# Graduate Algebra Symposium Fall 2017

The Graduate Algebra Symposium is a workshop organized by and for graduate students, the goal of which is to bring together students from several universities in the area to discuss research in algebra by graduate students. The symposium gives graduate students the opportunity to present their research to others in their field, gain experience giving conference-style talks, and network with local researchers. The Symposium is a collaborative effort by the University of North Texas, University of Texas at Arlington, Texas A&M University, Oklahoma State University, and the University of Oklahoma.

The Fall 2017 Symposium will be held at the University of Oklahoma in Norman, on Saturday October 21st. Bagels & coffee will be provided in the morning, as well as pizza for lunch.

## Schedule

October 21st

10-10:30am

### Lucas's Theorem - Binomial Coefficients Modulo a Prime

Chelsea Drescher
10:30-11am
UNT

#### Abstract

In this talk, I discuss the history of the research on the Binomial Coefficient modulo a prime, including Lucas's Theorem. I will give a proof of Lucas's Theorem, as well as provide examples and discuss some applications.

### Tools in Algebraic Geometry

Ian Lim
11:10-11:40am
UTA

#### Abstract

If $R$ is an arbitrary ring, the Prime Spectrum of $R$ is the set of all prime ideals contained in $R$. Throughout the talk, we will discuss how one can associate a geometric object to $R$ and how this idea can be useful in research.

11:40-1pm

### On the index of $\Gamma^+(\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/q\mathbb{Z},\pi)$ in $\text{Aut}^+(F_2)$

Thomas Morgan
1-1:30pm
OU

#### Abstract

For an epimorphsim $\pi: F_2\to G$ of the free group on two generators onto a finite group, $\Gamma(G,\pi)=\{\varphi\in\text{Aut}(F_2)\hspace{1mm}|\hspace{1mm} \pi\circ\varphi = \pi\}$ is a finite index subgroup of the automorphism group $\text{Aut}(F_2)$ called the principal congruence subgroup. We calculate the index of this group when $G$ is a non-abelian semi-direct product of cyclic groups.

### Supermathematics: Many Sign Changes, Few Capes, and One Particular Example

Dwight Williams II
1:40-2:10pm
UTA

#### Abstract

We reference certain series of Lie algebras and consider their supermathematics analog of Lie superalgebras.

### Annihilators of Representations of Lie Algebras

Tyler Kenefake
2:20-2:50pm
UNT

#### Abstract

Given a representation of a Lie algebra, we may extend the representation to an action by the universal enveloping algebra of the Lie algebra. The kernel of this extension is called the annihilator of the representation. We search for generators of the annihilator.

## Parking

Parking will be available for free on Saturday in the Elm Street Parking Garage, directly across Elm street from the Physical Science Center. It is suggested that you park there, though there are other (read: paid) options across campus.

## Organizers

Gordon Brown (OU), Chelsea Drescher (UNT), Derek Tomlin (UTA), and Jordan Wiebe (OU).