**Double-n Circular Societies**

**Students:** Edwin Baeza, Nikaya Smith, Sarah Yoseph

**Abstract:**

A society is a geometric space with a collection of subsets that represent voter preferences. We call this space the spectrum and these preference sets approval sets. The agreement proportion is the largest fraction of approval sets that intersect in a common point. Klawe et al. considered linear societies where approval sets are the disjoint union of two intervals, or double intervals. We examine arc-shaped double intervals on circular societies. We consider the case of pairwise-intersecting intervals of equal length and call these double-n circular societies. What is the minimal agreement proportion for double-n societies? We show that the asymptotic agreement proportion is bounded between 0.3333 and 0.3529and conjecture that the proportion approaches 1/3.

**An Analogue of the Median Voter Theorem for Approval Voting **

**Students:** Ethan Bush, Kyle Duke, Miles Stevens

**Abstract:**

The Median Voter Theorem is a well-known result in social choice theory for majority-rule elections.
We develop an analogue in the context of approval voting. We consider voters to have preference
sets that are intervals on a line, called approval sets, and the approval winner is a point on the
line that is contained in the most approval sets. We define median voter by considering the left
and right end points of each voters approval sets. We consider the case where approval sets are
equal length. We show that if the pairwise agreement proportion is at least 3/4, then the median
voter interval will contain the approval winner. We also prove that under an alternate geometric
condition, the median voter interval will contain the approval winner, and we investigate variants
of this result

**A Matroid Generalization of Sperner’s Lemma **

**Students:** Gabriel Andrade, Andres Rodriguez Rey, Alberto Ruiz

**Abstract:**

In a 1980 paper, Lov´asz generalized Sperner’s lemma for matroids. He claimed that a triangulation
of a d-simplex labeled with elements of a matroid M must contain at least one “basis simplex”. We
present a counterexample to Lov´asz’s claim when the matroid contains singleton dependent sets
and provide an additional sufficient condition that corrects Lov´asz’s result. Furthermore, we show
that under some conditions on the matroids, there is an improved lower bound on the number of
basis simplices. We present further work to sharpen this lower bound by looking at M’s lattice of
flats and by proving that there exists a group action on the simplex labeled by M with Sn.

**Committee Selection with Approval Voting and Hypercubes **

**Students:** Caleb Bugg, Gabriel Elvin

**Abstract:**

In this paper we will examine elections of the following form: a committee of size k is to be
elected with two candidates running for each position. Each voter submits a ballot with his or
her ideal committee, which generates their approval set. The approval sets of voters consist of
committees that are “close” to their ideal preference. We define this notion of closeness with
Hamming distance in a hypercube: the number of candidates by which a particular committee differs
from a voter’s ideal preference. We establish a tight lower bound for the popularity of the most
approved committee and consider restrictions on voter preferences that may increase that popularity.
Our approach considers both the combinatorial and geometric aspects of these elections.

**The Banquet Seating Problem **

**Students:** Michelle Rosado Perez, Ashley Scruse, A.J. Torre

**Abstract:**

Suppose you want to seat n = mk people around k tables with m people at each table. Each person
gives you a list of j people next to whom they would enjoy sitting. What is the smallest j for which
you can always make a seating arrangement that would seat each person next to one of the people
on their list? In this paper we show that j must be strictly more than half of n, the total number
of people. Our key tool is a particular ‘blue-green-red’ lemma that helps us construct ‘worst-case
scenario’ seating arrangements. We consider cases with two tables and more than two tables and
explore seating arrangements with particular kinds of preferences.

**A Volume Argument for Tucker’s Lemma **

**Students:** Beauttie Kuture, Oscar Leong, Christopher Loa

**Abstract:**

Sperner’s lemma is a statement about labeled triangulations of a simplex. McLennan and Tourky
(2007) provided a novel proof of Sperner’s Lemma using a volume argument and a piecewise linear
deformation of a triangulation. We adapt a similar argument to prove Tucker’s Lemma on a triangulated
cross-polytope P in the 2-dimensional case where vertices of P have different labels. The
McLennan-Tourky technique would not directly apply because the natural deformation distorts the
volume of P; we remedy this by inscribing P in its dual polytope, triangulating it, and considering
how the volumes of deformed simplices behave.