# Undergraduate Math Research

# at

# St. Norbert College

## Poss-Wroble Fellowship

The Summer Undergraduate Research Program in Mathematics at SNC consists of eight weeks of full time work open to St. Norbert College students with a declared mathematics major or minor. On the application form, students provide a list of their mathematics and other relevant course work, their research interests, and how a research experience would contribute to their professional goals. Students do not need to propose a project. Faculty members may already have projects in mind. The Mathematics Discipline reviews all applications and chooses which applicants will be awarded the summer positions. The research students (Poss-Wroble Fellows) are required to present their work at a national conference, which includes either MathFest in August or the Joint Mathematics Meetings in January, and at the Pi Mu Epsilon Regional Undergraduate Math Conference on the SNC campus in November. Interested students are encouraged to speak with a member of the Mathematics Discipline, either before or after completing an application. Faculty members can offer insight into past projects along with the research projects proposed for the upcoming summer program.

# CURRENT PROJECTS

### SUMMER 2022

# Measuring Partisan Bias with Randomly Generated District Maps

#### Alexandra Bennett with mentor Jonathan Dunbar

We aimed to create a sampling of district maps for the state of Iowa through random selection of adjacent counties. Once enough maps were created, we applied historical voting data to these maps and calculated the Partisan Gini score for each. This gave us a distribution of PG scores for comparison to maps actually used (or proposed to be used) by the state.

# Seats-Votes Curves and Partisan Bias in Decennial Redistricting

#### Crimson Groh with mentor Jonathan Dunbar

The debate between what is considered fair political redistricting has been around for years. Occasionally politicians will illegally manipulate boundaries in order to assist their party in elections. Due to this, there exist many methods to determine the level of inequality presented in political redistricting plans, one of which is to use the Partisan Gini Index. The Gini Index Score is originally used to find where income inequality exists around the globe, but it is very similar in finding potential gerrymandering in political districting plans. The goal of this project is to create and find the area of step function seats-votes curves while then using this area to explore what values work for the Gini Index Score.

# Angles, Trigonometry, Circles, and Chords in Taxicab Geometry

#### Nathaniel Woltman with mentor Jonathan Dunbar

When traveling through a city, one often cannot move directly from point A to point B, as this would necessitate passing through the walls of buildings. Instead, one must move along the grid-like system of streets, and a more practical measure of the distance between two points A and B is the distance one travels east-west plus the distance one travels north-south. Inspired by city travel, taxicab geometry is a type of non-Euclidean geometry that is based on the taxicab metric, which measures the distance between two points on the plane by the sum of the horizontal distance plus the vertical distance. There are two main types of taxicab geometry: traditional taxicab geometry and pure taxicab geometry. Traditional taxicab geometry only changes only the distance metric, while keeping Euclidean concepts such as angles the same. Pure taxicab changes the distance metric as explained above, but also introduces concepts such as angles which are native and natural to its geometry. Our analysis hopes to investigate how native angles and trigonometry behave in pure taxicab geometry, as well as investigate Euclid’s Book III propositions on circles and chords translate into taxicab geometry.