TITLE: What to answer when people ask what I do in mathematics
QUESTION [53 upvotes]: 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:

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.
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.
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.

REPLY [4 votes]: Come up with a simple way of describing what it is you study. I study combinatorics, so I usually say that I count structures.  This usually draws a blank look since it's so vague, so I give them a concrete example:  "How many ways can we pair off the people in that group over there into twos?" This gives them an idea of the sorts of things I think about.
A number theorist might say that they work with prime numbers, or an analyst might say think about things that change smoothly.  I don't know anything about operator algebras, but you could perhaps say "geometry with a lot of dimensions" or something like that.  They'll probably go cross-eyed if you even try to explain four dimensions, but most people are intrigued by that concept.  You could also use a very broad metaphor for how you think about what it is you are trying to do in a problem.  For example, to explain to someone about representation theory, I told him to imagine taking a complicated object and trying to break it into simple pieces.  This was the idea of decomposing a representation into irreducibles.
Here is the secret in these conversations:  unless the person you are speaking to has a math or science background, they are not interested in your subject area.  They are interested in you.  Even if they're just making polite conversation, they have some interest in what it is that makes you tick.  To many people, the idea of enjoying math for fun is a foreign concept.  So when someone asks you what it is you do, you're something of an ambassador for the entirety of the math (or STEM) world.  You need to give them some idea of  what it is about your work that intrigues you.  Your answer doesn't need to 100% accurate. It just needs to be honest on a human level.  If you are interested in the applications, you can explain them; but if you don't know or care about any possible applications, then be honest and explain that it's simply intriguing to you, like a good jigsaw puzzle or board game.  It's really no different than any other conversation you have when you're getting to know someone - you want to communicate what kind of things interest you.  The details aren't very important.