Layman's Terms

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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!

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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.