# 2016-2017 Abstracts

## May

5/2/17
Taylor Matsumura, Rufei Men, and Mattie Wiltbank
Math Senior Thesis: Efficient Mino Foldings

Our research is based in a field of origami which focuses on efficient foldings and the algorithms for finding them. The most efficient folding is one which contains the minimal amount of overlap. In this project, we employ origami folding techniques to efficiently fold polyominoes, working primarily with n x k rectangular strips. As an example Tetris pieces are tetrominoes, which are the 4 different arrangements of four squares, whereas pentominoes are the 11 different arrangements of five squares.

Jeremy Coste and Dane Miyata
Math Senior Thesis: Weighted k-Majority Tournaments
Suppose we have an election with 2k-1 voters. Each voter lists the candidates in the order of their preference. We portray each candidate as a vertex in a graph and draw arrows from one vertex to another only when the first candidate is listed above the second candidate by a majority, namely at least k of the voters. Finally we assign weights to each edge to represent the number of orders that realize that edge. This construction yields a type of graph called a weighted k-majority tournament.  In this talk we explore the properties of these graphs, specifically looking at particular subsets of the graphs called dominating sets.

## April

4/27/17
Katy Ohsiek and Ana Wright
Math Senior Thesis: Mountain Graphs in Origami Crease Patterns
Our work concerns the mathematics of flat folded origami. A graph is a mountain graph if there exists a valid crease pattern such that there exists a bijection between the vertices of the graph and the vertices of the crease pattern. We seek to classify graph families by their relationship to valid crease patterns. We outline the folding procedure followed to construct representations of mountain graphs and work towards understanding the sufficient and necessary conditions for a graph to be a mountain graph.

Samuel Coste and Boyuan Lyu
Math Senior Thesis: The Accountable Art Gallery Problem
Imagine a museum gallery that is filled with priceless paintings. Each painting needs to be seen by one guard so that it is not stolen. The museum is on a budget, and wants to cover the gallery with the minimum amount of guards. This is called the Art Gallery Problem. In our research, we explore a variation of the Art Gallery Problem, where each guard is covered by at least one other guard. We prove results on various families of polygons.

4/26/17
Megan Duff and Kees McGahan
Math Senior Thesis: Transport Pebbling
Transport pebbling is a variation of pebbling played on graphs where, given a single blue and some red pebbles, the blue pebble can be moved to the target vertex using the red pebbles. A pebbling move is defined by removing two pebbles from a vertex and placing one pebble on an adjacent vertex. The transport pebbling number is the minimum number of pebbles such that the blue pebble can be moved to any target vertex. This paper introduces this new category of pebbling, as well as investigates the properties of the transport pebbling number for specific families of graphs.

4/20/17
Colin Starr, Willamette University
Unipancyclic Graphs and Matroids, or I Thought This Would Be Easier
We will consider the problem of uniquely pancyclic (UPC) graphs, or graphs that have exactly one cycle of each possible size. Since the problem first appeared in 1973, very little progress has been made.  Indeed, only a handful of UPC graphs are known, so our exploration will mostly take place in the world of matroids, mathematical objects that combine and generalize notions from Graph Theory and Linear Algebra.  We'll see a "new" UPC matroid that doesn't come from a graph and investigate some properties of UPC matroids.

4/6/17
Richard Moy, Willamette University
What is Mathematical Research? A Former Undergrad's Tale
One of the most common questions a mathematician is asked is "How do you do research in math?" However, answering that question can be difficult even when your audience has a background in mathematics. In this talk, I will give a (hopefully humorous) description of the mathematical research process and show how it played out in a research project I began as an undergraduate. This project involved arithmetic progression free sets, a problem of Erdos, and a greedy algorithm. Come to the talk to find out more!

## March

3/23/17
Naiomi Cameron, Lewis & Clark College
Belyi Maps for Trees of a Given Passport
Dessin is short for dessin d’enfant, which is French for “child’s drawing.” Introduced in the late 1990s by A. Grothendieck, a dessin can be described as a connected bicolored graph where the edges around every vertex are cyclically ordered. Dessins can be realized by Belyi maps, which are meromorphic functions from a compact connected Riemann surface to the extended complex plane having at most three critical values. In this talk, I will discuss the origins, outcomes and future of an undergraduate research project designed to determine the Belyi maps that realize certain classes of bicolored trees embedded on the complex sphere.

3/16/17
Holly Swisher, Oregon State University
Ramanujan's Mock Theta Functions and Quantum Modular Forms
Nearly 100 years after his untimely death, Ramanujan's legacy is still intriguing mathematicians today. One of the last obsessions of Ramanujan were what he called mock theta functions. In this talk, we will begin by discussing Ramanujan's work on integer partitions and how they connect to objects called modular forms and mock theta functions. Then we will consider some recent work on the construction of a table of mock theta functions with some interesting properties, including what is called quantum modularity. Part of this work is joint with Sharon Garthwaite, Amanda Folsom, Soon-Yi Kang, and Stephanie Treneer. The rest is joint with Brian Diaz and Erin Ellefsen from their undergraduate REU project during summer 2016.

3/2/17
Vivek Pal, University of Oregon
Combinatorics and Magic Tricks
Combinatorics is the study of counting, it has many applications such as figuring out your probability of winning a poker hand. In this talk we'll uncover some more unique applications and also discover some magic (card) tricks.

## February

2/8/17
Derek Garton, Portland State University
Dynamically Distinguishing Polynomials
Any polynomial with integer coefficients yields a family of (discrete) dynamical systems indexed by primes as follows: for any prime p, reduce the polynomial's coefficients mod p, then consider its action on the set of congruence classes mod p. If there is a prime p such that a set of polynomials yield mutually distinct dynamical systems, we say this set of polynomials is "dynamically distinguishable modulo p." In this talk, based on work that is joint with Andrew Bridy, we will construct arbitrary large sets of polynomials that are dynamically distinguishable modulo most primes.

2/2/17
Fractals: Hunting the Hidden Dimension (Nova Video)
You may not know it, but fractals, like the air you breathe, are all around you. Their irregular, repeating shapes are found in cloud formations and tree limbs, in stalks of broccoli and craggy mountain ranges, even in the rhythm of the human heart. In this film, NOVA takes viewers on a fascinating quest with a group of maverick mathematicians determined to decipher the rules that govern fractal geometry.

For centuries, fractal-like irregular shapes were considered beyond the boundaries of mathematical understanding. Now, mathematicians have finally begun mapping this uncharted territory. Their remarkable findings are deepening our understanding of nature and stimulating a new wave of scientific, medical, and artistic innovation stretching from the ecology of the rainforest fashion design. The documentary highlights a host of filmmakers, fashion designers, physicians, and researchers who are using fractal geometry to innovate and inspire.

## December

12/8/16
Ana Wright

Mixing Times of the Generalized Rook's Walk
Using path coupling, a powerful probabilistic tool, we will find bounds on the mixing times of a class of Markov chains. The mixing time of a Markov chain measures the rate of convergence to its stationary distribution. This mixing time is of interest for sampling and simulations of random processes. The Markov chains we are investigating are restrictions on the random rook's walk on a d-dimensional chessboard, which can also be considered random walks on the Cartesian powers of certain groups of circulant graphs. We prove bounds on the mixing times of these Markov chains, extending and generalizing previous results for the unrestricted case of the rook's walk.

Dane Miyata
Neural Mapping using Gröbner Bases
We begin with a brief introduction to algebraic geometry, specifically with regards to monomial orderings and Gröber bases over polynomial rings.  Then, we show how algebraic geometry can be used to model how the brain processes spacial information via an algebraic object called the neural ideal. Neural ideals can be used to extract the stimulus space structure by computing a specific basis called the canonical form. Computation of the Gröbner bases is much quicker than that of the canonical form and so we will go over specific instances where Gröbner bases can instead be used to extract the stimulus space structure.

12/1/16
Evan Hedlund, Corban University
Applications of Local Fields in the Classification of Rational Periodic Points of the Map Qc(x)=x2+c
We begin with an introduction to arithmetic dynamical systems and the work of classifying rational periodic points of Qc(x) = x2 + c ∈ Q[x]. Through this, we will give a brief classification of rational period 1, 2, and 3 points of Qc. It is known that there are no rational periodic points of minimal periods 4 or 5. Our primary focus is introducing local fields and some tools of p-adic analysis to narrow down possible values of c ∈ Q which might yield rational periodic points under Qc.

This work presented is expository in nature, and stems from the work of Ralph Walde and Paula Russo in their article titled, “Rational Periodic Points of the Quadratic Function Qc(x) = x2 + c,” published by the MAA in 1994.

## November

11/17/16
Taylor Mutch '15
Data Visualization & Decision Support
The data deluge of the 21st century has necessitated the development of new ways to digest information for science, industry and government. More importantly it has come from the vast amounts of sensors and remote recording devices, yielding new insights and relationships previously unnoticed. The field of data visualization and its close relationship with decision making has greatly benefited from this advent of raw information, particularly in areas benefiting from geospatial analysis. Thus the creation of visualization and decision tools becomes paramount in guiding scientists, government officials, financiers, and students in their decisions.

11/10/16
Carolyn Yackel, Mercer University
Discrete Planar and Spherical Symmetries
Humans are innately drawn to symmetry! In this talk we will investigate three types of symmetry: the Frieze groups, the wallpaper groups, and the finite spherical symmetries. We will discuss these from a geometric perspective, as well as how to construct examples of some of the symmetry types so that the audience can fill their world with beauty.

11/3/16
McKenzie West, Reed College
Solutions of Polynomial Equations: Not So Easy After All
Polynomial equations and their solutions form a cornerstone of mathematics. Solutions with rational coordinates are particularly intriguing; a fantastic surprise is the great difficulty of determining the mere existence of a rational solution to a given equation (let alone the complete set). West will discuss this problem in two cases, diagonal cubic surfaces,

ax3 + by3 + cz3 + d = 0,
and degree 2 del Pezzo surfaces,
ax4 + by4 + cx2y2 + d = z2.

A surprising and successful modern approach, the BrauerManin obstruction, employs tools from linear algebra, geometry and non-commutative algebra. I will discuss a collection of interesting and motivating examples with simultaneous historical and modern interest, and also explain some of the tools and techniques that form the backbone of my research program.

## October

10/20/16
Breeann Flesch, Western Oregon University
Variations of Interval Graphs
A graph is interval if to every vertex v of G, we can assign an interval of the real line I_v, such that xy is an edge of G if and only if I_x intersects I_y.  Interval graphs were characterized by the absence of induced cycles larger than 3 and asteroidal triples by Lekkerkerker and Boland in 1962.  Subsequently variations on the interval theme have been introduced, including probe interval graphs and interval p-graphs, which are a generalization of interval bigraphs.  This talk focuses on the variations of interval graphs, discussing results and open problems.

10/13/16
Karl-Dieter Crisman, Gordon College
Connecting Voting Theory and Graph Theory

In studying mathematics connected to voting, we use any tools we find helpful.  Graph theory (which models connections abstractly) has recently proved very useful to analyze choices where there is a natural symmetry among the options we are voting for. In this talk, we will see some recent results about voting for committees and rank-orderings that use graphs, and even use them to explore questions you didn't know you had about how to seat people at a round table!  I also promise to somehow find a connection to Donald, Hillary, and the rest, because what would a talk about voting theory less than a month before the election be without one?

## September

9/22/16
Joshua Scott & Aimee Reynolds, McKay High School
Willamette Students and the McKay Algebra Academy
In 2009 McKay was one of the worst comprehensive high schools in the State of Oregon.  Passing rates in class and on standardized tests were extremely low, while violence and dropout rates were at an all-time high.  In 2010 McKay applied for and received the federal Student Improvement Grant (SIG), which provided additional funding for staffing, training and staff changes.  Over the course of the next four years McKay became a model school for change in the United States.  Passing rates on standardized tests tripled, student attendance improved, school violence disappeared and the dropout rate decreased to a single student.  This placed McKay in the top 1% of SIG schools in the entire country.

In 2015 our school was presented with a new challenge called the Smarter Balance Assessment Consortium, also known as the SBAC test.  The new testing model has been extremely challenging for students across the country and especially challenging for McKay students.  The test offers a unique challenge at McKay because many of the students have limited English skills and their math skills are often at a sixth grade level when they entire high school.  There are many challenges ahead for our current 9th grade students in order to graduate on time.  We would like to partner with Willamette students in order to help tutor and mentor our 9th grade students in Algebra I classes.  This is a great chance to volunteer and give back to the Salem community.

9/15/16  Dr. Martin Flashman, Emeritus Professor of Mathematics, Humboldt State University
The Role of Philosophy in Proof: Euclid's Proof of Proposition 1
It is widely believed that logic is at the heart of proof in mathematics. Professor Flashman suggests that students might be better served with an alternative view that connects notions of proof with philosophical discussions related to ontology and epistemology. Euclid's proof of Proposition 1 in his Elements, Book I, will be offered as a primary example to illustrate some possible changes in focus.

Willamette University