What to answer when people ask what I do in mathematics

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."

I finished my undergrad in pure math last May and have encountered some of this before. I especially like your description of solution $(1)$. I know exactly what you mean by "Usually people just look at me with horror..." As far as how to respond to your question, I usually break people into two categories.

Category one is the type of person who is on somewhat equal grounds as you. Someone who is genuinely interested in what you do and have to say about it, regardless of how much math they know. This type of person I will really do my best to give a good, honest answer. The fact that they have genuine interest in what I say/do is enough for me to respect their interest and answer their question as thoroughly as possible. For these people I would discuss pure math vs. applied math. I would talk about how pure math can be critiqued as "useless" since there isn't (always) a clear application. I'd also give examples of how applications can be found years after the math was created. Having a couple concrete examples of this to call on can be nice, like number theory's usefulness in cryptography. I don't think anyone can deny cryptography's importance to the world. Further, I'd argue that spending so much time with abstract logic allows one to develop incredible reasoning and problem-solving skills. This in itself is very useful for a vast number of fields. Engineering, physics, chemistry, computer science, biology, etc. are all fields that require strong problem solving skills and excellent ability to reason. These are all fields that offer tremendous benefit to mankind. So, while pure math itself may not always be directly important for the rest of the world, the study of pure math can give individuals the ability to move on and succeed in other fields that do benefit the rest of the world. I'd probably wrap up the conversation here with most category one people. But if it were another mathematician/STEM individual who wanted to continue discussing mathematics, I might also talk about "The unreasonable effectiveness of mathematics."

http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences

If the person is more artistic, I'd talk about the elegance and aesthetic appeal of pure math. I didn't write the following paragraph, but I think it is an excellent explanation of how some mathematicians feel about the beauty of pure mathematics:

You can think of [pure math] like you're going to a museum and you see Van Gogh, Picasso, Monet. Is learning these painting styles useful? No, but they were not conceived with practicality in mind. These are ways to explore different aspects of human culture and human thought. Painting explores the visual aesthetic and visual abstraction parts of humanity. Math explores the cognitive aesthetic and cognitive abstraction parts of humanity. A civilization with a high culture is characterized by people who have the means to freely explore their thoughts and ideas, outside the need of practicality. Early civilizations with high culture can be marked by how much art they produce and what math they have created. Math is a cultural profession akin to art, literature and music.

So that is how I would approach someone in category one. Next, Category two.

Category two is unfortunately the vast majority of people I encounter. This is the type of person who is just making small talk, who probably wouldn't understand a single word of a sentence you use to describe what you study, someone who already unwaveringly believes math is useless and/or carries an anti-intellectual attitude. To these people I will not do anything that I described above. Sometimes these people can be hostile and flat out tell me I have wasted my time. I'm not one to argue with someone who has no problem making a claim like that. I'll usually laugh it off or change the subject instead of using energy on explaining everything above. If they aren't hostile but are still category two, I'll say something vague like "you like your [insert piece of technology that this person cannot live without] right? Well, you wouldn't have it without advances in mathematics!" Often they will laugh and drop it, but if they continue to ask how specifically their piece of technology is in any way related to math, I'll just tell them they would first have to understand complex numbers and or cryptography to get a clear picture. I'll follow that up with "mathematics plays a role in essentially everything that is important in this world. Building anything requires math, as does understanding how to transmit and use electricity. You will have to take my word that pure math has its place as well." I've never encountered anyone who pushed past this point.