A Project-Based Guide to Undergraduate Research in Mathematics: Starting and Sustaining Accessible Undergraduate Research

Series Preface
Preface
Contents
Folding Words Around Trees: Models Inspired by RNA
	1 Introduction
	2 Catalan Numbers
	3 Valid Plane Trees
	4 Local Moves and the State Space Graph
	5 Enumerating Words with Only One Valid Plane Tree
	6 Enumeration of Valid Plane Trees
		6.1 Small Values in R(n,1)
		6.2 Large Values not in R(n,1)
	7 Wobble Pairs and Other Modifications to the Model
	8 Numerical Models of RNA Folding
		8.1 Structures of RNA
		8.2 The RNA Folding Problem
		8.3 The Inverse RNA Folding Problem
	9 Conclusion
	References
Phylogenetic Networks
	1 Introduction
	2 Combinatorics of Phylogenetic Networks
		2.1 Graphs and Trees
		2.2 Phylogenetic Networks
		2.3 Semi-Directed Networks
		2.4 Restrictions of Networks
	3 Algebra of Phylogenetic Trees and Networks
		3.1 Ideals Associated to Trees
		3.2 Ideals Associated to Sunlet Networks
		3.3 Beyond Sunlet Networks
	References
Tropical Geometry
	1 Tropical Mathematics
		1.1 Tropical Arithmetic and Tropical Linear Algebra
		1.2 Tropical Polynomials and Tropical Varieties
		1.3 Some Tropical Resources
	2 Tropical Curves in the Plane
		2.1 Convex Hulls and Newton Polygons
		2.2 Subdivisions and the Duality Theorem
		2.3 The Geometry of Tropical Plane Curves
		2.4 Skeletons of Tropical Plane Curves
	3 Tropical Geometry in Three Dimensions
		3.1 Tropical Surfaces and the Duality Theorem
		3.2 Tropical Curves in R3
	4 Tropicalization
		4.1 Fields with Valuation
		4.2 Two Ways to Tropicalize
		4.3 Tropical Intersections
	References
Chip-Firing Games and Critical Groups
	1 Critical Groups
		1.1 Definitions and Examples
		1.2 Divisors on a Graph and the Chip-Firing Game
		1.3 Smith Normal Forms
		1.4 Elements of the Critical Group
		1.5 Spanning Trees and the Matrix Tree Theorem
		1.6 How Does the Critical Group Change Under Graph Operations?
		1.7 Which Finite Abelian Groups Occur as the Critical Group of a Graph?
		1.8 Generators of Critical Groups
		1.9 Critical Groups of Random Graphs
		1.10 The Monodromy Pairing on Divisors
		1.11 Ranks of Divisors and Gonality of Graphs
		1.12 Chip-Firing on Directed Graphs
	2 Arithmetical Structures
		2.1 Definitions and Examples
		2.2 Counting Arithmetical Structures
		2.3 Critical Groups of Arithmetical Structures
	References1
Counting Tilings by Taking Walks in a Graph
	1 Introduction
		1.1 A First Example
		1.2 A Second Example
		1.3 A First Example, Revisited
			1.3.1 A Second Example, Revisited
	2 General Approach
		2.1 Using Linear Algebra Tools
			2.1.1 Recurrence Relationships and Generating Functions
			2.1.2 Closed-Form Solutions from Projection
	3 Using More Entries of the Matrix
	4 Tiling with Statistics
	5 Using Other Boards for Tiling
		5.1 Boards Formed from Triangles
		5.2 Boards Formed from Hexagons
		5.3 Three Dimensional Boards
	6 Using Abstract Tiles
		6.1 Squaring a Square
		6.2 Rectangling a Rectangle
		6.3 Tiling with Skinny Rectangles
	7 A Global Constraint on Tiling
	8 Concluding Remarks
	References
Beyond Coins, Stamps, and Chicken McNuggets: An Invitation to Numerical Semigroups
	1 Introduction
	2 A Crash Course on Numerical Semigroups
	3 Using Software to Guide Mathematical Inquisition
	4 Research Projects: Asymptotics of Factorizations
	5 Research Projects: Random Numerical Semigroups
	References
Lateral Movement in Undergraduate Research: Case Studies in Number Theory
	1 Introduction
	2 Guiding Principles
	3 Case Study I: Quotient Sets
	4 Case Study II: Primitive Roots, Prime Pairs, and the Bateman–Horn Conjecture
	5 Case Study III: Supercharacters and Exponential Sums
	6 Conclusion
	References
Projects in (t,r) Broadcast Domination
	1 Introduction
	2 Background
		2.1 Distance Domination
		2.2 (t,r) Broadcast Domination
	3 Projects in (t,r) Broadcast Domination
		3.1 Grid Graphs
		3.2 The Infinite Square Grid Graph
		3.3 Infinite Triangular Grid Graph
		3.4 Other Families of Graphs
	4 Developing Accessible Research Projects
		4.1 Sample Honors Contract
		4.2 Student Developed Projects
	References
Squigonometry: Trigonometry in the p-Norm
	1 Introduction
	2 Defining Trigonometric Functions
		2.1 Differential Equations Approach
		2.2 Unit Circle Approach
		2.3 Analytic Approach
	3 Squigonometric Functions
		3.1 Differential Equations Approach
		3.2 The Many Values of π
	4 The Geometry of p-Circles
		4.1 The Planar p-Norm
		4.2 Sines Everywhere
		4.3 Transcendental Functions
		4.4 π Redux
	5 Widening the Scope
		5.1 Duality
		5.2 Arclength Trigonometric Functions
		5.3 Other Parameterizations
	6 Onward
	References
Researching in Undergraduate Mathematics Education: Possible Directions for Both Undergraduate Students and Faculty
	1 Introduction
	2 Beginning RUME with a Question
	3 Common Methods of RUME Research
		3.1 Qualitative Research
			3.1.1 Literature Review
			3.1.2 Teaching Experiment
			3.1.3 Textbook/Material Analysis
		3.2 Quantitative Research
	4 A Sample Timeline for Mentoring Undergraduate Students in RUME
		4.1 Research Question
		4.2 Background Literature
		4.3 Institutional Review Board Application (If Needed)
		4.4 Data Collection and Analysis
		4.5 Writing or Presenting Findings
			4.5.1 Typical RUME Paper Sections
	5 Paxton\'s Undergraduate Research Experience
	6 Suggested Projects
	7 Conclusion
	References
Undergraduate Research in Mathematical Epidemiology
	1 Introduction
		1.1 Note to Students
		1.2 Note to Mentors
	2 Epidemiology Background
	3 Mathematical Modeling Problems
		3.1 Simple ODE Models
		3.2 Vector-Borne Diseases
		3.3 ABM and Epidemics
	4 Projects
	5 An Example: Gendered Zika Model
	References