"You do research in mathematics! Can you explain your research to me?"

If you're a research mathematician, and you have any contact with people outside of the mathematics community, I'm sure you've been asked this question many times. For years now, I've struggled to find a satisfying answer. I think an ideal answer to this question should:

- be accessible to someone who hasn't studied math since high school
- build intrigue and wonder
- honestly, albeit vaguely reflect your research
- only require a few sentences

(Of course, these guidelines will change depending on the audience and venue. For example, speaking with an engineer over a meal allows more time and technical language than would speaking with a stranger on a bus.)

I study the representation theory of algebraic groups and Lie algebras over fields of positive characteristic, so I usually say something along the following lines:

I work with two algebraic objects that are closely related called algebraic groups and Lie algebras. These objects can act on spaces (like three-dimensional space) by transforming them in a nice way, and I study these actions. One aspect of my work that is especially challenging is that I use number systems in which a chosen prime number is equal to zero.

Honestly, based on my guidelines above, I think this response is poor, but with so much to communicate in such limited terms with such limited time, the task seems nearly impossible.

Using my guidelines, how would you describe your own field of research? Or, if my guidelines are too strict, how would you deal with this question?