Welcome to the Math and Stats Group Comps Presentations for Winter 2021! This term students have recorded comps presentations that are viewable by a prerecorded video. Please refrain from posting these videos to any other online resource. Group comps students will be hosting a Q&A session by Zoom on February 23 and 25 at the times listed below. Q&A questions submitted by 2 pm the day of the presentation will receive full consideration. The Zoom link will be mailed prior to the event. We hope you all enjoy the comps presentations.

## Tuesday, February 23, 2021

**Time: 4:30-4:50 pm****Title: Dedekind Domains, Prime Factorization and Elliptic Class Groups**

Comps Advisor: Alex Barrios

Students: Jack Heinzel, Daniel Kleber, Matthew Mendiola

Comps Presentation Video

Abstract: We study the properties of unique factorization of prime ideals in Dedekind domains, and the class group of Dedekind domains. We ask the question of what abelian groups can be constructed as the class group of a Dedekind domain, and find that every abelian group is the class group of some Dedekind domain. In this talk, we consider the countable case. We use the theory of elliptic curves to generate a Dedekind domain with a prescribed countable abelian class group.

**Time: 5:00-5:20 pm****Title: Bernoulli Numbers and the Class Group of Cyclotomic Fields**

Comps Advisor: Caroline Turnage-Butterbaugh

Students: Marco Bommarito, Marcella Manivel, Gavin Peng, Marguerite Shaya

Comps Presentation Video

Abstract: In this talk, we explore the Bernoulli numbers: a sequence of rational numbers with connections to combinatorics, complex analysis, algebraic number theory, etc. Importantly, Bernoulli numbers may be used to define the B-regular and B-irregular primes, which have historical connections to investigations of Fermat’s Last Theorem. In 1850, Ernst Kummer related B-irregularity of primes to the class numbers of cyclotomic fields. Over the course of the talk, we build towards an understanding of this connection and its mathematical significance.

## Thursday, February 25, 2021

**Time: 4:00-4:20 pm****Title: Predicting COVID-19 using State-Space SIR Models**

Comps Advisor: Katie St. Clair

Students: Travis Brown, Vincent Gu, Marko Jurkovich, Andrew Vance

Comps Presentation Video

Abstract: What will COVID cases look like in one month? How many people have already been infected? What would be the efficacy of another lockdown? In our talk, we will give an introduction to statistical disease modeling using state-space SIR (Susceptible-Infected-Removed) models that aim to answer these questions. We then test two Bayesian models’ predictive accuracy in the pandemic using Israel as a case study.

**Time: 4:30-4:50 pmTitle: Northcott’s Theorem: An Adventure in Preperiodic Points of Rational Functions**

Comps Advisor: Rafe Jones

Students: Evan David, Ian Klein, Ben Richardson, Sameer Swarup

Comps Presentation Video

Abstract: What happens when you repeatedly apply a function to a rational number? Some numbers eventually loop back on themselves; Northcott’s theorem states that there are only finitely many of these “preperiodic” numbers for a given rational function. We introduce a notion of height to provide a surprising proof of Northcott’s theorem. During this proof, we make an emergency foray into algebra in the form of homogeneous polynomials and polynomial rings. We also introduce the canonical height, which measures how close a number is to being preperiodic.

**Time: 5:00-5:20 pmTitle:**

**Network Analysis in R: Two Case Studies**

Comps Advisor: Katie St. Clair

Students: Cindy Guo, Jenna Korobova, Erika Mino, Matt Zacharski, Daniel Zin

Comps Presentation Video

Abstract: Social media has become an increasingly common way for people around the globe to engage in discourse and seek information. Data that captures relationships between users on various platforms including followers, hashtags, likes, and retweets is conducive to representation in the form of a network. The structure and characteristics of these networks can be analyzed using combinations of mathematical and statistical approaches for graphs. In particular, data from Twitter is a rich source of information about users, topics, and communities that can be analyzed using the statistical methods in R. Our Twitter investigation considers two case studies: (i) a comparison pro- and anti-mask networks in various states during the COVID-19 pandemic and (ii) an analysis influential users and their characteristics in a network of retweets about the death of George Floyd. In the first case study, we found similar structural characteristics in both networks though the anti-mask communities were larger, and we found that the edge-betweenness and infomap algorithms for community detection yielded similar results. Finally, in the second case study, we found that shared sentiment amongst users was the strongest predictor of retweets, and that @YourAnonCentral and @AttorneyCrump were the most retweeted (influential) users.

**Time: 5:30-5:50 pm****Title: Virtual Edge Detection and Bone Measurements**

Comps Advisor: Rob Thompson

Students: Bat-Orgil Batjargal, Charlotte Clapham, Abby Loe, Maddie Kyhl

Comps Presentation Video

Abstract: Bones can inform our understanding of the past and make predictions for the future. Information is encoded in characteristics of the bone, and applying mathematical tools to bone analysis can provide robust and interesting insights. The AMAAZE group at the University of Minnesota specializes in these questions–particularly, questions about broken bones. One of their current projects is working to reassemble broken bone fragments and classify broken bones by how they were broken. In order to answer these questions and more, they need to collect information on the angle of the bone break, which can be measured using their virtual goniometer tool–a virtual tool capable of measuring these angles on 3D bone scans. This process introduces interesting mathematical questions. How does the curvature of the bone’s surfaces relate to this angle measurement–can we find a relationship to inform the measurement process? Also, how do we actually detect where the bone was broken inside this virtual tool? With our research, we hope to understand and provide new insight to AMAAZE’s measurement tool and larger mission.