# Math Colloquium Abstracts Archive '15 - '16

# Spring 2016

## April

**4/28/16 Colleen Chrisinger, Oregon Department of Revenue Taxing Marijuana and Other Adventures**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.

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**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 from mathematics - 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**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.

Prime Product Distance Graphs and Prime Power Distance Graphs

**2/11/16 Professor Josh Laison, Willamette University**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.

Variations of Graph Pebbling

**2/4/16 Professor Peter Otto, Willamette University**

**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.**

**Path Coupling Method to Bound Mixing Times of Markov Chains**

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# 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**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.

Willamette Students and the McKay Algebra Academy

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 9^{th} grade students in order to graduate on time. We would like to partner with Willamette students in order to help tutor and mentor our 9^{th} grade students in Algebra I classes. This is a great chance to volunteer and give back to the Salem community.

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## 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 16^{th}-19^{th} 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 20^{th} century and use this theory to prove, following Cartan, the unsolvability 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**

**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.**

*How Mathematicians Play Dot-to-Dot to Design Air-Foils and Other Useful Things*

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 airfoil developed by NASA.

**9/10/15 Ross Casebolt, Portland State University (WU Alumni '12)**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.

Linear algebra applications: using matrices to represent finite groups

** 9/3/15 Max Lipton, WU Mathematics Major**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-occurring 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.

Differential Equations on 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!