# Mathematics Colloquium

## September

**9/18/14 Spencer Sitton, Willamette University Mathematics Department
**

*Partial Differential Equations & Equivalence*

### Abstracts Archives 2013-14

##### Previous Abstracts

#### 2012-2013 2011-2012 2010-2011 2008-2009 2007-2008 2006-2007

# Fall 2014

## 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.