## The Connection Between Hopf Algebras and Groups

Hopf algebras are known to have a copious ammounts of structure which makes them useful in studying representations while, on the other hand, groups come equipt with little structure. In the following we will “realize” Hopf algebras as groups and comment on why we call quasitriangular Hopf algebras quantum groups. Hopefully, this will help demistify the confusion that comes along when studying Hopf algebras. For those of you who don’t know Hopf algebras, don’t worry! We will review them as well.

## Presentation: The Root Datum of Special Linear and Projective Linear Groups

This is the script for a presentation I gave for the Directed Reading Program at the University of Minnesota in the Fall of 2019. In it, I dicuss how the root datum is an extension of the root system, and how we can use it to distinguish between linear algebraic groups.

## Presentation: The Functional Equation for Riemann Zeta Over Function Fields

This is a presentation I gave for my Algebraic Number Theory class during the Spring 2019 semester. In it I prove the functional equation for the Riemann zeta function for global function fields of transcendence degree one.

## Presentation: Root Systems and Hecke Algebras

Yet another presentation! Below is a written version of a presentation I gave for the DRP program during the spring of 2019. The presentation is on root systems and Hecke algebras, the main idea being to introduce the flavor of these topics.

## The Arithmetic of Polynomial Rings over Finite Fields

Today we’re going to talk about something quite exciting and unexpected: the arithmetic of polynomial rings over finite fields and its similarity to the arithmetic of the integers. We’ll first run though some preliminary observations about the arithmetic of the polynomial rings which will suggest that it behaves similar to the arithmetic of the integers.

The material below is a presentation I gave at the start of February in 2019 about the abelianness of groups of order $p^{2}$ using representations.