This is not really a math question, but I think that every mathematician and student in math (specially pure) struggles with this at some point. Inevitably at some point when we're talking with family, friends, etc… which are not in academia, they ask something like
What do you study, and how is this important for the rest of the world?
I don't know what is the best answer for this question. So far I've tried the following solutions:
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Be completely honest. In my case, I answered something like "I don't know any applications, I don't care about applications, and I do math solely because it is fun for me". Usually people just look at me with horror for finding it fun and think my work is just useless.
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Try to come up with some application. In my case (I study mainly operator algebras), I tried saying that "there are some applications in quantum physics" (I just heard this somewhere). Usually people then ask what are the applications, in which case I have no idea, and in the end they just think my work is useless again.
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I explain as I would for a mathematician. I received some different reactions in this case: blank stare followed by an abrupt change in the subject; blank stare followed by asking to explain it "in simple terms", in which case we're back at the beggining; or people say some nonsense in order to try to look like they are very smart and understood everything I said.
These are the bad scenarios I've experienced (usually when talking with someone in academia, even from another department, simply saying that I don't care about applications is good enough).
I appreciate any comments.
Best Answer
My favorite answer to the question of "What do you actually work on?" was one used by a grad-school friend of mine. He'd say "Mostly word problems" (or "story problems", if you're from the 1990s). There were two possible reactions:
(a) "Wow! I could never do word problems!" after which he'd say something like "...and I could never really draw the way you do" or "work with customers the way you do" or whatever, and something about each of us having their own skills, and the conversation moved on, or
(b) "Yeah, right. I'm serious....are you in topology? Analysis? Algebra?" and the conversation would get more interesting.
I know that this doesn't completely address your question, but it could be worse. You could be Danny Ainge, where every bozo in the world thinks that he knows how to run a basketball team, and wants to tell you about it.
Sometimes it's best to say "I don't think much about possible applications, partly because of a long history in mathematics of the applications being discovered only decades after the work was done. For instance, Gauss thought a lot about modular arithmetic -- perhaps you called it 'clock arithmetic' in school, where you say things like 9 + 6 = 3 because if you add 6 hours to 9 AM you get 3PM, and so on. Well, Gauss thought a lot about that, and proved a bunch of interesting theorems, and mathematicians have tinkered with it ever since, but the main ideas Gauss developed are right at the core of almost every practical system of cryptography used anywhere in the world today. Cauchy studied complex numbers, and calculus with complex numbers, and now we use them everyday to solve problems in heat transfer (like making home insulation more efficient!) and electrical engineering. I'm not good at guessing the applications my work might someday have, but I'm good at doing the work itself, so I stick to that, and hope that mathematical history repeats itself as it has so often before."