# Fall 2017

## November

11/9
Matthew Nabity, Western Oregon University
Dimensional Analysis for All
Dimensional analysis is an often misunderstood and underutilized analytical tool in applied mathematics. What was once a standard approach for scientists like Fourier and Maxwell, is now often neglected in favor of the computational power in today's computers. We will explore the underlying mathematical theory and the resurgence of dimensional analysis.

11/2
Charles Dunn, Linfield College
Clique-Relaxed Graph Coloring
We consider a variation of the following game played on a finite graph G. Two players, Alice and Bob, alternate coloring the uncolored vertices of G from a set of r colors. At each step, the players must ensure that adjacent vertices receive different colors. Alice always goes first. She wins the game if the entire graph is eventually colored; otherwise, Bob wins if there comes a time such that there is an uncolored vertex that cannot be colored. The least r such that Alice has a winning strategy for this game on G is called the game chromatic number of G.

We will examine a variation of this game in which the players ensure that the subgraphs induced by the color classes have bounded clique size. Our focus with these variations will be on the classes of outer planar graphs and planar graphs.

11/1

Tutor Outreach at Willamette
This colloquium intends to showcase and bring awareness to the multi-faceted tutoring opportunities that are available at or through Willamette University.  Representatives from Community Service Learning, Willamette Academy, Chemawa Indian School Partnership Program, and Academic Support will present and answer questions about the various functions, components, and goals of their respective tutoring/mentorship opportunity.

## October

10/26
William DeLee/Lars Mayrand
The Ehrenfest Urn Model

The two-urn Ehrenfest model is a classical Markov chain that can be used to describe many natural processes such as gas diffusion. If a and b are two states of a Markov chain, the hitting time from state a to state b is the number of steps that the chain takes to reach state b for the first time when the chain initially starts at state a. While hitting time problems associated with the two-urn model are well-studied, we focus on a variety of hitting times under a generalized Ehrenfest urn model with two or more urns. In particular, we use the electric network approach to obtain simple closed-form expressions for the expected hitting times for filling and emptying urns.

10/12
Kyle Salois
Intersection graphs of Maximal Sub-Polygons of k-lizards
A k-lizard is a simply connected polygon with sides parallel to a regular 2k-gon. For a k-lizard P let S be the set of all maximal sub-polygons contained in P. A graph G is a k-maximal sub-polygon graph (or k-MSP graph) if there exists a k-lizard P and a one-to-one correspondence between vertices of G and polygons in S such that two vertices are adjacent in G if and only if their corresponding polygons in S intersect. We find separating examples of graphs that are k-MSP graphs but not j-MSP graphs for j≠k.

## September

9/21
Ross Casebolt '12; Pre-Med U of O
Symmetry Structures: Classifying Finite Planar
Groups

Subgroup graphs are a nice way to present the structure of a group in visual fashion. Certain finite groups have subgroup graphs which can be drawn without crossing edges, but most finite groups cannot. The abelian case was classified by Bohanon, Starr, and Turner, and the non-abelian case finished by Bohanon and Reid.

In particular, most finite groups have a subgroup graph which cannot be drawn without crossing edges, except for a few families without many prime factors; no such groups with four or more distinct prime factors exist.

We will discuss enough group theory to present the main proof techniques for the non-abelian case and detail a few short examples.

9/7
Allison Kerkhoff and Zechariah Hazel
LARC - Creating Problems
Creating Problems was an experiment-driven, studio-based Math/Art experience (with faculty Cayla Skillin-Brauchle and Josh Liaison). The LARC students and faculty first worked together on a series of exploratory mathematical problems and artistic prompts to create common ground, enable critical conversation and fuel collaboration. Their early meetings included lively conversations about formal mathematical definitions versus artistic interpretations (e.g. symmetry), the visualization of systems and patterns, and the spectrum from abstraction (e.g. theoretical) to realism (e.g. analytical). Students and faculty then pursued individual and collaborative projects based in a full spectrum of artistic and mathematical thinking.

Participants discovered that that artists and mathematicians use similar language and creative problem-solving processes, yet the different ways that they employ that language and those processes yield vastly different outcomes.

# Spring 2017

## 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 flatfolded 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 rain forestto fashion design. The documentary highlights a host of filmmakers, fashion designers, physicians, and researchers who are using fractal geometry to innovate and inspire.

# Fall 2016

## 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.

# Spring 2016

## April

4/28/16  Colleen Chrisinger, Oregon Department of Revenue
The transition that Oregon and other states are making to an open, legal, and taxed marijuana market is a complex and fascinating one, full of political, economic, environmental, administrative, and even mathematical choices. States are navigating questions such as: What is the optimal tax structure that will raise the desired revenue for schools and substance abuse treatment but will also discourage purchases from unauthorized sources? Which statistical methods and data sources should be used to predict marijuana tax revenues? How can marijuana businesses operate when federal regulations prohibit banks to serve them? This talk explores these topics as well as the sequence of events that led a Willamette mathematics alumna to become a tax policy research economist.

4/26/16  Albert Garcia, WU Math/Econ
Senior Thesis Presentation

Some species of Hawaiian Honeycreeper are found only on specific islands, while elsewhere in the archipelago, others are found. Is this due to competition between the species or simply random fluctuation? Markov Chain Monte Carlo may help us find an answer. Using convergence diagnostic procedures, we can find a burn-in period for the chain, and then use the Metropolis algorithm to generate otherwise unobtainable samples and reach a conclusion.

## March

3/31/16  John Hossler, Seattle Pacific University
Let’s Play!  Principles of Gamification in Higher Education, Especially in STEM Courses

While the word "gamification" may sound like it means playing games in class, it means something entirely different: the infusion of game principles into an otherwise non-game situation. Gamification is the addition of game elements, mechanics, and principles to non-game contexts--the classroom, for example. Gamified settings are becoming more and more popular in non-classroom contexts, and this research specifically looks at what it might take to gamify an undergraduate STEM course, including advantages, disadvantages, and challenges.  This talk will address some of the principles of gamification, as well as some of the details of its implementation; it will also discuss one popular software program for gamifying a course and provide some examples/ideas specific to STEM disciplines.

3/17/16  Marylesa Howard, National Securities Technology, LLC
The Need for Mathematics, Science, and Engineering in Nuclear Security
The Department of Energy employs scientists, mathematicians, and engineers to work on problems ranging from renewable energy resources to global climate change. However, unbeknownst to many people is the fact that the Department of Energy is also the nation’s overseer of our nuclear weapons program, nuclear non-proliferation, nuclear emergency response, and nuclear power for the U.S. Navy. In this presentation, some of the scientific research interests of the Department of Energy will be highlighted, with a focus on measurement diagnostics and analysis for subcritical experiments in support of the Stockpile Stewardship Program at the Nevada National Security Site: the nation’s premier explosives laboratory.

This work was done by National Security Technologies, LLC, under Contract No. DE-AC52-06NA25946 with the U.S. Department of Energy and supported by the Site-Directed Research and Development Program.

3/3/16  Christian Millichap, Linfield College
How Many Different Ways Can You Prove There Are Infinite Many Primes?
We have known that there are infinitely many primes since Euclid first gave a basic number theory proof in 300 B.C.  Since then, many other proofs have been developed using a variety of tools frommathematics - algebraic number theory, analytic number theory, calculus, and even topology.  In this talk, we shall go over three different proofs - Euclid's proof, Euler's proof which uses some basic results from calculus, and Furstenberg's proof which relies on point-set topology.  We'll also dive a bit deeper into Euclid's proof and analyze a recent result about prime numbers. This talk does not require any background in number theory or point-set topology, but rather, just an interest in seeing how different areas of mathematics can help lead to the same result.

## February

2/25/16  Professor Colin Starr, Willamette University
Prime Product Distance Graphs and Prime Power Distance Graphs
A graph $G$ is a $k$-prime product distance graph if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is the product of at most $k$ primes.  A graph has prime product number $\ppn(G)=k$ if it is a $k$-prime product graph but not a $(k-1)$-prime product graph.  Similarly, $G$ is a prime $k$th-power graph if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is the $k$th power of a prime. We prove that $\ppn(K_n) = \lceil \log_2(n)\rceil - 1$, and that if $G$ is $k$-chromatic $\ppn(G) = \lceil \log_2(k)\rceil - 1$ or $\ppn(G) = \lceil \log_2(k)\rceil$.  We also prove that $K_n$ is not a prime $k$th-power graph for any $k \geq 7$, even cycles are prime $k$th-power graphs for all positive integers $k$, and odd cycles are prime $k$th-power graphs for sufficiently large $k$. We find connections between prime product and prime power distance graphs and the Twin Prime Conjecture, the Green-Tao Theorem, and Fermat's Last Theorem.

2/11/16  Professor Josh Laison, Willamette University
Variations of Graph Pebbling
Graph pebbling is a fun game involving moving pebbles around on a piece of paper, an exciting field of graph theory, and a great source of student research problems.  There are many variations of the original pebbling game.  In this talk, we'll play variations of the game defined by two research teams I've worked with, and prove a few theorems.

2/4/16  Professor Peter Otto, Willamette University
Path Coupling Method to Bound Mixing Times of Markov Chains
In this talk, I’ll first introduce the idea of the mixing time of a Markov chain with a few examples, including the Rook’s walk.  Then we’ll discuss the probabilistic method called path coupling that yields an upper bound on the mixing time.  The talk will include work completed during the Willamette Mathematics Consortium REU during the summer of 2014.

# Fall 2015

## November

11/12/15  Shelbi Jenkins, Jacqueline Remmel, and N. Spencer Sitton
Differential Equations Rock!
It takes a lot of guts to climb on exposed rock features; it takes about as many guts to take on differential equations. In this talk, we're gonna do a little bit of both! Come learn how differential equations help us unlock the secrets of population changes, springs, and more. In particular, we use differential equations and linear algebra to optimize rock climbing gear so that it is as strong and safe as possible.

11/5/15  Joshua Scott, Aimee Reynolds, and N. Spencer Sitton
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.

## October

10/29/15  N. Spencer Sitton, Willamette University, Math Department
The Unsolvable Equations Whose Solutions Can Never Be Found
The great art of obtaining formulas describing the solutions to equations has enticed the efforts of the world’s greatest mathematicians and resulted in the creation of powerful theories that shape modern mathematics.

Solvability of algebraic equations was the great unsolved problem in mathematics during the 16th-19th centuries until 1824, when Abel brilliantly proved the unsolvability of the quintic. Soon after, the extraordinary creativity of Galois definitively answered the question of solvability of algebraic equations and the great art progressed to new class of equations called differential equations.

In this talk, we tour the history of solvability of equations from algebraic to differential. We introduce the geometric theory of differential equations as developed by Lie, Cartan, Goursat and others during the early 20th century and use this theory to prove, following Cartan, the insolvability of the celebrated Hilbert-Cartan equation.

10/8/15  N. Spencer Sitton, Willamette University, Math Department
The Great Art of Solving Equations
At the turn of the 14th century, Antonio Fior challenged Niccolo Fontana to a duel; however, this was not your average duel as the weapons were story problems whose solutions could only be found by solving cubic equations. During this battle of wits, Fontana discovered a method to solve certain cubics which lead to the swift demise of Fior.

Fontana's discovery marked the beginning of the great art of obtaining formulas describing the solutions to equations. The great art has enticed the efforts of the world's greatest mathematicians and their work culminated in the creation of the geometric theory of partial differential equations (PDEs).

In this talk, we consider a parameterized family of second-order PDEs first posed by Goursat in 1898 and then further considered by Cartan in 1910 in his groundbreaking 5-variables paper. This parameterized family of PDEs remained unsolved for the last century until now. To solve this family of equations, I use the geometric theory of PDEs to obtain the general solution.

## September

9/24/15 Jesse Walker, Ph.D., Intel Corporation
The Evolution of Cryptographic Hash Function Design
Cryptographic hash functions have become the workhorse of cryptography, used for authentication, key derivation, commitments, trusted computing, entropy extraction, and random mappings. This talk discusses what they are and how they came to be designed the way they are. It begins by reviewing the definition and important properties of hash functions, and then conducts a tour of key highlights in the evolution of hash function designs: Rabin’s hash function, the Davies-Meyer construction, the Merkle-Damgaard construction, and some of the flavor of modern approaches. At each step the talk examines some of the key attack techniques developed to think about and break the dominant design of the day.

9/17/15 Elton Graves, Rose-Hulman Institute of Technology
How Mathemeticians Play Dot-to-Dot to Design Air-Foils and Other Useful Things
In ancient times, craftsmen used a flexible rod called a spline (ship’s spline) to create the curves needed to design and build the hulls of ships.  In later years, splines were used to help in the building of airplane wings (air-foils).  The idea was that the curve had to go through (interpolate) some given data points (nodes), and the spline was used to draw the smooth curves needed to fit the data.

With the advent of computers in the mid 1900’s, mathematicians developed a method to interpolate a given set of data points using a set cubic polynomials.  This set of polynomials when treated as a piecewise continuous function acted like the craftsman’s spline.  Thus, the name cubic spline.

In this talk, we will use the concepts of cubic polynomials, piecewise continuous functions, parametric equations, and the idea of solving a system of n equations with n unknowns, to create cubic splines.  We will then show how cubic splines are used to interpolate a given set data points (play dot-to-dot) to actually design an air-foil developed by NASA.

9/10/15 Ross Casebolt, Portland State University (WU Alumn '12)
Linear algebra applications: using matrices to represent finite groups
Groups can be found in a wide range of math specialties, and group theory has many applications. Sometimes arbitrary groups can be difficult to work with compared to groups of matrices. Finite groups can be represented using homomorphisms (structure-preserving maps) to the general linear group (the multiplicative group of invertible matrices) of some vector space. This process allows us to use all the powerful tools of linear algebra to tease out information about the group that is represented. In this talk, I will give a brief introduction into groups, field characteristics and modules, and conclude with Maschke's Theorem.

9/3/15 Max Lipton, WU Mathematics Major
Differential Equations on Fractals
Many sciences involve the use of differential equations defined on simple domains like a plane or a sphere, but the real world is not so forgiving. Many naturally-occuring objects are rough, jagged, but surprisingly self-similar. In the 1980s, Jun Kigami and Robert Strichartz developed the theory of differential equations on certain self-similar fractals. In this talk, I will illustrate this theory by explaining how they are defined on the Sierpinski Triangle, one of the most recognizable self-similar fractals.

In the second half of the talk, I will explore the construction of fractals generated in $\mathbb{R}^3$ where the operation of component-wise addition is replaced with the operation of the Heisenberg group, a special non-commutative addition in the $z$-axis with applications to quantum physics. Many familiar fractals like the Koch Snowflake and Twin Dragon Curve have corresponding Heisenberg fractals with fundamental geometric distinctions that could provide the basis for a new theory of differential equations. This work was conducted at Cornell University's 2015 Summer Program for Undergraduate Research (SPUR) under the direction of Professor Robert Strichartz.

A modest understanding of real analysis and abstract algebra will be needed to comprehend the full details of the theory, but come anyways because there are guaranteed to be pretty pictures of fractals!

# Spring 2015

## March

3/31/15 Prof. Erin McNicholas
Mathematics Course Preview 2015
Come learn about several of the exciting courses offered next fall by the math department. This special preview is open only to current Willamette students, their friends and family, and any other interested parties. Come enjoy the treats, learn a little more about such exotic topics as knot theory, and get all your math major/minor/course related questions answered. The only problem will be limiting yourself to at most 4 math classes in the fall. But never fear, if your schedule does not allow you to take every math course offered next semester, many of these courses will be offered again.

3/19/15 Heidi Andersen '11
Fantastic Groups and Where to Find Them
Starting with the fundamental concept of a group that one encounters in a first abstract algebra course, this talk aims to provide the undergraduate listener with a broader, graduate-level perspective on the huge role groups play in many other fields of math (with a focus on topology and geometry). Beyond the elementary, pretty examples like the dihedral and symmetric groups, groups also act on topological spaces and yield new manifolds in the form of quotient manifolds (also called orbit spaces), and groups themselves sometimes admit topological and/or geometric structure. Many examples will be provided.

3/12/15 Prof. Inga Johnson
All Tangled Up: Conway's Classification of Rational Tangles
Tangles are of interest to both mathematicians and biologist due to their applications in the study of DNA. We will look and a subset of tangles called rational tangles with 2 strands. These tangles are basically 2 pieces of string that are twisted together in certain specified ways. Rational tangles have many interesting properties and structures associated to them which we will explore. These properties and structures are the key ingredients to an elegant proof of Conway's Classification Theorem of Rational Tangles due to J. R. Goldman and L. H. Kaufmann ('96).

3/5/15 Prof. Josh Laison
Modern Board Games and the Math Behind Them
More people than ever are playing games, and many of those people are secretly doing math! In this talk I'll introduce the exciting world of modern board games, and give some examples of their many connections to mathematical ideas and research.

# Fall 2014

## December

12/4/14 Jeremy Coste and Kees McGahan
Cops and Robbers on Graphs
Join us as we explore the game of Cops and Robbers on graphs. We will take a look at cop-win and robber-win graphs, as well as finding algorithms for computing the cop number. Furthermore, we will learn some variations of the game with firefighters and helicopters!

## November

11/24/14 Jared Nishikawa, Willamette '10
Hash Functions, A Soft Intro
Number theorists often talk about functions with "nice" properties (additive, multiplicative, periodic, symmetric, and so on). Hash functions are, in this sense, a mathematician's nightmare. But, in terms of cryptography and security, they are very important. This talk will gently introduce what hash functions are, their applications to cryptography (have you heard of bitcoins?), and current and ongoing work. The content will be accessible to both math and computer science majors.

11/13/14 Professor Benjamin Young, University of Oregon
Tiling an Aztec Diamond
An Aztec diamond is a diamond-shaped region of the plane, which can be completely covered with nonoverlapping dominos. We'll work out the number of ways in which this can be done, and look into what a typical tiling of a large Aztec diamond looks like.

11/6/14 Bob Milnikel, Kenyon College
A New Angle on an Old Construction
It's well known that exact straightedge-and-compass construction of a regular n-gon is impossible for most values of n, but that didn't keep people from needing to construct such polygons in the days when straightedge and compass were the principal tools of drafting. I'll introduce a historical technique for approximating a regular n-gon that works (more or less) for any value of n. Finally, I'll introduce a slight variation -- original as far as I know -- that improves the construction's accuracy. The material is very accessible! The only mathematical background needed is a little high school algebra and trigonometry.

## October

10/30/14 Lexi Scheel & Eric Samelson
Lexi & Eric's Summer Research
This past summer, Willamette Math Majors Lexi Scheel and Eric Samelson, participated in mathematics summer research experiences. Lexi worked with a team of researchers at the University of Hawaii at Hilo, and Eric worked with Linfield’s research team. Lexi and Eric will share their research results and discuss the process of applying for and participating in a summer math research experience.

10/23/14 MegaMenger Mania!
We’re in the home stretch! With 4 big days of Menger, now’s your chance to participate—and take part in the big finish.

ØThursday at 4:00 (Ford 204) we’ll fold MegaMenger cubes, while listening to members of the Math department talk about fractals, and watch cool fractal movies!
ØFriday at 3:30, during Friday Floats we’ll continue MegaMenger cube building while enjoying our usual root beer floats
ØThen Saturday & Sunday join us at 2pm (Math Hearth) to assemble the final Level 3! You’ll also have the opportunity to talk with the organizers of global project via Google Hangout!

Partial Differential Equations & Equivalence
Professor Sitton gives a brief, not-too-technical introduction to the geometric theory of partial differential equations (PDEs) as developed by Lie, Cartan, Goursat, Darboux, and others during the early 20th century. This theory allows us to define various geometric properties, including the notion of equivalence, of PDEs. In particular, we consider the following PDEs introduced and studied by Cartan, Goursat, (and Sitton):

9u2xx + 12uxxu3xy + 36uxxuxyuxy – 12u2xyu2xy – 32u3xy = 0
8u3xx + 24u2xxu2xy + 18uxxu4xy - 108uxxu2xyuxy – 18u2xyu3xy + 81u4xy = 0

9/11/14 Dr. Elton Graves, Rose-Hulman Institute of Technology
See the Wave: A Mathematical Simulation of the Waller Violin"
The card game SET is played with a special deck of 81 cards.

The Waller Violin, and in fact, all stringed instruments work on the basic principle that a string of length L is pinned at both ends and is under tension. The string is plucked and begins to vibrate causing a sound, usually nice music. Because of wind resistance the string will eventually stop vibrating and the music will cease. Mathematically the vibrating string is known as the “wave equation.” Our task in this talk is to simulate, mathematically, the movement of the string.

This talk will take the listener on a tour of the undergraduate mathematics needed to solve the “wave equation”. The talk will weave together topics from integration by parts, simple differential equations, along with a little matrix theory, and least squares (linear regression). These mathematical concepts will be the stepping stones which lead to the concept of the Fourier series, which is the ultimate mathematical tool used to solve the “wave equation”.

The talk will also show the derivation of the mathematical model of the “wave equation” using elementary vector addition. Once the “wave equation” had been derived, the talk will show the techniques used in solving the “wave equation” for a simulated string, using the elementary mathematics and Fourier series discussed during the talk.
The talk will conclude with a computer graphics animation which actually shows the vibrations of the string we are simulating.

# Spring 2014

## April

4/17 Liz McMahon, Lafayette College
Mathematics in the Game of Set
The card game SET is played with a special deck of 81 cards.

The game has a lot of mathematics hidden within. We’ll look at questions in combinatorics, probability, linear algebra, and especially geometry. The deck is an excellent model for a finite affine geometry, and we will use the game to explore that geometry. If you’d like some practice before the talk, go to www.setgame.com for the rules and a Daily Puzzle.
(If you saw talks on SET last year, this talk will contain new information.)

4/10 Gary Gordon, Lafayette College
Pick a Tree, Any Tree
Trees are an extremely important and useful topic in graph theory and network design. I'll talk about some of the motivation and history of the subject, including Cayley's famous formula that counts the number of spanning trees of a complete graph. Then we'll use that formula to figure out the probability that a randomly chosen subtree of a complete graph is a spanning tree. This is joint work with Alex Chin, Kelly MacPhee and Charles Vincent, three undergraduates in Lafayette College's REU program last summer. No prior knowledge of graph theory will be assumed.

## March

3/13 Paul Cull, Computer Science, Oregon State University
Solving Towers of Hanoi and Related Puzzles
We start by solving the well-known Towers of Hanoi puzzle. Then we solve a lesser known puzzle, Spin-Out. We notice that these puzzles can be described as graphs and define a family of graphs, the {\it iterated complete} graphs which generalize these puzzle graphs. Generalized Towers of Hanoi puzzles correspond to these graphs with odd dimension, and generalized Spin-Out puzzles correspond to these graphs with dimension a power of 2. By “crossing” these puzzles, we obtain combination puzzles for every natural number bigger than 1. We show that these combination puzzles can be solved in essentially the same way as Towers of Hanoi and Spin-Out. We also show how to compute the number of moves between any two configurations of these puzzles. Our iterated complete graphs have a number of remarkable properties. For example, they have Hamiltonian paths and perfect one-error-correcting codes – properties that are NP-complete for general graphs. We also discuss computational complexity and show that many calculations on our graphs. We also discuss computational complexity and show that many calculations on our graphs and puzzles can be carried out by finite state machines.

3/7 Matt Anderson
A Prime Producing Polynomial
To me, prime numbers are interesting. Although there are not as many practical applications like in statistics, physics, and engineering; there is a certain mystery and challenge in their study. My study of prime numbers has revealed many unsolved problems. For example, although it is known that many linear functions with integer coefficients and integer input variables will produce a sequence with an infinite number of prime numbers in it (Dirichlet’s Theorem), it is not known if this is the case for polynomials of degree 2 or more. This is the Bouniakowsky Conjecture. This talk will focus on a quadratic polynomial, namely x^2 + x + 41. It is my finding that many restrictions on x will yield an infinite sequence of composite numbers.

## February

2/20 K. Tucker (a.k.a. k-TUCK)
Enumeration and Projection Dependence of 1-Singular Knots
I will describe the methods of enumerating knots with a lone singularity developed during the James Madison University Knot Theory REU, methods we used to distinguish these one-singular knots, and surprising difficulties encountered along the way. These surprises include the projection dependence on the classic knots from which one-singular knots are obtained, even when the projections are both minimal in terms of crossing number. We also show that the two standard projections of (p,q)-torus knots yield different one-singular sets if p < 3q/2.

2/20 R. Robinson (a.k.a. Ray-Robins)
Convergence of Sequences of Polygons
In 1932, Martin Rosenman proposed the following problem in the American Mathematical Monthly:

Let Pi be a closed polygon in the plane with vertices z_0, z_1,...,z_{k-1}. Denote by z_0^(1), z_1^(1),...,z_{k-1}^(1) the midpoints of the sides. Using z_0^(1), z_1^(1),...,z_{k-1}^(1) as vertices, we derive a new polygon, denoted by Pi^(1). Apply the same procedure to derive the polygon Pi^(2). After n constructions, we obtain polygon Pi^(n). Show that Pi^(n) converges, as n approaches infinity to the centroid of the original points.

I will present various approaches to the solution of this and related problems.

# Fall 2013

## December

12/5 Jordan Purdy, Mathematics Dept
Spatial Statistics - Logistic Regression, the Autologistic Model and Mountain Pine Beetle
When information on a binary response variable is collected for many observational units, the logistic model is commonly implemented to describe the probability of “success” as a function of one or more explanatory variables. As long as the response variables are independent, such a paradigm is appropriate. However, when binary responses on a regular lattice are observed in space and/or time, spatio-temporal dependencies typically exist and the logistic model is rendered invalid. Thespatio-temporal autologistic model is an intuitive extension of the logistic model that accommodates such a lack of independence. In this talk we will review the logistic model and introduce the spatio-temporal autologistic model along with the inherent challenges associated with its implementation.

Data on the spread of Mountain Pine Beetle in Montana will be used to motivate the generalization of the logistic model into the space-time domain.

12/4 Samantha Reynolds, Willamette University '14
College Entrance Exam Firms, Nonprofit Efficiency, and Testing Fees
College entrance exam companies such as the College Board or the ACT claim nonprofit status. Theoretically these companies should not have high costs and considering that they aren’t profit driven, we would expect to see low testing fees. In reality this is not the case and many would claim that it stems from the inefficiency of the nonprofit. I analyzed whether high test fees could be the result of a company’s primary mission rather than inefficiency. Using the team incentive problem and the role of a budget breaker, I showed that nonprofits can induce workers to provide an effort level that minimizes costs in order to maximize net revenue. Assuming the firm has idealistic workers, the model can be extended where we still maximize net revenue without a principal playing the role of a budget breaker. The primary mission of nonprofits takes the form of a publicly valued good or service and that by maximizing revenue they can maximize the amount allocated to producing the public good. This implies that test takers may pay high fees not because the firm necessarily is inefficient but because the firm is trying to maximize how much of the public good is produced.

## November

11/14 Professor Inga Johnson, Math Department
Topology, Homology, and Applications to Data
Topology is the subfield of mathematics that is concerned with the study of shape. Mathematicians have studied topological questions for the past 250 years. In the past few years a new interdisciplinary field has blossomed bringing together topologists, statisticians, computer scientists, engineers and others, to use topological ideas to study data sets in new and exciting ways. We will discuss one of the new topological tools that has been developed called persistence homology.

This talk will be an introduction to topology and the concept of homology. We will then use homology to a look at examples of how topological ideas can be used to give new and surprising insight towards understanding data. This talk will emphasize examples and concepts. Prerequisites will be minimal.

## October

10/31 Jeff Schreiner-McGraw and Will Agnew-Svoboda
Unipancyclic Matroids
A unipancyclic (UPC) graph is a graph containing exactly one cycle of every possible size. Only a handful of these are known to exist, although searches have been performed through all graphs with 56 or fewer vertices. We generalized this problem by seeking to find and characterize UPC matroids. There are UPC matroids that are not graphic, so this does result in a larger family. In this talk, we will discuss the progress from the summer's research program.

10/24 Nancy Ann Neudauer, Pacific University
What is a Matroid? Investigations of asymptotic enumeration in matroids
In 1933, three Harvard junior-fellows tied together recurring themes in mathematics into what Gian Carlo Rota called one of the most important ideas of our day. They were finding independence everywhere they looked. Do you? We find that matroids are everywhere: Vector spaces are matroids; We can define matroids on a graph. Matroids are useful in situations that are modeled by both graphs and matrices. We consider how we can ask research questions about matroids, and look into results from a student's investigation.

Two matroids are commonly defined on a graph: the familiar cycle matroid and the more rarely-encountered bicircular matroid. The bases of the cycle matroid are the spanning trees of the associated graph; the bases of the bicircular matroid are all subgraphs of the graph, each of whose connected components contain exactly one cycle and (possibly) other edges. We enumerate the bases of the bicircular matroid for several classes of graphs. For a given graph, usually there are more bases of the bicircular matroid than of the cycle matroid. We ask when these numbers are the same. We also consider when there are more bases of the cycle matroid, and what this translates to in terms of the structure of the graph. No prior knowledge of matroids or graphs is needed!

10/3 Yumi Li, Math Major
Put Your Thinking CAPS On (Exploring Finite Geometry in the Card Game SET®)
Besides being a great card game, SET® serves as an excellent model for the ﬁnite geometry AG(n,3). Using the SET® cards as a visual representation, we will explore the structure of maximal caps and how we can manipulate them to discover new properties and substructures of AG(n,3). This work was done at the Research Experience for Undergraduates program at Lafayette College.

## September

9/19 Ryan Wright, Janrain Inc.
Computing the Coming Robot Apocalypse: The math behind Artificial Intelligence and Machine Learning
Let’s face it, it’s only a matter of time before machines rise up and take over the world. From image recognition, to Netflix recommendations, to predicting the future, Machine Learning and Artificial Intelligence are at the heart of some of the coolest technology being developed today. We give a quick introduction to how these technologies work and explain why math is how we welcome our future robot overlords.