How do you describe your mathematical research in layman's terms?

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.

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.

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.