# 2015-2016 Colloquium Schedule

## Fall 2016

### A Geometric & Combinatorial Approach to Weighted Voting

• Speaker: Jason Parsley (Wake Forest University)
• October 6, 2016 at 4:40 pm in Robinson Hall 105
• Refreshments in Robinson Hall foyer at 4:20
• Abstract: Weighted voting refers to the situation where n players, each with a certain weight, vote on a yes or no motion. For one side to win, the weights of its players must reach a certain fixed quota q. A natural example is a corporation: each stockholder is a player with weight equal to the shares of stock he or she owns. In joint work with Sarah Mason, we describe several combinatorial properties of weighted games. We introduce a geometric approach to weighted voting by constructing a polyhedron for each weighted game. We also form a partially ordered set (a poset) containing all weighted games. Connecting these two perspectives, we prove that the poset is a blueprint for how the polyhedra fit together to form Cn, the configuration region of all weights and quotas for n players. This talk assumes no background information and is suitable for any undergraduate.

### What We Did Last Summer (Parents and Family Weekend Event)

• Speakers: W&L Math students Holly Paige Chaos, Margaret Kallus, Max Rezek and Saimon Islam
• September 30th at 4:40 pm in Robinson Hall 105
• Refreshments at 4:20 pm in Robinson Hall foyer
• Abstract: The students will talk about their summer research.

## Winter 2016

### Monochromatic Solutions of Linear Equations

• Speaker: Mitchel T. Keller (Washington & Lee University) — Pi Mu Epsilon Induction
• March 24, 2016 at 3:35 pm in Robinson Hall 105
• Followed by refreshments in Robinson Hall foyer
• Abstract: A coloring of the positive integers is a function $f$ from the positive integers to some finite set of colors. For example, if the set of colors is $\{\text{red}, \text{green}, \text{blue}\}$, we might say that $1$ is colored red, $2$ is colored blue, $3$ is colored blue, $4$ is colored blue, ..., $8$ is colored green, $9$ is colored blue, $10$ is colored red, and so on. Given a linear equation with integer coefficients, say $x+y-z = 0$ or $x+y-3z=0$, a monochromatic solution is a solution to the equation in which all of the values are the same color. For example, although $1+2 - 3 = 0$, $(1,2,3)$ is not a monochromatic solution to $x+y-z=0$ in our coloring above, since we have $1$ colored red and $2$ colored blue. However, $(3,9,4)$ is a monochromatic solution to $x+y-3z=0$, since $3$, $4$, and $9$ are all colored blue and $3+9-3\cdot 4=0$. In this talk, we will explore some classical results that help us identify which linear equations with integer coefficients must have a monochromatic solution in every coloring of the positive integers.

## Fall 2015

### What I Did Last Summer (Parents and Family Weekend Event)

• Speakers: W&L Math students Trenten Babcock, Elliot Emadian, Emily Jaekle, Ryan McDonnell, Luke Quigley, and Lewis Sears.
• October 2, 2015 at 4:40 pm in Robinson Hall 105
• Refreshments at 4:20 pm in Robinson Hall foyer
• Abstract: The students will talk about their summer research.