# Math Colloquium Abstracts Archive '17 - '18

# Spring 2018

## April

**4/10Stephanie Partlow (WU Alum), Woodburn Wellness, Business and Sports School**

*From Calculus to Lightsabers: Fostering Student Success and Interest in Advanced Mathematics in a Culturally and Economically Diverse High School*In this talk I will share my journey through my first five years of teaching, focusing on supporting vulnerable student groups, increasing student success in calculus, and fostering interest in post-secondary mathematics topics. We will look at stories of students flourishing despite obstacles and adversity and the lasting impact teachers have on students and students have on teachers.

**4/12Emerald Stacy, Oregon State University**

*Totally P-Adic Numbers of Small Height*

An algebraic number is a root of a polynomial with integer coefficients. An algebraic number is totally real if its minimal polynomial factors completely over the real numbers. For example, sqrt(2) is totally real since its minimal polynomial is x^2-2, with roots +sqrt(2) and -sqrt(2). In contrast, 2^(1/3) is not totally real, since x^3-2 has two nonreal roots. We extend this idea to p-adic numbers, and say a number is totally p-adic if its minimal polynomial splits completely over Q_p. In particular, we’ll investigate totally p-adic numbers of small height. This talk will assume no prior knowledge of p-adic numbers or height functions.

**4/5**

**Hannah Jackson, Clark College**

*Mathematics Education at Two-Year Colleges: Benefits and Challenges of Teaching a Diverse Population*

Want to be a professor? Have you considered teaching at a community college? Math educator Hannah Jackson will discuss her experience moving from Willamette University math major to tenure-track mathematics instructor at Clark College, a two-year school in Vancouver, WA.

Benefits to teaching at a two-year college include a fast-tracked career as compared to a tenure-track position at a four-year institution (no need to get a PhD! shorter tenure processes!), and the opportunity to work with highly motivated students from a wide variety of backgrounds. With diverse student populations however comes issues of inequity in learning outcomes, and STEM fields are not immune to bias in and out of the classroom. Hannah will share research highlighting the impact of implicit bias in the community college classroom and provide tools for recognizing and combating those biases as an educator.

This talk will be valuable for individuals considering a career in education or anyone who wants to face their own implicit biases and start taking steps towards counteracting them.

## March

**3/15**

**Everett Sullivan, Lewis & Clark College**

*Linear Cord Diagrams With Long Chords*

A linear chord diagram of size n is a partition of first 2n integers into sets of size two. Geometrically, we consider it as a choice of parings of 2n ordered points. Linear chord diagrams have shown up in permutation patterns, knot theory, and full rook placements on Ferrers boards. The length of a chord is the difference between its start and end point.

We explore a restriction on the set of linear chord diagrams by requiring each chord to have a minimum length k. After constructing a table of counting the number of linear chord diagrams of degree n such that every chord has length at least k, we observe that if we proceed far enough along the diagonals, that they are given by a geometric sequence. We prove that this holds for all diagonals, as well as when the effect starts. We also explore how to compute the columns of the table and the various tools used to do so.

3/8**Richard Moy, Willamette University**

*Factoring Polynomials: You Mean There's More Than The Quadratic Formula?*

Most students learn how to factor polynomials in high school using techniques such as the quadratic formula. However, factoring polynomials becomes more complicated when they have large degree. And what if a polynomial doesn't factor, i.e. it is irreducible? How does one go about showing that a particular polynomial is irreducible? We’ll address these questions and more!

**3/1Vladimir Bulatov, ShapewaysVisualization of Hyperbolic Groups and Art**

*Abstract not available*

## February

**2/22Kyle Hart and Michael Lasarev, Oregon Health Sciences University**

*What is Biostatistics?*When you read a New York Times article about a new drug or special diet that reduces risk of heart disease, a newly discovered gene that increases risk for diabetes, or the discovery that the proportion of children not vaccinated for a certain disease may cause an outbreak, it’s not just biologists doing the work. These breakthroughs come from collaborations, where biostatisticians play a key role in designing studies, analyzing data, and interpreting the results.

Statistics offers interesting and exciting work in diverse areas along with many opportunities to make a positive difference; it’s also a rapidly expanding field with jobs projected to grow much faster than average: 34% between 2016 and 2026, according to U.S. Bureau of Labor Statistics. Statistical analysis and data mining was listed as #2 in the “Top Skills Companies Need Most in 2018” by LinkedIn, #2 best STEM and #1 Business jobs in 2018 by US News, and Fortune magazine ranked statistics and biostatistics among the top graduate degrees based on salary, growth, and job satisfaction.

OHSU Biostatisticians will give an overview of the field of biostatistics and talk about research they do and how they use their analytical skills to collaborate with many types of scientists and clinicians on exciting and meaningful research problems and clinical studies. They will also discuss the Biostatistics graduate programs at OHSU.

**2/1**

**Allison Arnold-Roksandich, Oregon State University**

*Counting ETA-Quotients of Prime Level*

It is known that all modular forms on SL

_{2}(Z) can be expressed as a rational function in η(z), η(2z) and η(4z). This inspired research to create a similar theorem for a subgroup, Γ(N), of SL

_{2}(Z). By using known theorems, and calculating the order of vanishing, it is possible to compute the eta-quotients for a given level. Thus, we can count them, and utilize this count to start to create a basis for all modular forms on Γ(N). In this talk, we discuss the case where N=p a prime.

## January

**1/25Kathryn Nyman, Willamette University**

*Cutting Cakes with Combinatorial Fixed Point*

Imagine you and your friends are sitting down to share a delicious cake topped with cherries on one side, and frosting roses on the corner. It might not be possible to give everyone an identical piece, but is there a way to cut the cake in such a way that everyone is happy with their allocated piece?

We look at several cake-cutting questions including the division of multiple cakes in which the piece you prefer in the second cake may depend on the piece you get in the first cake, and consensus halving, in which we try to cut a cake in two pieces that two players agree are of equal value. Sperner's Lemma, a combinatorial analogue of the Brouwer Fixed Point Theorem, plays a delightful role in the solution of several of these problems.

# Fall 2017

## December

**12/4Leo Goldstein, Willamette University**

*Philosophies of Math: Why Should I Care?*

Many mathematicians and students of mathematics devote their focus entirely on the math itself, without considering the philosophy behind what they are doing. In this talk, we will explore different ways mathematicians and philosophers have thought about math throughout history and use those perspectives to demonstrate that the ways we think about math have a real and meaningful effect on the ways we do math.

## November

**11/30Evan Hedlund, Corban University**

*When Arithmetic Dynamics and Cryptanalysis Collide*

**In modern society, where computers are an integral component in the exchange of information from the mundane to ultra secret, encryption systems are vital to securely communicate ideas and figures through insecure channels such as the internet. Numerous cryptosystems are based on the notion that factoring numbers with large prime factors is hard or the notion that the discrete logarithm problem (DLP) is difficult. Among these are RSA and El Gamal.**

Knowing where potential vulnerabilities lie in these is important to develop stronger encryption standards. For this, we examine some applications of arithmetic dynamics in the construction of “efficient” attacks on factoring and the DLP. In particular, we will consider Pollard’s ρ algorithm, a type of collision algorithm, applied to factoring and the DLP. The birthday paradox and development of cryptology over the last 50 years motivate our exploration.

** **

**11/16Mathew Titus, Oregon State University**

*A Smattering of Applications of the Laplace Operator*

**Humans have studied the Laplace operator (∆) for longer than the United States has been an independent nation, and during that time it has served scientists and mathematicians in multifarious roles. Despite its innocuous appearance in the one-dimensional euclidean setting (∆ = d^2/dx^2), the operator has proven fundamental in problems from partial differential equations to spectral geometry, and from cohomology to modern-day data analysis. In this talk we will discuss a handful of techniques and problems (from among eigenfunction decomposition, the Karhunen-Loève theorem, fractal fields, dispersion equations, and manifold learning), which all have their roots in ∆.**

**11/9Matthew 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**

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.

*Clique-Relaxed Graph Coloring*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

**10/26William 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 OSymmetry 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 HazelLARC - 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.