$F\langle 1,i,j,k \rangle/(i^2=a$, $j^2=b$, $ij=-ji=k)$

## Hey there!

My name is Jordan Wiebe and I am a graduate student in the Department of Mathematics at the University of Oklahoma. I work with Dr. Kimball Martin on arithmetic in quaternion algebras and related topics in automorphic forms. I also teach various undergraduate math courses, from algebra to calculus. I am currently teaching Accelerated Calc 2 with Dr. Tao.

I am originally from Lincoln, Nebraska, where I completed my undergraduate degree in mathematics at the University of Nebraska-Lincoln. My wife Abby and I love backpacking and are currently planning a few excursions in Arkansas, Colorado, and Alaska.

I truly love mathematics, and I'm eager to share my excitement with you!

## Research

I am currently studying orders of higher level in quaternions algebras, and their applications to modular forms. In particular, my most recent result gives an explicit formula for orders of level $N=p^{2k+1}M$ over $\mathbb{Q}$. This can be applied (for instance) to A. Pizer's algorithm for computing modular forms via Brandt matrices outlined in his paper "An Algorithm for Computing Modular Forms on $\Gamma_0(N)$". I am currently working on direct computations of modular forms using my previous result and Magma.

I am also interested in orders with level of the form $N=p^{2k}M$, which requires extending the definition of level to include this case. I'd like to analyze the arithmetic in this case further, expand my previous result to this new case, and apply these to modular forms.

I'm also interested in constructing quaternionic modular forms, something Dr. Martin has begun to pursue recently.

### Arithmetic in Quaternion Algebras (Speed Talk)

In this speed talk, I present the fundamentals of quaternion algebras, orders, level, and present a special case of my general result. I also briefly mention connections to computing modular forms via Pizer's algorithm.

Conference: Automorphic Forms Workshop

### Arithmetic in Quaternion Algebras

In this talk, I discuss in detail the roots of quaternion algebras. I develop orders and local levels, and provide a number of examples. Last, I discuss the local-global connection we can see for $\mathbb{Q}$, and indicate my work on the global case.

### An Algorithm for Computing Modular Forms on $\Gamma_0(N)$

In this talk, I present A. Pizer's paper of the same title. In particular, I develop the quaternion algebras and orders, and apply these to compute the Brandt matrices which yield our modular forms. I present the algorithm for accomplishing these tasks.

I will be attending the Texas-Oklahoma Representation theory and Automorphic forms conference (TORA VIII) at the end of March, as well as the Graduate Algebra Symposium in late April.

## Layman's Terms

So let's say you don't have a degree in mathematics but want to get the gist of what my research is about. First, let's talk about advanced mathematics as a whole. How do we even do math research? Isn't it all "figured out"? This is a very common thought about mathematics, and is sadly not corrected (even in college-level courses). The power of mathematics comes from the fact that the things that we know a lot about (say whole numbers, or fractions, or functions, or derivatives and integrals) can actually be developed and applied to more interesting structures.

Let's do a quick example. Say we're looking at a clock. We know that there are 24 hours in a day, but it would be hard to represent all 24 in one clock face, so we have 12 hours represented, right? The thing to think about here is that an hour after 12 is 1. So somehow when we do 12+1, we get 1. But that makes it seem like 12=0, right? What's happening here is something called modular arithmetic, and it works by considering the hour's remainder after division by 12. Another way of thinking about this is the following:

So we can glue 12 and 0 together, since 12 has remainder zero when divided by 12!

So we can take the basic ideas of mathematics and do interesting things with them, and study the resulting objects. Let's get back to the real question: what am I studying? If you're familiar with imaginary numbers, you know that if we write $i^2=-1$ and "add" this to the real numbers, we get the complex numbers $\mathbb{C}$. So we have

$\mathbb{R}\to\mathbb{C}$ via $i=\sqrt{-1}$.

I study quaternion algebras, which are similar: generally, set $i^2=a$ and $j^2=b$, so that we have two "roots" $\sqrt{a}$ and $\sqrt{b}$. Then add these to, say, fractions (we'll denote fractions by $\mathbb{Q}$) and we have a quaternion algebra:

$\mathbb{Q}\to\left(\frac{a,b}{\mathbb{Q}}\right)=\mathbb{Q}\langle 1,i,j,k\rangle/(i^2=a,j^2=b,ij=-ji=k)$

So quaternion algebras are a 4-dimensional analog of the 2-dimensional situation we have for $\mathbb{C}$. Inside these algebras, I study something called orders, which mimic the properties of the integers (whole numbers). My work lies in the area called number theory.

## Teaching

I am currently teaching Accelerated Calc 2 with Dr. Tao. If you'd like more information about the current course, please check out the course page on Canvas. I have previously taught the following courses:

• College Algebra
• Precalculus & Trigonometry
• Calculus 1 and 2 (and their accelerated variants)
• Business Calculus 1 and 2

I consistently receive positive feedback from students, and continue to do my very best to provide an environment that facilitates learning and excitement about mathematics. My hope is that every person in each of my classes leaves class every day a little more excited about mathematics than when they came.

I'm honored to have received teaching awards for my performance:

Provost's Certificate of Distinction in Teaching

Fall 2016

Provost's Certificate of Distinction in Teaching

Spring 2015

If you have any questions about class, suggestions for the future, or comments, please feel free to contact me.

## A Brief History

I've been in school most of my life, and am what some call a professional student. Over the years I've had a variety of accomplishments, a few of which I've listed below.

#### Bachelor of Science

University of Nebraska - Lincoln, 2012
• Regent's Scholar