# Fall 2011

### September -

9/22 Inga Johnson, Willamette University
We will learn about and play a new topological combinatorial game called the Link Smoothing Game. The game is played on the shadow of a link diagram and legal moves consist of smoothing precrossings. One player wants to keep the diagram connected while the other player wants to disconnect the shadow. We make significant progress towards a complete classification of link shadows into outcome classes by capitalizing on the relationship between link shadows and the planar graphs associated to their checkerboard coloring.

9/29 Peter Otto, Willamette University
Mathematics of Mixing Things Up

Suppose a company has developed a new card shuffling machine and is hoping it’ll be used in many casinos. The company comes to you and asks you to help them determine if their new machine is any good. How do you go about answering this question? What does a “good” shuffling machine mean? This question has been well-defined mathematically and is the basis of an important theory of probability called mixing times of Markov chains. In this talk, I will give an overview of this (relatively) new area of mathematics and if time permits, introduce some of my own research work in this field.

### October -

10/6 Michael Dorff, Department of Mathematics, Brigham Young University
Soap Bubbles, Shortest Paths, and When Does the Derivative of the Area Equal Perimeter?
In high school geometry, we learn that the shortest path between two points is a line. In this talk, we will explore this idea in several settings. First, we will apply this idea to finding the shortest path connecting four points. Then we will move this idea up a dimension and look at a few equivalent ideas in terms of surfaces in 3-dimensional space. Surprisingly, these first two settings are connected through soap films that result when a wire frame is dipped in soap solution. We will use a hands-on approach to look at the geometry of some specific soap films or “minimal surfaces”. As we explore this idea ofmoving through dimensions, we note that the derivative of the area of a circle with respect to its radius equals its circumference. This leads to the question “for what geometric shapes does the derivative of its area equal it’s perimeter?” We will end with some interesting examples of this and some open questions.

10/7 Michael Dorff, Department of Mathematics, Brigham Young University

Toy Story 3, the “real” Iron Man Suit, and advising the President of the United States
Have you ever been asked “What can you do with a degree in math?” Besides teaching, many people are clueless on what you can do with strong math skills. For the past three years, I have been hosting a “Careers in Mathematics” seminar and inviting mathematicians to talk about how they use math in their careers from research scientist at Pixar Animation Studios to operations research analyst at the Pentagon in Washington DC. In this talk, we will present some highlights from these mathematicians and their careers.

### November -

11/3 Colin Starr, Associate Professor of Mathematics
The Nine-Point Circle and Feuerbach's Theorem
We will explore one of my very favorite theorems from anywhere in mathematics.  This one happens to come from geometry and is about some beautiful relationships among some of the special points of a triangle.

11/10 Kathryn Nyman, Assistant Professor of Mathematics
The Peak Algebra of the Symmetric Group
If you take a permutation on the integers 1, ..., n, and look at the places where the numbers increase and then decrease, you will have spotted the "peaks" of the permutation.  If you then group permutations based on where they have peaks, you can form an algebra (known as the peak algebra).

We will take a look at the structure of this (combinatorially described) peak algebra as well as a couple places where peaks and descents of permutations naturally occur. One such place is in the arrangements that result from shuffling cards. Warning: this talk requires some linear algebra (and knowing some abstract algebra would be helpful).

11/17 Erin McNicholas, Assistant Professor of Mathematics
Spring Course Preview

### December -

12/1 Kristine Pelatt, University of Oregon
The Hairy Ball Theorem
If you have a ball, completely covered with hair, can the hair be combed without a cowlick?  We will start by investigating some properties of triangulations of polygons. Using what we learn, we will conclude that the answer to this question is no - every time you comb the hair there will be cowlick.  We will end with examples of ``hairy'' shapes which can be combed.

# Spring 2012

### February-

2/23 Josh Laison, Associate Professor of Mathematics
Variations of Graph Pebbling
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 the game, and variations defined by two different research teams I’ve worked with. We might even have time to prove a couple theorems! Pebbles will be provided.

### March-

3/15 Pete L. Clark, University of Georgia
From cockroaches to marriage via graph derangements
I will relate the true story of how a puzzle involving cockroaches that a friend told me (and several other mathematicians) on the patio behind the Normaltown Bar in Athens, Georgia on a hot night last summer inspired me to create a new subfield of graph theory: graph derangements.  No, seriously.  Starting from scratch (or rather, from cockroaches skittering on a kitchen floor) I will introduce the notion of a graph derangement and discuss the two main problems: their existence and their classification.  The first problem will be solved by an appeal to a famous theorem in combinatorics.  The second problem largely remains open and seems like it could be a nice research topic for mathematicians of all ages.

### April -

4/9 Francis Edward Su, Harvey Mudd College

Voting in Agreeable Societies
When do majorities exist?  How does the geometry of the political spectrum influence the outcome?  What does mathematics have to say about how people behave?  When mathematical objects have a social interpretation, the associated theorems have social applications.  We give examples of situations where sets model preferences, and prove extensions of classical theorems on convex sets such as Helly's theorem and Turan's theorem that can be used in the analysis of voting in "agreeable" societies.  This talk also features research with undergraduates.

4/12 Nicole Seaders, Visiting Professor of Mathematics
Using Group Theory to Generalize Diatonic and Pentatonic Scales
In Western Music, the equal-tempered 12-tone system has become the norm. Within the 12-tone system, traditional music often uses the diatonic and pentatonic scales. In non-Western music, other tuning systems have been used like the 22-tone system (not equal tempered) in India, with 72 different scales. Today, some Western composers are exploring alternate tuning systems, creating what is called microtonal music. In this talk we will use group theory to understand some of the structure of the diatonic and pentatonic scales in a 12-tone system, and generalize these properties to other equal-tempered systems.

4/19 Hanna Bliden
Investing in Inequality? Education Subsidies and Income Inequality
Education is commonly believed to be a stepping stone to success, leading to a constant effort to make higher education affordable for everyone. But what if education subsidies actually do more harm than good? Using an economic model presented by Hendel, Shapiro, and Willen we can explore how education functions in the labor market and how education subsidies may actually increase income inequality.