UW Bothell and Seattle University join forces to host a 2015 research symposium that will showcase a plenary talk and the results from various groups from the National Science Foundation Research Experiences for Undergraduates (REU) sites at that two campuses. The full program for August 13 is below.

9:00-9:20 Light breakfast – DISC 1st floor atrium

9:30-10:20 Invited talk, DISC collaboratory

**Mathematical Fights! The seedy underbelly of mathematical history**

Dominic Klyve, Central Washington University

Abstract: Although students are often led to believe that mathematics is a purely rational, unemotional, and orderly field of study, history shows that this is often not true. This talk will discuss some of the greatest fights in the history of mathematics. We will hear stories of friendships destroyed and national rivalries heightened because of disagreements about underlying mathematics. We will also look at what these fights teach us about the nature of mathematics, and we will learn some interesting math on the way.

10:30-10:50 DISC collaboratory

**Momentum Term for the Modified Spectral Projected Subgradient Method (MSPS)
**

Samantha Clapp, Charles Cratty, Breanna Page

Abstract: The phenomenon of Zigzagging of Kind I is present in pure subgradient optimization algorithms when at an iterate p_k, the subgradient direction s_k forms an obtuse angle with the previous movement m_k. The main objective of this project is to identify and correct the Zigzagging phenomenon of Kind I for MSPS. We do this by adding a proportion of m_k to s_k; this proportion is called the momentum term and is denoted by t. We create a function that defines t such that Zigzagging of Kind I is eliminated. We track zigzagging, conduct numerical experimentation using Matlab, and compare our numerical results to those of the original MSPS algorithm.

11:00-11:20 DISC collaboratory

**Unilateral and Equitransitive Tilings by Equilateral Triangles of n Different Sizes**

Rebekah Aduddell, Morgan Ascanio, Adam Deaton

Abstract: A tiling is said to be unilateral if no two equal sides of polygons meet corner to corner, and equitransitive if any tile can be mapped via a symmetry of the tiling to any other congruent tile. It has been shown that a unilateral and equitransitive (UE) tiling can be made with any arbitrary number of squares. We show that there are exactly two UE tilings by equilateral triangles: one with two sizes of triangles and one with three sizes of triangles.

11:30-11:50 DISC collaboratory

**Quadratic Prime-Generating Polynomials Over Z[i]**

Frank Fuentes and Monta Meirose

Abstract: The quadratic polynomial x^2+x+41 is prime for x = 0, 1, …, 39. For this reason, it is called a prime-generating polynomial. Many other prime-generating polynomials have been discovered by computer searches, and their efficiency at producing primes can be predicted in some special cases. In this talk, we find and classify prime-generating polynomials f(z), where the variable and coefficients are permitted to be Gaussian integers. Many of the same criteria for efficiency may be generalized from integer polynomials, though without a natural ordering of Gaussian integers there are some surprising differences. Since Gaussian polynomials live in a two-dimensional space, some symmetry can be observed—rotations and reflections, as well as translations, dilations, and combinations of these create more complicated families of polynomials. Our results so far have led us to polynomials that have a high efficiency on a relatively small region near 0.

12:00-1:00 Lunch – DISC 1st floor atrium

Pick up Panera lunch boxes and enjoy the sun outside!

1:00-1:20 DISC collaboratory

**Bertrand’s Postulate for the Gaussian Integers**

Maiya Loucks, Samantha Meek, and Levi Overcast

Abstract: Bertrand’s postulate states that for all n>1 there exists a prime, p, such that n-1 < p < 2n. We explore an extension of Bertrand’s postulate to the Gaussian integers. The Gaussian integers are the complex numbers where the real and imaginary parts are integers. We define notions of doubling and betweenness for the Gaussian integers and conjecture that there exists a Gaussian prime between any Gaussian integer and its double.

1:30-1:50 DISC collaboratory

**Prime Labeling of Trees with Gaussian Integers**

Hunter Lehmann and Andrew Park

Abstract: A graph of n vertices is said to admit a prime labeling if we can label its vertices with the first n natural numbers such that any two adjacent vertices have relatively prime labels. Here we extend the idea of prime labeling to the Gaussian integers which are the complex numbers whose real and imaginary parts are both integers. We begin by defining an order on the Gaussian integers that lie in the first quadrant. Using this ordering, we show that several families of trees as well as all trees of order at most 54 admit a prime labeling with the Gaussian integers.

2:00-2:20 DISC collaboratory

**It’s Knot All Fun and Games: An Introduction to the Region Unknotting Game**

Sarah Brown and Gianni Gibbs

Abstract: We will present some basic definitions and theorems from knot theory. In particular, we will introduce the idea of a region crossing change, an unknotting operation on a knot diagram. We will introduce a region crossing change knot game that we discovered to explore properties of families of knots.

2:30-2:50 DISC collaboratory

**A Guaranteed Win: Optimal Strategies for the Region Unknotting Game**

Franky Cabrera and Riley Evans

Abstract: Expanding on the previous talk, we will present winning strategies of knot games for certain knot diagrams, including diagrams of twist and torus knots. We will also generalize some of our findings to infinite families of twist knot diagrams.

3:00-3:30 Tea Time – DISC 1st floor atrium

3:30-3:50 DISC collaboratory

**Regions, Crossings, and Alternating Knots**

Ra’Jene Martin, Colin Murphy, and McKenna Renn

Abstract: Colin Adams introduced the notion of almost alternating knots which require one crossing change to create an alternating diagram. We extend this notion in conjunction with Ayaka Shimizu’s work on region crossing changes to develop the idea of a region almost alternating knot. This is defined as a knot where there exists a diagram such that a single region crossing change will produce an alternating diagram. In our talk, we discuss families of knots that are region almost alternating and their characteristics, such as their relation to almost alternating knots, their behavior in a connect sum, bounds on region dealternating numbers, and warping span. Along the way, we answer an open question about the maximum warping span of knots.

4:00-4:30 DISC collaboratory

**Lattice Sphere Numbers**

Izaak Berg, Tyler Campbell, Mike Emerick-Cayton, Zonia Menendez

Abstract: We define the lattice sphere number of a lattice, L, to be the minimum radius of a lattice sphere containing a non-trivial knot. We use a variety of arguments to establish the lattice sphere numbers for the simple cubic lattice (sc), face-centered cubic lattice (fcc), two variations of the body-centered cubic lattice (bcc-14 and bcc-8), and the simple hexagonal lattice (sh).