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A Project-Based Guide to Undergraduate Research in Mathematics

Folding Words Around Trees: Models Inspired by RNA

At the intersection of mathematics and biology, we find mathematical models built to address biological questions as well as new mathematical theories inspired by biological structures. In this chapter, we explore a combinatorial model for folding words around plane trees which is inspired by the bonds that form between nucleotides in a single-stranded RNA molecule. This chapter walks the reader through the construction of valid plane trees, structures formed by folding a word in a complementary alphabet around a plane tree, and enumerates the class of words with exactly one such folding. Valid plane trees are relatively unexplored combined objects, and while we present several potential research projects, a careful reader can come up with many additional directions for further study.
Elizabeth Drellich, Heather C. Smith

Phylogenetic Networks

Phylogenetics is the study of the evolutionary relationships between organisms. One of the main challenges in the field is to take biological data for a group of organisms and to infer an evolutionary tree, a graph that represents these relationships. Developing practical and efficient methods for inferring phylogenetic trees has led to a number of interesting mathematical questions across a variety of fields. However, due to hybridization and gene flow, a phylogenetic network may be a better representation of the evolutionary history of some groups of organisms. In this chapter, we introduce some of the basic concepts in phylogenetics and present related research projects on phylogenetic networks that touch on areas of graph theory and abstract algebra. In the first section, we describe several open research questions related to the combinatorics of phylogenetic networks. In the second, we describe problems related to understanding phylogenetic statistical models as algebraic varieties. These problems fit broadly in the realm of algebra, but could be more accurately classified as problems in algebraic statistics or applied algebraic geometry.
Elizabeth Gross, Colby Long, Joseph Rusinko

Tropical Geometry

Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials built with these new operations. These equations define piecewise-linear geometric objects called tropical varieties. We explore these tropical varieties in two and three dimensions, building up discrete tools for studying them and determining their geometric properties. We then discuss the relationship between tropical geometry and algebraic geometry, which considers shapes defined by usual polynomial equations.

Chip-Firing Games and Critical Groups

In this note we introduce a finite abelian group that can be associated with any finite connected graph. This group can be defined in an elementary combinatorial way in terms of chip-firing operations, and has been an object of interest in combinatorics, algebraic geometry, statistical physics, and several other areas of mathematics. We will begin with basic definitions and examples and develop a number of properties that can be derived by looking at this group from different angles. Throughout, we will give exercises, some of which are straightforward and some of which are open questions. We will also highlight some of the many contributions to this area made by undergraduate students.
Darren Glass, Nathan Kaplan

Counting Tilings by Taking Walks in a Graph

Given a region and a collection of basic shapes (tiles), a natural question is to look at how many ways there are to cover the region using the tiles where no pair of tiles overlaps in their interiors. We show how to transform some problems of this type into counting walks on graphs. In the latter setting, there are well-known and efficient methods to count these for small cases, and in many cases recurrences and closed-form expressions can be found. We explore variations of these problems, and get to the point where the reader can set off to explore problems of this type.
Steve Butler, Jason Ekstrand, Steven Osborne

Beyond Coins, Stamps, and Chicken McNuggets: An Invitation to Numerical Semigroups

We give a self-contained introduction to numerical semigroups and present several open problems centered on their factorization properties.
Scott Chapman, Rebecca Garcia, Christopher O'Neill

Lateral Movement in Undergraduate Research: Case Studies in Number Theory

We explore the thought processes, strategies, and pitfalls involved in entering new territory, developing novel projects, and seeing them through to publication. We propose twenty-one general principles for developing a sustainable undergraduate research pipeline and we illustrate those ideas in three case studies.

Projects in (t, r) Broadcast Domination

Domination theory is a subfield within graph theory that aims to describe subsets of the vertices of a graph which satisfy certain distance properties. The original domination problem asked one to find subsets of the vertices of a graph (with minimal cardinality) so that every vertex in the graph was either in the set or adjacent to a vertex in the set. Since its development, thousands of papers on domination theory and its many variants have appeared in the literature. We focus our study on (t, r) broadcast domination, a variant with a connection to the placement of cellphone towers, where some vertices send out a signal to nearby vertices (with the signal decaying linearly along edges according to distance), and where all vertices must receive a minimum predetermined amount of this signal. The overall goal is to minimize the number of tower vertices needed to have all vertices receive the appropriate amount of signal reception. We summarize our past work with students and present many remaining open problems in this field. We end the chapter by providing some advice on how we continue to develop new research projects with and for students; although the mathematical content of the chapter is in domination theory, the suggestions can be implemented in any area.
Pamela E. Harris, Erik Insko, Katie Johnson

Squigonometry: Trigonometry in the p-norm

We can define the traditional trigonometric functions in several different ways: via differential equations, via an arclength definition on the unit circle x2 + y2 = 1, or via an analytic approach. In this project, we adapt these approaches to define analogous functions for a unit squircle | x |p + | y ​​|p = 1, p ≥ 1. As we develop these functions using only elementary calculus, we will ponder the importance and role of π, and glimpse some very deep ideas in elliptic integrals, special functions, non-Euclidean geometry, number theory, and complex analysis.
William E. Wood, Robert D. Poodiack

Researching in Undergraduate Mathematics Education: Possible Directions for Both Undergraduate Students and Faculty

Research in Undergraduate Mathematics Education (RUME) is a new field to both mathematics and mathematics education. It borrows theory and methodology from other disciplines including psychology, sociology, and neurology. At its core, RUME is attempting to find out about the teaching and learning of undergraduate mathematics education in order to improve it. In this book chapter, I attempt to give a quick overview on how to conduct RUME with undergraduate students. I pull from my experiences as a mentor of ten undergraduate projects. There is also a suggested timeline of RUME in a semester, some ways to generate RUME open questions, and a large amount of open questions conjectured by others. My hope is that this book chapter has information for both mentors and undergraduates alike.

Undergraduate Research in Mathematical Epidemiology

The spread of diseases remains an important issue in public health. The use of mathematics in predicting and understanding epidemics is not new, but still relevant and useful. In this chapter we provide relevant resources and useful exercises for undergraduate students and their mentors. We describe two different modeling techniques which require different backgrounds. For agent based modeling, we suggest students who are either comfortable with programming or willing to learn and who have basic knowledge of probability. For the differential equation approach, we suggest students who have taken at least Calculus 2. Students with a differential equation background will advance faster and can do a more theoretical analysis of the system. A student who might be willing to spend more time working on this topic can model the same disease outbreak using different modeling techniques which will allow for comparison and much deeper analysis of both the mathematics and the biological and public policy implications. Additionally, we include a sample project, developed and written by an undergraduate student who co-authors this chapter. Finally, we provide four different projects that students and their mentors can work on.
Selenne Bañuelos, Mathew Bush, Marco V. Martinez, Alicia Prieto-Langarica