What to answer when people ask what I do in mathematics 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.
 A: I'll try to be brief and to the point--when asked, "What do you study, and how is this important for the rest of the world?", I do not involve the questioner in a debate about semantics and the meaning of the word important and all that entails. Instead, I like to consider a basic math problem that the questioner can be an active participant in. 


*

*An answer I gave to the very popular "Favorite Proof Accessible to a General Audience" question posed three weeks ago is one that you can actively involve someone in and then explain how this circle morphing idea can be extended to explain the staggered running positions people assume in track races (even further extensions provided by Ravi Vakil's paper The Mathematics of Doodling). 

*Then there are very basic problems that can blow peoples' minds. For example, an old textbook problem stated (I've updated the information here with accurate numbers) that the average ocean depth is $3.7\times 10^3 \rm m$ and the area of the oceans is $3.6\times 10^{14} \rm m^2$. The question in the text was what was the volume of the ocean in liters? I like to take the rather apparent answer, $1.332\times 10^{21}$ cubic meters, and make it somewhat interesting. I ask the person asking me your question about whether or not they have been to Niagra Falls and know how much water pours over the falls each second--$2400$ cubic meters every second or roughly $630000$ gallons a second. With only the $630000$ gallons per second and image of Niagra Falls in mind (I do not give the aforementioned volume), I ask the questioner how long they think it would take to fill the oceans. I have never had a guess even come close to the correct answer of $17.6$ million years. That will blow some minds. So I tell the person I fiddle with numbers and ideas and make them engaging and entertaining and useful. 

*This next one is something I am actually working on right now. Martin Gardner proposed an interesting magic track that produces a magic square when the magician and spectator play tic-tac-toe. Despite the spectator shuffling and choosing where to play, etc., the game ends in a tie and when the cards are flipped over, the participant sees that a magic square is the result (the magician plays cards face down and spectator face up). The original trick was rather shoddy, but I saw a way to greatly improve it using some basic abstract algebra (exploiting properties of the dihedral group $D_8$ and a few other things). It is now to a point where a spectator can look at all the cards beforehand, shuffle them, and then you can play four games of tic-tac-toe and all of them end in a magic square. 
Of course, there are so many other applications of mathematics, but it does not do much good unless you actually show the person. So the next time someone asks you what you do with mathematics and how it's important--show them. It can make for a very memorable experience. 
A: I personally think that, ultimately, the importance of studying mathematics does lie in its application. However, I do not think that it is essential that an application already exists or even that one will exist. I think the point is to develop tools to solve problems. Some of these tools will be used, some perhaps not, but providing new ways to think about problems promises the possibility of a novel solution when every well known technique fails.
A few examples of very abstract, pure mathematical objects which now have many applications in physics are:


*

*Manifolds   

*Vector bundles and K theory   

*Operator algebras which act on the sections of these bundles   

*Representation theory and equivariant cohomology   

*Topos theory and higher topos theory


The list continues, but the point is that many of these objects were studied outside of their application to field theories and quantum field theories. It wasn't until recently that physicists became interested in these objects to better describe the universe around us. It is now becoming more transparent that we are reaching the end of what older, more well known, mathematical tools (such as analysis, linear algebra and even classical differential geometry) can do to solve today's open problems in physics. 
So to answer the question, I would say that you are providing new ways to solve the big problems in applied science. 
Also, I would recommend becoming more acquainted with the applications of your field of study. In my experience, it can really enrich the way you look at the problems your trying to solve. I have even experienced a few occasions where just understanding the application pointed me in the right direction when looking for a solution to the problem.
A: This may be a bit controversial but I would suggest lying. The fun part is to see how far you can stretch the lie and make them believe you. After all, we mainly learn math because we enjoy it, and wish to understand how the theories fit together. So why not have some fun with somebody who is not serious about mathematics and will never be? Tell him something like, "I work with manifolds, currently the physicists have build a collider, and need mathematicians to tell them which type of manifolds are stable to prevent the collider from blowing up." 
Of course, one has to be honest with those who are genuinely interested in mathematics. But with the regular commoner, who has no interest in mathematics, and is glad that he no longer sees it, why not have fun at their expense? 
A: 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.
A: 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." 
A: 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. 
