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? 
 A: Layman's terms first; you can try to guess the mathematical problem. 
There are two tennis players. They decide to play each other with a new set of rules: They play point by point. The referee counts the points. The first time that one of them is three points ahead he's the winner. 
Now you'll notice that this game could go on forever, if they play equally well. If whoever is two points ahead always loses the next point, you never have a winner. So they hire a second referee: The second referee only counts every second point. And if one player is three points ahead according to his count, that player is the winner. 
Turns out they can still play forever. So they hire another referee who counts every third point. And another referee who counts every fourth point. And so on. But they are dubious: Could the game still go on forever? So they ask a mathematician. And the mathematician says: Guys, you are lucky. Just a week ago some mathematicians proved that after 1161 points, you'll have a winner. Guaranteed. It's possible to play 1160 points without a winner, and they have found out how, but not 1161. 
As an exercise: Prove that if they play until one player is two points ahead, the game is over after 12 rounds. And: Prove that the referees never disagree about the winner. 
A: Sam sat with his eyes closed for several minutes, then said softly:
"I have many names, and none of them matter." He opened his eyes slightly then, but he did not move his head. He looked upon nothing in particular. 
"Names are not important," he said. "To speak is to name names, but to speak is not important. A thing happens once that has never happened before. Seeing it, a man looks on reality. He cannot tell others what he has seen. Others wish to know, however, so they question him saying, 'What is it like, this thing you have seen?' So he tries to tell them. Perhaps he has seen the very first fire in the world. He tells them, 'It is red, like a poppy, but through it dance other colors. It has no form, like water, flowing everywhere. It is warm, like the sun of summer, only warmer. It exists for a time on a piece of wood, and then the wood is gone, as though it were eaten, leaving behind that which is black and can be sifted like sand. When the wood is gone, it too is gone.' Therefore, the hearers must think reality is like a poppy, like water, like the sun, like that which eats and excretes. They think it is like to anything that they are told it is like by the man who has known it. But they have not looked upon fire. They cannot really know it. They can only know of it. But fire comes again into the world, many times. More men look upon fire. After a time, fire is as common as grass and clouds and the air they breathe. They see that, while it is like a poppy, it is not a poppy, while it is like water, it is not water, while it is like the sun, it is not the sun, and while it is like that which eats and passes wastes, it is not  that which eats and passes wastes, but something different from each of these apart or all of these together. So they look upon this new thing and they make a new word to call it. They call it 'fire.'
"If they come upon one who still has not seen it and they speak to him of fire, he does not know what they mean. So they, in turn, fall back upon telling him what fire is like. As they do, they know from their own experience that what they are telling him is not the truth, but only a part of it. They know that this man will never know reality from their words, though all the words in the world are theirs to use. He must look upon the fire, smell of it, warm his hands by it, stare into its heart, or remain forever ignorant. Therefore, 'fire' does not matter, 'earth' and 'air' and 'water' do not matter. 'I' do not matter. No word matters. But man forgets reality and remembers words. The more words he remembers, the cleverer do his fellows esteem him. He looks upon the great transformations of the world, but he does not see them as they were seen when man looked upon reality for the first time. Their names come to his lips and he smiles as he tastes them, thinking he knows them in the naming. The thing that has never happened before is still happening. It is still a miracle. The great burning blossom squats, flowing, upon the limb of the world, excreting the ash of the world, and being none of these things I have named and at the same time all of them, and this is reality--the Nameless. 
A: My problem is in some sense the opposite of the one described by the OP.  Although my background is in pure mathematics, my current work is in statistics.  The problem I have is that when I tell people what I do, they think they know about the subject when they really don't.  In a way, that's a much harder issue to deal with--to dispel preconceived notions of the nature of the discipline, as opposed to informing others in layman's terms.  But these issues are like two sides of the same coin.
When people are only superficially interested in what I do, it is appropriate to leave any discussion as brief as possible.  I might say, "I am interested in biostatistics, as it pertains to epidemiology and public health.  For example, I might be interested in case-control studies to determine if there is an association between vaccines and autism."
If they ask about more information, or if they come at me with statements like, "oh, you guys are the ones who manipulate numbers to say whatever you want," then I might go into more detail, but usually I just shrug it off.  But I definitely try to avoid being pedantic unless they ask specific technical questions.
The general public does not have a grasp of what professional mathematicians or statisticians do, let alone how they do it.  Explaining it in layman's terms doesn't really address that gap in knowledge, but it does tend to leave them with the impression that what you do is a lot simpler or naive than it really is, which is equally unfortunate.  And if you attempt to delve into the abstraction, their eyes immediately glaze over because you are literally speaking a different language.  It's sort of a no-win situation:  if you make it accessible, they think it's simple; if you speak informatively, they don't actually care because they don't get it.  I do not see it as our obligation to be ambassadors for our field.  The bottom line is, the vast, vast majority of non-mathematical people do not actually care to know, because if they did, they would have studied mathematics more seriously.  They want to stereotype us because doing so makes them feel less uncomfortable about the huge knowledge gap they know exists.
A: Thanks for asking this question. As a layman (but fan of math) -- and one with recent experience asking mathematicians what they do -- I think I have a pretty relevant perspective to offer on this. 
Let me first suggest what not to do (not that I assume you do this, but just in case, and for anyone else). Almost every mathematician or scientist I've asked about what they do says  something like "oh, geez, it's really complicated." This is patronizing and basically pointless. Of course it's complicated; you don't have to remind us. And we don't have to know EXACTLY what you're studying. We're not a thesis committee. 
Anyhow, your idea of preparing a bite-sized, "dumbed down" description is absolutely in the right direction; I would just go even further with it. For instance, instead of your description (which is basically meaningless to me), I would just say something like "I do algebra." I think this is a) immediately comprehensible to almost everyone, and b) kind of funny (i.e. it sounds like you do homework from middle school, when of course that can't actually be the case). It prompts the other person (if they're actually interested) to ask for further details, e.g. "What kind of algebra?" 
But even then, instead of launching into your full description, just take the next step to narrow it down, like the fact that you work with 3-dimensional spaces, transforming 3-D spaces, etc. I think at that point, most people will go "oh, cool," and that'll be that!
I would also make sure not to mention any jargon or terminology like "Lie algebras" that aren't at least somewhat commonly known. If you must (if the person is really interested), try to think of a way to paraphrase such terms Simple English Wikipedia style. 
Hope that helps!
A: As a laymen myself, let me shed some light on this, and perhaps prevent this potentially awkward conversation from happening. The problem is you're trying to explain your research in a few sentences and induce wonder and so on. There is no solution.
You'd have a better response if, instead of explaining what you're doing or how you're doing it, you told them why it's important to you.
Tell a story instead. Explain the problems that motivated people into looking into algebraic geometry. Explain how the roots of your field are ancient, going back to the Greeks, to Arab mathematicians in the 10th century. Boast, tell them how problems that would leave people scratching their heads 300 years ago are routinely solved by undergrads with mathematical tools your kind of research produced. Be humble, tell them how -- for all a mathematician's vaunted intellect -- some problems are still far beyond your reach. Go off on a tangent, explain to them what motivated you to choose mathematical research.
If you're feeling creative and adventurous, talk about previously theoretical math that now directly impacts the listener. Make it personal to them.
Ex.
Cryptography, information theory - without it, governments can read your emails!
Non-euclidean geometry - ever use a GPS to go anywhere?
Graph theory - the paper that started Google, and international shipping, a really hard problem.
For a person with no technical background, hearing about the history, the problems, the struggle to find the solution and the people involved is far more interesting than a truncated Abstract Algebra 101.
Above all, I implore you to be relatable. Because mathematics certainly isn't; by its very nature it is an abstract endeavour.
You're allowed to be tangential. You're allowed to talk about subjects far from your universe of discourse, like you're some kind of expert in everything mathematical. Allowing yourself these freedoms gives you all kinds of tools to make the subsequent talking less of a chore.
A: Simply describe outcomes. Tell why it matters in the big picture. Folks that want details will certainly ask.
A: A colleague of mine once said: "I do stuff with numbers that you put in a short and wide box to make the world a better place."  (Frame Theory: generalization of Linear Algebra with overcomplete bases)
See this blog entry for more: My research explained, using only the 1000 most common English words (Duston G. Mixon)
A: 
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.

Can you make a nice analogy that illustrates what a "space" and what "transforming in a nice way" looks like in everyday life? Can you explain where mathematical research fits into science they might better understand (theories, predictions, experiments, etc)? Can you tell about a concrete (to the average person) result from mathematical research?
For example -- and I'm totally winging it on this since I don't know what you're talking about -- could you say something like:
"You know how X-rays let us see what's happening in your body without having to cut you open? It's obviously useful, and folks have invented MRI's, ultrasounds, and all kinds of variations on the idea of looking inside of us to see things we couldn't otherwise see from the outside. I do research in a mathematical area where we do something like X-rays for mathematical objects. It lets us look at them in ways that reveal things we'd never guess otherwise."
"That sounds all theoretical, but think about a science like chemistry: it's all based on mathematics. So if we develop the proper mathematical transformation -- the underlying principals have technical names like Lie groups and Lie algebras -- we can help chemists, or other scientists, to see things that they couldn't see otherwise. We can know things that would have taken many experiments -- if they knew what they were looking for -- to capture."
"It's not always that directly applicable, and sometimes it takes many developments and decades of thought before we know how to apply these transformations to specific problems, but I'm one of the researchers that lays the groundwork for those advances in science."
A: I work in a branch of mathematical logic.  Although I'm a mathematician, logic is multidisciplinary, and is also studied by philosophers, computer scientists, and linguists.  
The essential idea in logic is the relationship between syntax and semantics.  Syntax refers to symbols and language.  Most of mathematics is a linguistic exercise: we manipulate language to reach our conclusions.  For example, the notion of proof is a syntactic concept, since a proof is a sequence of expressions, each of which follows in some way from those that have come before. A language generally consists of a set of grammatical rules for building expressions, and a set of rules for drawing inferences (you're probably familiar with classical Boolean logic, but there are other logics out there).
Semantics, on other hand, focuses on meaning: the actual objects (numbers, functions, geometric objects etc...) that are under discussion.  For example, truth is a semantic concept.  The primary semantic object is an interpretation or model - a relationship between the syntactic objects of a language and actual (usually mathematical) objects.
The first goal of any logical analysis is to see how provability and truth are related.  For example, a soundness theorem shows that in a particular context, anything that is provable is also true.  If you cannot prove a soundness theorem, you probably need to revisit the language you are using to study the objects you are considering.  A completeness theorem works in the other direction: a logic (together with a range of relevant interpretations) is complete if everything that is true in all models can be proven.  One of the central results in logic (Gödel's incompleteness theorem) shows that most suitably rich languages are incomplete.  
My own work is in an obscure corner of logic called topos theory. I have started with some objects that have semantic significance in a certain class of models (these objects and models come from a branch of mathematics known as measure theory, which is closely related to probability theory).  I have been looking at ways to describe these objects using a rather weak language and then use these descriptions to find analogous objects in other models.  The goal has been to use the machinery of topos theory to translate traditional measure theory into more general settings than that in which it was originally developed.

It's a little long, but if needed it can be halted after paragraph 1, paragraph 3, or paragraph 4, assuming that I remember to take note of how quickly the other person's eyes are glazing over.
A: I think it was Einstein who said, "If you can't explain it to a six year old you don't understand it yourself."
I find that it is always best to stick to the very basics, omitting superfluous details that matter only to someone working in the field.
A: I realize I'm late to this party, but I'd note that the other answers try to describe a protocol best for the recipient.  What's best for you?
I fly on planes a lot (usually within the United States).  I study both Computer Science and Abstract Mathematics (primarily separately).  Often, when I sit down, the person sitting next to me will ask me what I do.  So, I cover the basics: "Well, I'm an academic at such-and-such a university; I'm flying to X for reason Y."  Then, they'll typically ask what I study.
If I'm feeling sociable, I'll say "Computer Science".  It turns out that the vast majority of people think that CS is the same as Software Engineering and have no idea that a dedicated field for studying algorithms and complexity even exists.
If I'm feeling antisocial, I'll say "Mathematics".  It turns out that many people who don't have the right hate math.  They'll usually just go "Oh, that's nice," and then never speak again for the rest of the flight.  It would be after this, if they still have interest, that I would consider explaining my research in the manners above.  I don't like explaining anything to someone who doesn't care.
A: First, I think it's important to consider the person's motivation for asking this question.  If they're just asking to be polite, then I think your answer, or basically any answer, would be fine.  They're not going to take away much from whatever few words you say, so those words aren't really important.
If they truly are interested in your work, then it's important to broach your topic in an accessible way.  For me at least, an illuminating example is worth far more than an abstract algebraic object that satisfies some list of unmotivated axioms.  I think it's also important to have some visual aids to help you explain the concepts.  Don't be afraid to scribble something on the back of an envelope or a napkin.  After all, when mathematicians talk about math to other mathematicians, there's usually a blackboard or at least a piece paper to write down ideas.
If you want to explain what an algebraic group is, consider explaining what an elliptic curve is.  Grab a napkin and draw a picture of $y^2 = x^3 - x$ (over $\mathbb{R}$, of course) and explain how to add points on the curve.  You can remark on the properties of this structure: there is an additive identity that acts like $0$; the order of addition doesn't matter, i.e. addition is commutative, etc.  I think the geometric simplicity of the group operation is accessible to most people.
If you want to explain what characteristic $p$ means, then consider doing some modular arithmetic.  Draw the numbers $1$ through $n$ (but actually choose a value for $n$) in a circle.  Show that strange things can happen when $n$ is composite (e.g., zero divisors) but that things work out nicely when $n$ is prime.  Maybe verify for $n=5$ that every nonzero element has a multiplicative inverse.
If you want to explain what a representation is, take the group $\mathbb{Z}/3\mathbb{Z}$ and show how it can be embedded on the unit circle in the complex plane by sending
\begin{align*}
0 &\mapsto 1,& &1 \mapsto e^{2 \pi i /3},& &2 \mapsto e^{4\pi i/3}
\end{align*}
(By the way, just pick one topic to approach unless the person is truly rapt.)
I guess my point is that you're never going to be able to accurately describe what you are actually working on--I'm guessing it took you years to acquire the needed background.  What you can do is show them some basic examples that they can understand and will hopefully pique their interest.  After all, these examples are what spawned the general theory. 
A: On my field:

  
*
  
*I study curvature.
  
*I study higher-dimensional shapes.
  
*I study a mix of calculus and geometry.  But not the geometry of flat things like triangles and squares, but smooth curvy wavy things.
  

On (one aspect of) my subfield:

I study shapes that are like soap films.  Like, if you dip a wire in a soapy water, then the bubble will have the smallest possible area, because of surface tension.  I study shapes like that, but in higher dimensions [at this point I wave my hands up and down like a crazy person].

On what mathematicians do all day:

We prove things.  Everything that is considered "known," everything in the math textbooks, someone had to prove those -- otherwise it's not considered that we know it.  And it turns out that there are lots of things we don't know!  [At this point, I mention some really easy-to-state open problem.]
So, most of mathematics research amounts to just sitting in a room with pen and paper, trying to come up with new ideas.  It's very creative!

A: Last time I tried to explain it went like this:
"So, mathematicians sometimes like to have a theory on hand from which you can derive as much of mathematics as possible, something to check your intuitions against when the stuff you're studying gets really flighty. There are a few alternate formulations of such a theory, and I study one of the weirder ones."
Usually that's about as far as I go unless they ask why it's weird; then I hand wave about Cantor's paradox and universal sets and non-Cantorian sets. If they haven't gone into a coma I might try to outline the category theory part.
Usually they're very impressed by the time I finish and don't catch on to all the terrible mathematical bumbling I actually do :D
A: 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.
A: Look at some Richard Feynman video.
He is great at explaining and visualizing concept, he makes imagination works for the layman, not just pure logic.
However, when asked some difficult question he can't respond in layman terms, he says "I can't explain it, because it is not familiar to anything you have seen and don't have the necessary vocabulary to understand it". (The question asked was about the nature of electro magnetical forces)
He puts a clear line about the things he can explain with common vocabulary, and things he can't.
I would say it is more easy for a physicist than a mathematician. But Mathematician works is always applied somewhere, or it would not be funded. Find who earns money thanks to your math, and explain to the layman how you helped.
The trick is that if he can see, you'll get a long line of "why why why why" that the person might ask until the point where the last why he asked is the purpose of yourwork... which you can finally respond : "I'm figuring out and paid for to respond that question".
A: 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. 

A: "There are only two kinds of math books: Those you cannot read beyond the first sentence, and those you cannot read beyond the first page." -- C.N. Yang
One of my favourite sayings from a Nobel laureate mathematician.  In my view, the ability to communicate complex mathematical ideas in a simple way is a very special gift that some people have and others perhaps develop with time and great effort.  Even what appears to be simple mathematics is difficult to explain to the lay audience.
For example, we recently did an analysis of data and discovered that certain experimental results, when placed under the certainty of the laws of probability, fell into an impossible region, meaning that the entire basis of the analysis was invalid.  This was using statistical analysis and applying foundational probability theory to it, and it was related to an arbitration claim involving quite a lot of money and professional reputation.
The audience were lay persons with almost no background in mathematics, much less probability theory.  How to explain this?  I came up with the idea of describing a probability density as being the same as a balloon.  What I did was to blow up the balloon, tie it, and explain that the law of probability is the same, that all results of an experiment must be contained under the curve, just like air is contained in a balloon.  Now, what happens if we pinch one end of the balloon?  The balloon must bulge somewhere else.  In the same way, if we have an experiment, and the values of that experiment cannot be below zero, we are pinching the curve and it must bulge somewhere else. Just like a balloon.  Therefore the analysis done by the other party could not be valid, as the curve they used, leads to a region of negative values, that cannot occur.  This is mathematically impossible, and breaks the laws of probability.
This was a slam dunk as far as the case was concerned.  The audience could not help but get it.
So, if possible try find some kind of physical analogy that an audience can visualize.  Or at least something that they can relate to, even if it is conceptual.  It doesn't have to perfect, it takes imagination, and it is not easily done, but it is certainly worth the effort.  Clearly communicating your work is just as important as the work itself.
