One of the difficulties of mathematics over other sciences is that our problems are harder to motivate to a general audience. A biologist studying a particular pathway in the body can say that he's looking to understand or cure some disease, a physicist at CERN can say he's trying to understand why particles have mass, a string theorist can say he's trying to find a single coherent picture that explains both the physics of the very small and very large. However, a lot of mathematics seems far removed from motivating problems that can be appreciated by non-mathematicians.
So what are some examples of research level mathematics that can be motivated to a non-mathematician in a paragraph or two? Ideally, I would like examples of motivation for things you have personally worked on.
A caveats. I don't want examples where the motivation is "the problem is old and unsolved, and the challenge of it captured my imagination."
To make this easier, I am happy with a motivation that would have to be fleshed out. Perhaps a paragraph aimed at a mathematician that would allow them to motivate the problem to a layperson in a 10 minute conversation?
Edit: In response to a comment, I would like to contrast what I am looking for with Not especially famous, long-open problems which anyone can understand.
First, I do not care about whether a problem has been long opens. In fact, as I explicitly stated above, I am NOT looking for motivation based on how hard a problem is (or the contrast between how hard a problem is and how hard it initially seems). If you are working to explore the properties of a newly defined algebraic object, for example, that would be perfectly fine, so long as you can explain why someone who doesn't know math might view that as a worthwhile pursuit. You don't have to be able to perfectly explain what the object is, just hint at a context of where it lives and what you hope would happen if it were better understood. This could be applications of the understanding, in a traditional sense, or something more nebulous, like a belief that there is a connection between two seemingly unconnected ideas and that the research might give "insight" into that connection.
Second, I am not necessarily for specific problems. What do you tell your mother when she asks why you study cluster algebras or string topology or loop erased random walks? You wouldn't drive down into the specifics of your research, you would pull back towards a general description of the landscape and where you are hoping to travel within it.
I would also like to contrast what I am looking for with How does one justify funding for mathematics research?. While I am looking for answers in that vein, I want things that are more pointed. Not "How does one justify funding for pure research at large" but "How does one justify funding for your particular field of research." What motivation do you jump to when intrinsic motivation wanes, when existential doubt begins to set in, or when you realize that your cousin isn't going to pass the mashed potatoes until he feels that he appreciates why you do what you do?
Best Answer
I'd claim that one should effectively refuse to respond too-directly to a question that otherwise purports to ask "why do you work on what you do?" Almost entirely, I think it is fair to say that we work on technical issues that arise in thinking about special cases of prototypes of models of ... relating to larger, more communicable issues. So, seriously, don't get embroiled talking about the literal technical issues that occupy your days, but, rather, about the "big issue(s)" which created the issue that created the issue that created the ... issue you address.
I do think that this approach assists comprehension even with more-expert people. Certainly grad students, postdocs, ... and surely more-seasoned people who've maybe lost sight of the motivating issue(s).
There is some general hazard in getting distracted by commodifiable, hype-able stuff... While some of this is actually a genuine account of "why we care", some of it is a highly corrupted version, that simply observes the possible modes-of-caring, and jumps in to bluster impressively thereupon. Probably wouldn't want to lie like that to your relatives.
And, on another tack, my blanket explanation is "to advance collective human understanding of the world/universe". (E.g., specifically not embarking on dubious claims about Platonic whatevers/shadows... No one cares.) This in fact has the additional genuine benefit of allowing one to explain that such progress is very incremental, and depends on the "small" (unheroic, unromantic, undramatic) work of many people...
In fact, in many cases, a wholesome (and probably unsurprising, given the ambient biases) political/ideological explanation will set off other discussions that will take attention away from your own personal technical project... and probably rightly so. :)