### Combinatorics

**Peter Kagey, Oregon State University****Random Walks and Recreational Mathematics**

Recreational Mathematics in the style of Scientific American's Martin Gardner inspire mathematical curiosity and lead to some unexpected places. This accessible presentation will look at random walks on the plane, on the number line, and on connected graphs through the lenses of numerical analysis, combinatorics, and graph theory.**Kuai Yu, David Perkinson, Qiaoyu Yang, Reed College****Parking functions and tree inversions**

A depth-first search version of Dhar's burning algorithm is used to give a bijection between the parking functions of a graph and labeled spanning trees, relating the degree of the parking function with the number of inversions of the spanning tree. Specializing to the complete graph answers a problem posed by R. Stanley.**Andrew Fry, Gautam Webb, Luis Garcia-Puente, Western Oregon University****The Abelian Sandpile and its Avalanche Polynomials**

Imagine yourself on a beach, playing in the sand. You begin to make a sandpile by adding handfuls of sand. Now you consider dropping another grain of sand onto the pile but you donâ€™t know what will happen. It may cause nothing to happen or it may cause the entire pile to collapse in a massive slide. This is the idea behind the Abelian sandpile model. We accomplish this task by using uses directed graphs where we denote one vertex as the sink and at all other vertices we have a nonnegative integer. These integers represent the number of grains of sand placed in that sandpile. When the pile gets too big, meaning the number gets too large, an avalanche occurs sending grains along each edge adjacent to the toppling vertex. We measure the sizes of these topplings and build what is called the avalanche polynomial. This topic is based on graph theory, group theory, and enumerative combinatorics.**Jeff Schreiner-McGraw, Willamette Univeristy****Unipancyclic Graphs and Matroid Theory**

Matroid Theory is an abstraction of Graph Theory and Linear Algebra. The Willamette REU project extended the definition of Unipancyclic Graphs into Matroid Theory and attempted to prove new results about Unipancyclic objects. Where a Unipancyclic Graph G has exactly one cycle of each size from 3 to r+1, a Unipancyclic Matroid M has exactly one circuit of each size from 3 to r+1. Also, all Unipancyclic Graphs can be described as a Unipancyclic Matroid. This talk will introduce listeners to the basics of Matroid Theory, share interesting results about Unipancyclic Matroids and discuss the application of these results to solving problems with Unipancyclic Graphs.

### COMAP MCM

**Lacey Mesia, Corey Bennett, Eastern Oregon University****COMAP Problem B- Coach of the Century**

The Consortium for Mathematics and Its Applications holds the Mathematical Contest in Modelling every year in February. Two problems with concentrations in continuous and discrete modelling are proposed and teams have a total of roughly four days to complete a solution to one of the problems. Our chosen problem was in the discrete concentration. We talk about the solution we reached and the methods to get there that were employed.**Kyle Evitts, Kara Grant, Nathan Mills, Linfield College****Dukes of Hazzardous Lane Changes**

In this talk we present the work we did for the COMAP-MCM competition. Our paper examined the effects of various freeway passing schemes on traffic congestion and driver safety by using a computer simulation. The three rules we considered are the common keep-right-unless-passing, and slower-traffic-keep-right rules, as well as a no passing rule as a control. Overall, our data suggested to us that the slower-traffic-keep-right-rule seems to be marginally better at reducing traffic congestion. We found that driver compliance was an important factor in traffic flow.**Peter Kagey, Daniel Takamori, Oregon State University****Modeling and Optimizing Traffic**

Results from the 2014 Consortium for Mathematics and Its Applications (COMAP) Mathematical Contest in Modeling. Model of cars as discrete stochastic automata.

### Geometry

**Andrew Bishop, Willamette University****Intersection Graphs of Translated Regular Polygons**

Intersection graphs are graphs in which each vertex represents a set and two vertices are adjacent if and only if the sets intersect. Our research focuses on intersection graphs in which the sets in question are translated regular polygons in the plane. There a variety of methods for answering questions about these graphs, which draw from both geometry and graph theory. We prove that for odd $n$, the graphs representable with regular $n$-gons are exactly those representable with regular $2n$-gons. We also show that there are graphs representable with triangles (and hexagons) but not squares, and discuss our investigation as to whether there are graphs representable with squares but not triangles.**Jem Kloor, Southern Oregon University****Fractals: Infinite Iteration**

Fractal geometry is a fascinating and relatively new branch of mathematics which studies complicated, iterative figures known as fractals. Unlike the smooth regularity of classical mathematics, fractal geometry allows us to understand and model the chaotic and irregular natural world. Fractals display numerous curious properties and range widely from the simple Sierpinski Triangle to the complex Julia Set. Fractal geometry not only provides rich new insights into mathematics, but it also suggests that perhaps there isn't anything mathematics can't explain.**Thomas Pitts, Oregon State University****Euclidean Tessellations and Regular Polygons in the Taxicab Plane**

Tessellations are planes made from repetitions of geometric shapes without gaps or overlaps. Tessellations exist in any geometry in which polygons exist, including elliptic, hyperbolic and Euclidean geometries, but tessellations in Euclidean geometry are perhaps the most widely known and explored. Of particular interest are regular tessellations, tessellations which, in Euclidean geometry, are made of equilateral and equiangular polygons. We have an intuitive notion of what it means to be equilateral under the usual Euclidean metric, but how does a change of metric affect equilateral and regular Euclidean polygons? Here we explore Euclidean tessellations viewed under a different metric, namely the taxicab metric. First, we examine how the three regular Euclidean tessellations change when viewed under the taxicab metric. After observing that regularity is not preserved by our change of metric, we redefine regular polygons to be simply equilateral polygons under the taxicab metric. With this definition in hand, we explore new options for regular tessellations, and in particular show that we are not limited to only tessellations of triangles, squares, and hexagons as we are under the Euclidean metric.

### Probability and Statistics

**Grayson Penfield, University of Portland****A Closer Look At The Ratings Percentage Index**

In the 1980's, the Ratings Percentage Index (RPI) was introduced as the standard method for ranking professional sports teams. Since its introduction, the RPI formula has been targeted as a poor method for providing rankings that accurately capture the true talent of a team. Many people have proposed their own ranking algorithms, and a handful of these algorithms have become notable for their simplicity and accuracy. To this day the RPI still remains as the number one ranking algorithm. In my presentation I will discuss the main problems with the RPI and offer a small modification to the RPI formula that allows for easier calculation, eliminate completely one of the main problems with the formula and may improve its ability to capture the true ability of a team.

### Algebraic and Complex Geometry

**Sebastian Bozlee, Aaron Wootton, University of Portland****Asymptotic Equivalence of Automorphism Groups of Surfaces and Riemann-Hurwitz Solutions**

An automorphism group of a compact Riemann surface is often described by a tuple $(h; m_1, ..., m_s)$ called its signature which encodes the topological data of the group. There are certain number theoretical conditions on a tuple necessary for it to be the signature of an action: the Riemann-Hurwitz formula is satisfied and each $m_i$ equals the order of a non-trivial group element. Our main result is that asymptotically speaking, satisfaction of these two conditions is sufficient for a tuple to be a signature.

### Mathematics Applications in the Sciences

**Yongyi Li, Oregon State University****Application of Mathematics to Traffic Flow**

As the development of automobile industry, more and more people have their own cars. Recent years, the cars price are getting cheaper all the time as the automobile companies produce more and more vehicles with advanced automobile industry. People can get a car as a lower price than past. For this reason, the number of people who own cars is on the rise. Some people have bought cars of their own, and others are planning to buy cars. By the website of the department of transportation of United States, there are approximately 240 million automobile in United State and the number continues to grow. The number of vehicle is meant that there is eight persons in ten having their own cars. Cars are bringing more convenience to human being. However, there are more trouble cause the increasing number of vehicles. If you are living large cities, such as New York or Los Angeles, you would face to huge traffic flow and traffic jam. Therefore, how to control the traffic flow, reduce traffic congestion is a popular topic on the modern society.**Steven Reeves, Southern Oregon University****Modelling Tsunamis with the Shallow Water Equations**

Tsunami waves are infrequent but spectacularly devastating events. Tsunamis can be mathematically modelled using the so-called Shallow Water Equations, which are derived from the Navierâ€“Stokes Equations with appropriate assumptions on the wave height, depth, and length. Numerical integration of the Shallow Water Equations can then be used to simulate tsunamis for a variety of bathymetries (shore geometries).

### Number Theory

**Taylor Matyasz, Pacific University****Counting Minimal Solutions to Diophantine Inequalities**

Diophantine equations, those in which only integer solutions are of interest, have been studied since the 3rd century. Also of great interest are Diophantine inequalities. Indeed, these inequalities arose even in senior capstone projects. In 2013, Evan Cooper encountered a Diophantine inequality in his work to improve upon a college football ranking system. Cooper needed to determine the so-called minimal solutions to his inequality. Inspired by his work, we seek to determine the number of minimal solutions to a general linear Diophantine inequality. We begin by investigating the case of a two dimensional, linear, Diophantine inequality, and find a simple closed expression for the number of minimal solutions. For higher dimensional problems, we are able to determine a recursive formula for the number of minimal solutions. Finally, we make use of Ehrhart polynomials to find a closed form for both an upper and lower bound on the number of minimal solutions to a general linear Diophantine inequality in n variables.

### Mathematical Aspects of Computer Science

**Georgia Mayfield, Erik Holmes, Dennis Moritz, Kathryn Adamyk, Marion Scheepers, Willamette University****Game Theory and Algebraic Structures**

Ciliates, a single celled organism, upgrade their genome by reordering the encrypted DNA of their micro nuclei into readable strands. The decryption process uses context guided operations which can be modeled on permutations. Using a graphical representation of the permutation we have characterized which strings can be decrypted using one of these operations. Strands that cannot be decrypted by this operation can be analyzed using finite, determined games between two players. Utilizing graphical representation, we have found criteria for deciding which of the two players has the winning strategy for certain permutations**Nyki Anderson, Eastern Oregon University****Using Visualization Tools to Analyze Automated Flight Conflict Resolution Concepts**

Air traffic is expected to increase in the coming years. Currently, air traffic controllers are responsible for maintaining a safe and efficient National Airspace System. However, it is unlikely that human cognitive abilities will be able to keep up with the continuous increase in traffic. In response, automated ground-based and airborne separation assurance algorithms have been developed. The safety of the NAS is of the utmost importance and therefore, meticulous evaluation of the algorithms is a crucial step in implementation of these concepts. Thus, it has become necessary to develop external visualization tools designed to verify, validate, and improve the existing algorithms by making them more robust. These visualization tools allow researchers perfecting the algorithms to run simulations without having to run entire programs or filter through vast amounts of data.

### Topology

**Jonathan David Evenboer, Oregon State University****Topological Invariance of Surfaces of Constant Curvature in Euclidean 3-space**

We explore the topic of topological invariance by investigating aspects of Gauss-Bonnet Theorem related to surfaces of constant curvature embedded in Euclidean 3-space. We informally show that closed, compact, simply connected 2-manifolds without boundary in Euclidean 3-space are all homeomorphic to the sphere.

### Other (Specify in Comments)

**Kyle Evitts, Levi Altringer, Amanda Bright, Gregory Clark, Brain Keating, Brian Whetter, Linfield College****Counting Tilings of Annular Regions**

In this talk, we present work done during an REU this past summer, in which we investigated tiling questions for rectangular annular regions using the set of T and skew tetrominoes. We briefly discuss our main result classifying which of these regions can be tiled, and then present the proof of a short corollary describing the number of ways one can tile some of these regions.