I have considered some of the lack of knowledge and mistaken impressions of mathematics (and scientific research in general) that can be held by nonmathematicians. In mathematics, there's nothing to prove, just things to calculate. Or all of the important things to prove were established ages ago. Or if there are things still to prove, it's because the remaining questions are incredibly complicated, or incomprehensibly abstract. Or remaining questions could be mystical ones with no right answer, only opinions. Maybe contemporary mathematicians are much smarter than their predecessors because they have powerful computers. In any case, applications — technology, health, some other science that's actually interesting — could be the serious reason to do mathematics. Or if not that, it could be sheer ego and hero worship.
To be fair, a lot of nonmathematicians don't have any such depressing view of our profession. However, they can adopt it very quickly in response to bad explanations. Certainly most nonmathematicians have little sense of the basic coin of research in pure mathematics: theorems, conjectures, proofs, definitions, conjectures, open problems. They also generally don't know that mathematics was already sophisticated in the 19th century, that a vast amount was accomplished in the first half of the 20th century, and that there are plenty of open problems left. (19th century mathematics is largely invisible in newspapers. On the one hand, very few readers or journalists know any of it; on the other hand, it certainly isn't news.)
To counter every side of that, I like to discuss questions that are not only accessible and fun, but also have a historical narrative. The narrative can go from an easy question, to some 19th or early 20th century result, to open problems. It can also cite great results from mathematicians other than the most famous heroes. I think that this can be done in lots of ways, but it is important to stick to clear explanations. Here is an example and a half:
Knot theory. Is an overhand (a trefoil) different from a nothing (an unknot)? Is a right-handed overhand different from a left-handed one? Yes and yes, according to Heegaard, Tietze, and Dehn from a century ago. Are there knots that aren't handed? For example, the figure-eight. Are there non-invertible knots? Yes, but that's harder; it was only established in 1964 by Trotter. Is it possible to distinguish any pair of knots? Yes, but that's a really hard and modern resultas was first proved by Haken in the late 1960s. How would you do it? The current best way is with Thurston's ideas, using hyperbolic geometry. (which Hyperbolic geometry is then another big topic). topic.) How many crossings do you need to switch to convert one knot to another? That is a big open problem, although it has been solved in many interesting cases. What's the easiest solution to the first of this whole chain of questions? Reidemeister moves and 3-colorings. Etc.
Real algebraic curves. Hyperbolas, parabolas, and ellipses are all quadratic curves. A hyperbola has two branches, but these are halves of one oval that passes through infinity. How many ovals can you have in higher degrees? An indirect argument tells you that you can't have an unbounded number in any fixed degree. In degree 4 you can have 4 ovals; in degree 6 you can have 11 ovals. Harnack discovered and proved these upper bounds in the 19th century. Can the ovals nest any way you like? No...
