How do you describe your mathematical research in layman's terms? "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?
I'm a little disappointed by the comments. Granted, it's hard to explain mathematics, but having the attitude that you're not even going to try is not doing mathematics PR any favors. We can't in good faith expect the public to fund our research if we're not even going to try to tell them what we're doing with their money.

First, I hope you won't take this the wrong way, but I'd like to spend a bit talking about why I agree that your proposed answer is not good:


  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.


The general problem is that you are trying much too hard to be accurate. (In other words, I think specification #3 is the least important of your specifications and should mostly be discarded.) The technical terms you're using mean nothing to a layperson, and depending on the layperson, "algebraic object," "transforming," and "prime number" could all be technical terms. 

I would aim much lower than where you're trying to aim, and settle for communicating some intuitive aspect of the already exciting general idea of group theory, symmetry, and representation theory. I would not even try to mention positive characteristic without more time. I would try to relate the ideas to concrete experiences the layperson has had or at least familiar ideas from other areas. For example: 


  I study special kinds of symmetries. For example, think of the symmetries of a sphere [here I would pretend to hold a sphere in my hands while rotating it around]; I study symmetries that are like these but more complicated. The idea of symmetry has applications all across mathematics and physics; in the case of symmetries of a sphere, you can use these symmetries to predict certain properties of the periodic table.
Despite not being a mathematician myself, I would like to contribute a bit as in my field I share the same problem often (depending on the project I am working on). What I have learned is that what works best is to bypass the entire description of what you're actually doing and just jump straight ahead to applications even if you will never get anywhere close to them.

So, if for example you're doing anything with number theory you can just say 'I am contributing to theories which one day might be used by others to improve the security of facebook or your bank for example' or with lot's of mathematics one might be contributing to the systems/engines on which you will play their games, or maybe for astronomers to understanding the universe and how it functions, or for electrical engineers to make your next mobile phones, or it can be something that will help economist to understand (model) the economy. Or any crazy combination of those :D . Like, I dunno, from as far as I know mostly all math has some application nowadays, especially with computers being as complex and diverse as they are, so just find something cool :P .

Now, I am perfectly aware that you might not even be remotely aware of what your research will be used for, so in those cases I would just describe your sub field in general or make an educated guess, but in the end I do think this qualifies with all your guidelines. Sure, from the point of an insider spec 3 won't be met, but you have given an outsider far more of an understanding than with your description.