities for instance). How does one assign each student to a university to make everyone happy?

Each student cannot get in their favorite university because the number of spots in each university is limited. Therefore, one would like each student to be assigned to a university that they ranked high but that still has a spot for them. But that is not all, one would not like to have a student $a$ assigned to a university $A$ and a student $b$ assigned to a university $B$ while $a$ would have preferred to be admitted to $B$ and $b$ would have preferred to be admitted to $A$ (based on their respective rankings).

One should thus try to achieve stable matches in which case the above situation does not happen. In this project, we will study how to create these stable matches, their properties and code some examples using python.

This research proposal has the potential for continued research after the program.

References:

- D. Knuth. Stable marriage and its relation to other combinatorial problems: An introduction to the mathematical analysis of algorithms, Volume 10. American Mathematical Soc., 1997.

## Session 2: Mon, Jul 18, 2022 – Sat, Aug 6, 2022

### Unavoidable Induced Subgraphs of Graphs with Large Rooted Matchings [Participants: 3/4]

- Topic area
- Combinatorics (Calculus not required)
- Instructor
- Samuel Weiner - Louisiana State University

In the field of graph theory, a graph is a collection of vertices and edges. Graphs are powerful tools that can be used to model any real-world system, from electrical grids to transportation networks, and even computer storage systems. Some very important systems are characterized by “matchings,” or objects which must exist in pairs. Think of a large group of people, all of whom are married; using graphs, we can model each person as a vertex, where two vertices are connected by an edge if the two corresponding people know each other. Suppose also that if two people are married, then the edge connecting them is colored red. This research project seeks to characterize all of the possible relationships between the people in this group. It is possible that each person only knows their spouse; it is also possible that everybody knows everybody else in the group. But what about all of the in-between scenarios? How many different configurations are there?

If discovered, the result would provide us with knowledge about any system in which we know matchings exist. Such research would be instrumental in improving upon existing computer algorithms relating to search engines, data storage, and more.

References:

- R. Diestel, Graph Theory, 5th ed., Graduate Texts in Mathematics, 2016.
- Bogdan Oporowski, James Oxley, and Robin Thomas, Typical Subgraphs of 3- and 4 connected graphs, Journal of Combinatorial Theory, Series B57 (1993) 239-257.

### Comparing the SIR and SEIR models using COVID-19 dynamics [Participants: 0/4]

- Topic area
- Epidemiology (Calculus required)
- Instructor
- Jacob Davis - Southern Methodist University

The COVID-19 Pandemic has caused a boom in the fields of epidemiology and disease dynamics. Mathematicians have used different methods to track and model past pandemics for years. Researchers finally have new relevant data to understand, and now high schoolers get their chance.

The SIR and SEIR model are classic systems of equations used to model disease dynamics. In these models, we have four distinct classes: Susceptible ($S$), who can catch the disease; Infectives ($I$), who have the disease and can transmit it; Removed ($R$), who have recovered, are immune, or isolated until recovered; and $E$, the people incubating the disease.

Using these two models, we will ask the following questions:

- Do these models accurately predict the first 100 days of the pandemic?
- Can we then use these models to predict the remainder of the pandemic?
- How do these two models compare?
- In which situations is one better than the other?

References:

- J.D. Murray, Mathematical Biology I: An Introduction, 3rd ed., Springer, 2002.

### Completeness of Distributions [Participants: 1/4]

- Topic area
- Probability Theory (Calculus not required)
- Instructor
- Christian Ennis - Louisiana State University

In the space of probability distributions on a countable space, such as the integers, we can easily define metrics within the space. The set of probability distributions is a subset of the infinite-dimensional space $\mathbb{R}^\infty$, and while the space it resides in can be made into a vector space under pointwise addition, the set of probability distributions is easily seen to not be a subspace. However, we can still discuss the “closeness” of vectors residing in the set of distributions.

The concept of completeness, in a metric space, is something often studied by those getting into the introductory analysis. Being that the set of probability distributions on the integers can be made into a metric space, we will attempt to determine whether the set is complete under a given metric. That is, for a Cauchy sequence of distributions $a_n$ in the set of distributions equipped with the metric $\rho$, does the property \[ \lim_{m,n\to\infty} \rho(a_{n}, a_{m}) \to 0 \] imply that \[ \lim_{n\to\infty} a_{n} \to a \] for a distribution $a$ in the set of probability distributions? We will start with the supremum metric. That is the metric $\rho$, defined on vectors $a$ and $b$ with \[ \rho(a,b)=\sup_k |a_k-b_k| \] where $a_k$ and $b_k$ are the entries in the $k$-th coordinate of the vectors $a$ and $b$ respectively. Once a more thorough understanding of metrics is gained, we will determine what is not a metric on the set, and examine the completeness of other metrics on the set. Introductory topics in linear algebra, such as vector spaces, will be discussed as well.

This research proposal has the potential for continued research after the program.

References:

- Protter, M.H. and Morrey, C.B. (1991) A First Course in Real Analysis, 2nd ed.. Springer-Verlag, New York.
- Larson, Ron. (2017) Elementary Linear Algebra. 8th ed., Cengage, USA.

### Studying Memory Effects Using Fractional Derivatives [Participants: 3/4]

- Topic area
- Fractional Calculus (Calculus required)
- Instructor
- Zachary Bradshaw - Tulane University

In calculus, one learns about the derivative and integral operators and how to apply them successively to achieve an order n operator for any whole number n. A natural question is whether these operators can be extended to allow fractional orders. In fact, this was discussed at least as early as 1695 in a correspondence between Leibniz and l'Hôpital, where they wrote of the possibility of extending the derivative operator to a fractional order. Thus, the field of fractional calculus was born. Though, it failed to develop until recently, with the discovery of many physical applications. One of the earliest applications is due to Abel in his 1823 treatment of a generalized version of the tautochrone problem. In this work, Abel laid the foundation for what would become the Riemann-Liouville fractional integral and the Caputo fractional derivative.

Given a fractional derivative, what physical models can it describe? Some of these operators, such as the Riemann-Liouville derivative, are nonlocal, meaning that the state of a system at a particular time depends on its state on an earlier time interval. This is known as a memory effect [1,3]. Since classical derivatives are defined locally, they are inherently incapable of describing such a system. Therefore, the project being proposed here is to study various systems with memory effects using several different fractional derivatives. We then compare the results to find the best fitting model.

This research proposal has the potential for continued research after the program.

References:

- K. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, 1974.
- R. Khalil et al. A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics, 2014.
- V. Tarasov, Generalized Memory: Fractional Calculus Approach, Fractal and Fractional, 2018.
- J. Stewart Calculus: Early Transcendentals, 8th ed., Cengage Learning, 2016.