For some reason, many mathematicians have trouble with the idea that when some layman asks them about their work, the appropriate response is not to try to figure out how to describe the latest theorem you've proved. We seem to feel like we're "selling out" unless we try to describe all the technical subtleties of our most recent work. Some other professions seem much better at this: when a psychologist is asked about what they do, they don't reply with their intricate struggles to minimize bias in their latest experiment, even if that's what's occupying most of their attention that week. (On the other hand psychologists can say "I devise experiments to study the long-term effects of alcohol abuse on cognition" and be reasonably confident that this will make at least some superficial amount of sense to a generic college-educated person. If I say "I study period-index problems in the Galois cohomology of abelian vareties" then notwithstanding the considerable syntactic similarities between these sentences, the social effect could hardly be more different: I might as well say "Please go away".)

I have to say that I feel privileged never to have had to prepare a ten minute (or less!) précis of my work to a general audience in any kind of formalized setting. I agree that that does not sound like much fun -- for me or the audience -- and if asked to do so at the point in my life I would raise my eyebrow and begin to question (inwardly at least) the assumptions and goals of the person who wanted me to do so.

However, one of the necessary evils of socializing with people outside of the mathematical sciences is that you are inevitably asked "What do you do?" in very informal settings. Usually my first answer is that I'm a mathematician, and my second answer is either (depending upon my mood?) that I'm a number theorist or that I'm an arithmetic geometer. The second answer is more ambitious, because after the expected "What's that?" I have to explain that I work in sort of a hybrid of two fields, number theory -- the study of properties of the whole numbers like primes and divisibility -- and algebraic geometry -- the study of the curves, surfaces and higher dimensional objects that arise as solution sets to polynomial equations. When I'm on and the other person cares I can get all this out in a couple of minutes without causing any obvious trauma.

If they want to hear more than this I often state Fermat's Two Squares Theorem. I think this is nice because it's specific and it's relatively simple but certainly not obvious: indeed it's a little window into how pleasantly surprising mathematics can be: why should there be such a nice, clean pattern like this? Of course this is number theory of 350 years ago not of today, but this was the theorem that attracted me to number theory in the first place, when I first learned about it at the age of 16. (Actually, in my more recent career I have spent time thinking about different proofs and generalizations of exactly this result. But while speaking to a layman I probably wouldn't even remember that.) I should say that sometimes I get completely cut off in my statement of the two squares theorem -- I mean cut off in the middle of a sentence. And then I have often gone on to have quite a pleasant conversation on something else entirely.

In fact, my most negative experiences in the "What do you do?" game have come from non-math people who have insisted on hearing about exactly what I've been working on, with all the technical terminology. For instance, shortly before I received my PhD I went to a bar with my cousin and somehow found myself at a table full of medical students and residents. The above gambits were not sufficient for them. At one point one of them demanded to know the title of my thesis. "All right: it's `Rational points on Atkin-Lehner quotients of Shimura curves'." His reponse? "Okay. So basically you study points on curves." He said this with the smug pleasure of someone who had demonstrated that once all the big words had been stripped away, the Harvard PhD student was actually studying something very simple and childish. Of course having omitted all the big words, even an expert wouldn't have the slightest clue as to what the title meant. What a jerk.